diff --git "a/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt" "b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/0dAzT4oBgHgl3EQfefzh/content/tmp_files/load_file.txt" @@ -0,0 +1,966 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf,len=965 +page_content='Convergence of Adaptive Mixed Interior Penalty Dis- continuous Galerkin Methods for H(cur l)-Elliptic Problems K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Liu1, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Tang2,, X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Xing2 and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Zhong2 1 School of Sciece, East China University of Technology, Nanchang, 330013, China 2 School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In this paper, we study the convergence of adaptive mixed interior penalty discontinuous Galerkin method for H(cur l)-elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We first get the mixed model of H(cur l)-elliptic problem by introducing a new intermediate variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we discuss the continuous variational problem and discrete variational problem, which based on interior penalty discontinuous Galerkin approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we construct the corresponding posteriori error indicator, and prove the contraction of the summation of the energy error and the scaled error indicator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' At last, we confirm and illustrate the theoretical result through some numerical experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' AMS subject classifications: 65M15,65N12,65N30 Key words: Adaptive mixed interior penalty discontinuous Galerkin methods, Convergence, H(cur l)- elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Introduction Let Ω ⊂ �3 be Lipschitz bounded polygonal domain with a single connected boundary ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We consider the following H(cur l)-elliptic problem ∇ × µ∇ × u + κu = f in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) u × n = 0 on ∂ Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) where n is the unit normal vector of the boundary ∂ Ω, f ∈ L2(Ω), µ and κ are piecewise constants is consistent with the initial partition �0 for Ω and satisfy µ1 < µ < µ2 and κ1 < κ < κ2, here, µi and κi(i = 1,2) are positive constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By introducing an auxiliary ∗Corresponding author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Email addresses: liukai@ecut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Liu), mingtang@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Tang),xingxq@scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn(X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Xing), zhong@m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='scnu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='cn (L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Zhong) 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='01439v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='NA] 4 Jan 2023 2 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' variable p = µ∇ × u, then we get the mixed scheme with the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) p = µ∇ × u in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) ∇ × p + κu = f in Ω, (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) The mixed finite element method is very convenient for processing high-order equations and equations containing two or more unknown functions, which has attracted widespread attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For mixed finite element method, there are only few research results for Maxwell problem [13] and Maxwell’s eigenvalue problem [12,14,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive finite element method automatically refines and optimizes meshes accord- ing to the singularity of solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' It is a highly reliable and efficient numerical calculation method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' At present, the convergence analysis research of the adaptive mixed finite element method for the elliptic equation is relatively complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Chen, Holst and Xu [7] proved the convergence analysis of the adaptive mixed finite element algorithm for elliptic equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Du and Xie [10] proved the convergence analysis of the adaptive mixed finite element algorithm for the convection diffusion equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' However, there are only few research results on the posterior error estimator of Maxwell’s equations for the adaptive mixed fi- nite element method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For example, Carstensen and Ma [5] establishes the convergence of adaptive mixed finite element methods for second-order linear non-self-adjoint indefinite elliptic problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Carstensen, Hoppe, Sharma and Warburton [4] designs and analyzes the posterior error estimation of the adaptive hybrid conforming finite element method of H(cur l)-elliptic problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Recently, Chung, Yuen and Zhong [8] present a-posteriori error analysis for the staggered discontinuous Galerkin method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' As far as we know, there are not any published literatures for the convergence analysis of the adaptive mixed finite element method for the boundary value problem(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Our contributions in this paper are to construct a new error estimator, which does not include the negative power of the local mesh size in the jump term for the traditional DG method;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' get the convergence of the Adaptive Mixed Interior Penalty Discontinuous Galerkin (AMIPDG) method by using the similar technique used in [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' However, this tech- nique in [2] can not be used directly for mixed forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We present our main result in the following theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let {�k,Uk,Qk, uk, pk,η(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k)}k≥0 be the sequence of meshes, finite element space, mixed discrete solution and posterior error estimate indicator produced by the AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exist constants ρ > 0 and δ ∈ (0,1), which depend on marking parameter and the shape regularity of the initial mesh �0, such that ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, for a given precision, the AMIPDG method will terminate after a finite number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 3 For convenience, we let C denote a generic positive constant which may be different at different occurrences and adopt the following notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The subscripted constant Ci represents a particularly important constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' a ≲ b means a ≤ C b for some constants C which are independent of mesh sizes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 2, we first present the contin- uous variational problem, the discrete variational problem, and the procedure of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 3, we first show the upper bound estimate of the error, which is key to the con- vergence analysis, then we prove the indicator reduction and the convergence of AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Section 4, we provide some numerical experiments to illustrate the effective- ness of the AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive Mixed interior penalty discontinuous Galerkin method In this section, we introduce the continuous variational problem, the discrete variational problem of mixed internal penalty discontinuous finite element method, and the procedure of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Continuous variational problem For an open and connected bounded domain D ⊂ �3, we denote by L2(D) (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' L2(D) := (L2(D))3) the spaces of square-integrable functions (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' vector fields) on D with inner product (·,·)0,D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We define the spaces H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {u ∈ L2(D) : ∇ × u ∈ L2(D)}, H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {u ∈ L2(D) : ∇ · u ∈ L2(D)}, with (u, v)cur l,D := (u, v)0,D + (∇ × u,∇ × v)0,D, ∀u, v ∈ H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D), (u, v)div,D := (u, v)0,D + (∇ · u,∇ · v)0,D, ∀u, v ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D), and the induced norm as: ∥u∥2 cur l,D := ∥u∥2 0,D + ∥∇ × u∥2 0,D, ∀u ∈ H(cur l, D), ∥u∥2 div,D := ∥u∥2 0,D + ∥∇ · u∥2 0,D, ∀u ∈ H(div, D), respectively, where ∥ · ∥L2(D) := (·,·)1/2 D denotes the norm of the space L2(D) or L2(D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We also define H0(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) = {v ∈ H(cur l;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' D) : v × n = 0 on ∂ D} in the trace sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we first define two space U := H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω),Q := L2(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, the mixed vari- ational problem of the mixed boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) reads as: find (u, p) ∈ U × Q such that: a(p,q) − b(u,q) = ℓ1(q), ∀q ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) d(v, p) + c(u, v) = ℓ2(v), ∀v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) 4 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The bilinear forms a, b, c and the functionals ℓ1(·),ℓ2(·) are given by a(p,q) := (p,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) b(u,q) := (µ∇ × u,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) c(u, v) := (κu, v), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) d(v, p) := (∇ × v, p) (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) ℓ1(q) := 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) ℓ2(v) := ( f , v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) The operator-theoretic framework involves operator � : (U × Q) → (U × Q)∗ defined by (� (u, p))(v,q) := a(p,q) − b(u,q) + d(v, p) + c(u, v),∀u, v ∈ U, p,q ∈ Q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) where (Q × U)∗ is the dual spaces of (Q × U).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we can rewrite (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) as (� (u, p))(v,q) = ℓ(v,q), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10) with ℓ(v,q) = ℓ1(q) + ℓ2(v), and ℓi are given by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we state the well-posedness of the variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) in the follow- ing lemma, and it can be found in section 3 of [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Under the assumptions on the problem of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), � is a continuous and bijective linear operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Hence, for any ℓ = (ℓ1,ℓ2) ∈ (Q×U)∗, the mixed variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) has a unique solution (u, p) ∈ (U × Q), which satisfy the following continuously ∥(u, p)∥U×Q := (∥u∥2 curl,Ω + ∥p∥2 0)1/2 ≲ ∥ℓ1∥Q∗ + ∥ℓ2∥U∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Discrete variational problem We suppose that �h is a family of shape regularity, quasi-uniform and conform tetrahe- dral generation on Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let hτ = |τ|1/3 denote the mesh size with |τ| being the volume of τ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Define the discontinuous finite element function space �(�h) as: �(�h) = {v ∈ L2(Ω) : vτ = v|τ ∈ (Pl(τ))3, ∀τ ∈ �h}, where Pl(τ) is the set of polynomials defined in the volume τ whose degree does not exceed l, where l ≥ 1 is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let �h, � 0 h and � ∂ h denote the set of the all faces of its volumes, and the set of internal faces, and the set of boundary faces, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, �h = � 0 h � � ∂ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the space of piecewise Sobolev functions defined by H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = � v ∈ L2(Ω) : vτ = v|τ ∈ H1(τ), ∀ τ ∈ �h � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 5 and H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = (H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h))3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let L2(�h) be the set of L2 functions defined on �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' More- over, we define the following inner products (v, w)� ′ h = � τ∈� ′ h � τ v · wdx, ∀v, w ∈ L2(Ω), ∀� ′ h ⊂ �h, < v, w >� ′ h = � f ∈� ′ h � f v · wds, ∀v, w ∈ L2(�h), ∀� ′ h ⊂ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For f ∈ � 0 h , we have τi ∈ �h(i = 1,2), such that f = ∂ τ1 ∩ ∂ τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we denote the jump and average of v as: [[v]] = v1 × n1 + v2 × n2, ∀v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), {{v}} = v1 + v2 2 , ∀v ∈ H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), where v i denote the values of v on v|τi(i = 1,2) and ni denote the out unit normal vectors on f exterior v|τi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For f ∈ � ∂ h , we have τ ∈ �h, such that f = ∂ τ ∩ ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we denote the jump and average of v as: [[v]] = vτ × n∂ Ω, {{v}} = vτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) Next, we give the corresponding discrete scheme of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Firstly, we define the corresponding discrete space as follow Uh := {vh ∈ �(�h)| [[vh]]|f = 0,∀f ∈ � ∂ h }, Qh := �(�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, the formulation of the discrete Mixed Interior Penalty Discontinuous Galerkin (MIPDG) method reads: find (uh, ph) ∈ (Uh,Qh) such that ah(ph,qh) − bh(uh,qh) = ℓ1,h(qh) + d1,h(uh,qh), ∀qh ∈ Qh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) dh(vh, ph) + ch(uh, vh) = ℓ2,h(vh) + d2,h(uh, vh), ∀vh ∈ Uh, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) where ah(ph,qh) := (ph,qh)�h, bh(uh,qh) := (µ∇ × uh,qh)�h, ch(uh, vh) := (κuh, vh)�h, dh(vh, ph) := (∇ × vh, ph)�h, ℓ1,h(qh) := 0, ℓ2,h(vh) := ( f , vh)�h, d1,h(uh,qh) := − < {{µqh}},[[uh]] >�h, d2,h(uh, vh) :=< ({{µ∇ × uh}} − αh−1 f [[uh]]),[[vh]] >�h, 6 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' here the constant α > 0 denote the penalty parameter, hf denote the diameter of the circumcircle of f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus hτ ≈ hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The calculation of ∇ × uh in the bilinear terms are piecewise derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The standard symmetric Interior Penalty Discontinuous Galerkin (IPDG) method of the boundary value problem (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) is to find uh ∈ Uh, such that aIP(uh, vh) := (κuh, vh)�h + (µ∇ × uh,∇ × vh)�h− < {{µ∇ × vh}},[[uh]] >�h − < {{µ∇ × uh}},[[vh]] >�h +αh−1 f < [[uh]],[[vh]] >�h (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) = ( f , vh)�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The following lemma shows that the discrete variational problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ [3], Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1] The formulations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) are formally equivalent in the following sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' If (uh, ph) ∈ (Uh,Qh) are the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), then uh ∈ Uh solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Conversely, if uh ∈ Uh solves (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15), then there exists some ph ∈ Qh such that (uh, ph) ∈ (Uh,Qh) are the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Ayuso de Dios, Hiptmair and Pagliantini proved the well-posedness of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) in section 2 of [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, by combining Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we obtain the well-posedness of discrete variational problems (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Adaptive Mixed Interior Penalty Discontinuous Galerkin method(AMIPDG) Our adaptive cycle can be implemented by the following algorithm: Next, we will discuss each step in AEFEM in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure SOLVE For f ∈ L2(Ω), and a shape regular mesh �k, Let (uk, pk) be the exact MIPDG solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Here, we assume that the solutions (uk, pk) can be solved accurately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure ESTIMATE A posteriori error indicator is an essential ingredient of adaptivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' They are computable quantities depending on the computed solution(s) and data that provide information about the quality of approximation and may consequently be used to make judicious mesh modi- fications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Here, we design a new posteriori error estimation indicator for equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), which is similar to that in [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For τ ∈ �h, f ∈ �h and (vh,qh) ∈ Uh × Qh, the residual a posteriori error estimator for the symmetric AMIPDG method is given by η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ) : = ∥R1(vh,qh)∥2 L2(τ) + h2 τ � ∥R2(vh,qh)∥2 L2(τ) + ∥R3(vh)∥2 L2(τ) � + � f ∈∂ τ hf � ∥J1(qh)∥2 L2(f ) + ∥J2(vh)∥2 L2(f ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16) Convergence of AMIPDG methods for H(cur l)-elliptic problems 7 Algorithm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 Adaptive Mixed Interior Penalty Discontinuous Galerkin Method (AMIPDG) cycle Input initial triangulation �0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' data f ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' tolerance tol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' marking parameter θ ∈ (0,1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Output a triangulation �J;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MIPDG solution (uJ, pJ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' η = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' k = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' while η ≥ tol SOLVE solve discrete varational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) on �k to get the solution (uk, pk);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' ESTIMATE compute the posterior error estimator η = η(uk, pk,�k) by using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' MARK seek a minimum cardinality �k ⊂ �k such that η2 � uk, pk,�k � ≥ θη2 � uk, pk,�k � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' REFINE bisect elements in �k and the neighboring elements to form a conforming �k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' k = k + 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' end uJ = uk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pJ = pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' �J = �k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' They consist of the element residuals and face jump residuals as R1(vh,qh)|τ := qh|τ − µ∇ × vh|τ, R2(vh,qh)|τ := f |τ − (∇ × qh + κvh)|τ, R3(vh)|τ := ∇ · ( f |τ − κvh|τ), J1(qh)|f := [[qh]], J2(vh)|f := [[(f − κvh)]].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' where hf denote the diameter of the circumcircle of f , and hτ ≈ hf .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any set � ′ h ⊆ �h, the error indicator is defined as η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='� ′ h ) = � τ∈� ′ h η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure MARK We use the Dörfler mark which was proposed by Dörfler [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Set marking parameter θ ∈ (0,1), the module MARK outputs a subset of marked elements �k ⊂ �k with minimal cardinality, such that η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≥ θη2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Procedure REFINE Our implementation of REFINE uses the longest edge bisection strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' A detailed intro- duction about the longest edge bisection strategy was provided in [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' To avoid confusion, the relationship between the two tetrahedral meshes �h and �H that are nested into each 8 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' other is defined as: �h is the new mesh division of �H after one cycle of the above cycle process, abbreviated as �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG algorithm In this section, we establish the upper bound estimate of the error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Subsequently, we demonstrate that the sum of the energy error and the error estimator between two consec- utive adaptive loops is a contraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, we proof that the AMIPDG is convergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The upper bound estimate of the error In this subsection, before establishing the reliability of a posteriori error estimator, we need to define the corresponding DG norm, for any (v,q) ∈ U × Q and (vh,qh) ∈ Uh × Qh, ∥(v,q) − (vh,qh)∥2 DG := ∥q − qh∥2 L2(Ω) + ∥κ(v − vh)∥2 L2(Ω) + � τ∈�h ∥µ∇ × (v − vh)∥2 L2(τ) + � f ∈�h αh−1 f < [[vh]],[[vh]] >f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any v ∈ U and vh ∈ Uh, we have ∥[[vh]]∥2 L2(f ) = ∥[[(v − vh)]]∥2 L2(f ), ∀f ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact, v ∈ U implies that [[v]]|f = 0 (see Chapter 5 of [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We summarize our main result in this subsection as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U×Q and (uh, ph) ∈ Uh ×Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the residual error indicator of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we have the following estimate ∥(u, p) − (uh, ph)∥2 DG ≤ C1η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) where the constant C1 depending on the shape regularity of mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), similarly to [4], we introduce the nonconformity of the MSIPDG method results in some consistency error: ζ := min ˜vh∈U � � τ∈�h (∥uh − ˜vh∥2 L2(τ) + ∥∇ × (uh − ˜vh)∥2 L2(τ)) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3) We denote that ˜uh ∈ U is the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), namely ˜ζ = � � τ∈�h (∥uh − ˜uh∥2 L2(τ) + ∥∇ × (uh − ˜uh)∥2 L2(τ)) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4) Convergence of AMIPDG methods for H(cur l)-elliptic problems 9 Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively, let ˜uh be the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), then ∥(u − ˜uh, p − ph)∥U×Q = (∥u − ˜uh∥2 curl,Ω + ∥p − ph∥2 0)1/2 ≲ ∥˜ℓ1∥Q∗ + ∥˜ℓ2∥U∗, where the residuals ˜ℓ1 ∈ Q∗ and ˜ℓ2 ∈ U∗ defined by ˜ℓ1(q) = ℓ1(q) − a(ph,q) + b(˜uh,q), ∀q ∈ Q, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) ˜ℓ2(v) = ℓ2(v) − d(v, ph) − c(˜uh, v), ∀v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any q1,q2,q ∈ Q and any v1, v2, v ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we have the following property by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) (� (v1 + v2,q1 + q2))(v,q) = a(q1 + q2,q) − b(v1 + v2,q) + d(v,q1 + q2) + c(v1 + v2, v) = a(q1,q) − b(v1,q) + d(v,q1) + c(v1, v) +a(q2,q) − b(v2,q) + d(v,q2) + c(v2, v) = (� (v1,q1))(v,q) + (� (v2,q2))(v,q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, (� (u − ˜uh, p − ph))(v,q) = (� (u, p))(v,q) − (� (˜uh, ph))(v,q) = (ℓ1(q) + ℓ2(v)) − (a(ph,q) − b(˜uh,q) + d(v, ph) + c(˜uh, v)) = ˜ℓ1(q) + ˜ℓ2(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact that (u − ˜uh, p −ph) ∈ U ×Q and combining the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 can concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we will provide upper bounds for ∥˜ℓ1∥Q∗ and ∥˜ℓ2∥U∗ in Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ˜uh be the unique minimizer of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then we get the estimate of the linear functional ˜ℓ1 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5) as following ∥˜ℓ1∥Q∗ ≲ � � τ∈�h ∥R1(uh, ph)∥2 L2(τ) �1/2 + � � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) �1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any q ∈ Q, by the definition of ˜ℓ1, we have ˜ℓ1(q) = ��� τ∈�h � τ � (µ∇ × uh − ph) + µ∇ × (˜uh − uh) � qdx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 10 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then applying the Hölder inequality and the Cauchy-Schwarz inequality, |˜ℓ1(q)| ≤ � τ∈�h ∥µ∇ × uh − ph∥L2(τ)∥q∥L2(Ω) + � τ∈�h ∥µ∇ × (˜uh − uh)∥L2(τ)∥q∥L2(Ω) ≲ �� � τ∈�h ∥R1(uh, ph)∥2 L2(τ) �1/2 + � � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) �1/2� ∥q∥L2(Ω), conclude the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Before estimating the term ∥˜ℓ2∥U∗, we need to introduce the following interpolation operator with the corresponding approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ [19], Theorem 1] Let Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) be the lowest order edge elements of Nédélec first family.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exists an operator Πh : H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω) → Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) with the following properties: For every v ∈ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω), there exist ϕ ∈ H1 0(Ω) and z ∈ H1 0(Ω), such that v − Πhv = ∇ϕ + z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' And for any τ ∈ �h and f ∈ �h, we have h−1 τ ∥ϕ∥L2(τ) + ∥∇ϕ∥L2(τ) ≲ hτ∥v∥L2(Ωτ), h−1 τ ∥z∥L2(τ) + ∥∇z∥L2(τ) ≲ hτ∥∇ × v∥L2(Ωτ), where Ωτ = � f ∈τ Ωf , Ωf = {τ′ ∈ �h, f ∈ τ′}, and the constants depending on the shape regularity of the mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ˜uh be the unique solution of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then the linear functional ˜ℓ2 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) satisfies the following estimate ∥˜ℓ2∥U∗ ≲ � � τ∈� h2 τ(∥R2(uh, ph)∥2 L2(τ) + ∥R2(uh)∥2 L2(τ)) + � f ∈� hf (∥J1(ph)∥2 L2(f ) + ∥J2(uh)∥2 L2(f )) + � τ∈� ∥uh − ˜uh∥2 L2(τ) �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any v ∈ U and Πh given by Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3, we have v − Πhv = ∇ϕ + z, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9) where ϕ ∈ H1 0(Ω) and z ∈ H1 0(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' According to linearity of the operator ˜ℓ2 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9), we have ˜ℓ2(v) = ˜ℓ2(Πhv) + ˜ℓ2(v − Πhv) = ˜ℓ2(Πhv) + ˜ℓ2(∇ϕ) + ˜ℓ2(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10) We will next estimate the three terms on the right hand side of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 11 For the first term ˜ℓ2(Πhv) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), using the definition of ˜ℓ2, we have ˜ℓ2(Πhv) = ℓ2(Πhv) − d(Πhv, ph) − c(˜uh,Πhv) = ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) + c(uh − ˜uh,Πhv).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that Πhv ∈ Nd1 0(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ⊆ Uh has zero jumps, and combining (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), we have ℓ2(Πhv) − d(Πhv, ph) − c(uh,Πhv) = ℓ2,h(Πhv) − dh(Πhv, ph) − ch(uh,Πhv) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, we have ˜ℓ2(Πhv) = c(vh − ˜uh,Πhv) = c(vh − ˜uh, v) + c(vh − ˜uh,Πhv − v) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥Πhv − v∥0,�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9), triangle inequality and Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3, we get ˜ℓ2(Πhv) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ + z∥0,�h) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h(∥v∥0,�h + ∥∇ϕ∥0,�h + ∥z∥0,�h) ≤ ∥κ∥0,∞∥vh − ˜uh∥0,�h∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11) For the second term ˜ℓ2(∇ϕ) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), using the definition of ˜ℓ2, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6) and the fact ∇ × ∇ϕ = 0, which implies ˜ℓ2(∇ϕ) = ℓ2(∇ϕ) − d(∇ϕ, ph) − c(˜uh,∇ϕ) = ( f ,∇ϕ) − (∇ × ∇ϕ, ph) − (κ˜uh,∇ϕ) = ( f ,∇ϕ) − (κ˜uh,∇ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='12) and Green’s formula, we have ˜ℓ2(∇ϕ) = ( f ,∇ϕ) − (κuh,∇ϕ) + (κ(uh − ˜uh),∇ϕ) ≤ � τ∈�h (R3(uh),ϕ)0,τ + � f ∈�h < J2(uh),ϕ >0,f +(κ(uh − ˜uh),∇ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Applying the Cauchy-Schwarz inequality, Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3 and trace inequality, we have ˜ℓ2(∇ϕ) ≤ � � τ∈�h h2 τ∥R3(uh)∥2 0,τ + � f ∈�h hf ∥J2(uh)∥2 0,f + � τ∈�h ∥κ∥0,∞∥uh − ˜uh∥2 0,τ �1/2 ∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) 12 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, for the third term ˜ℓ2(z) of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), we have ˜ℓ2(z) = ( f , z) − (∇ × z, ph) − (κ˜uh, z) = ( f , z) − (∇ × z, ph) − (κuh, z) + (κ(uh − ˜uh), z) ≤ � � τ∈�h h2 τ∥R2(uh, ph)∥2 0,τ + � f ∈�h hf ∥J1(ph)∥2 0,f + � τ∈�h ∥κ∥0,∞∥uh − ˜uh∥2 0,τ �1/2 ∥v∥curl,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) Substituting (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='11), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) into (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10), the proof is completed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Notice that both (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8) are related to the terms � τ∈�h ∥∇ × (˜uh − uh)∥2 L2(τ) and � τ∈� ∥uh − ˜uh∥2 L2(τ), which are a part of ˜ζ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, we prove upper bounds for ˜ζ in the following Lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14) and ˜ζ be consistency error of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), we have ˜ζ2 ≲ η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any vh ∈ Uh, there exit an interpolation operator �h : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) → Uc h, such that(see Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5 of [11]) ∥vh − �hvh∥2 L2(Ω) ≲ � f ∈�h hf ∥[[vh]]∥2 L2(f ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16) � τ∈�h ∥∇ × (vh − �hvh)∥2 L2(τ) ≲ � f ∈�h h−1 f ∥[[vh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17) Then, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='17), and the fact hf < 1, we get ˜ζ2 = � τ∈�h (∥uh − ˜uh∥2 L2(τ) + ∥∇ × (uh − ˜uh)∥2 L2(τ)) ≤ � τ∈�h (∥uh − �huh∥2 L2(τ) + ∥∇ × (uh − �huh)∥2 L2(τ)) ≲ � f ∈�h hf ∥[[uh]]∥2 L2(f ) + � f ∈�h h−1 f ∥[[uh]]∥2 L2(f ) ≲ � f ∈�h h−1 f ∥[[uh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) Noting that (uh, ph) ∈ Uh × Qh is the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)- (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then by using Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we know that uh is the solution of discrete variational problem (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Hence, we have ( see Lemma 5 of [20]) α∥h−1/2 f [[uh]]∥L2(�h) ≲ η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19) Convergence of AMIPDG methods for H(cur l)-elliptic problems 13 At last, combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we have ˜ζ2 ≲ η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='� ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Combining Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5, we will prove Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' [ Proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1:] By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1), the triangle inequality, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4), Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5 and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we get ∥(u, p) − (uh, ph)∥2 DG ≲ ∥p − ph∥2 L2(Ω) + ∥κ(u − uh)∥2 L2(Ω) + � τ∈�h ∥∇ × µ(u − uh)∥2 L2(τ) + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≲ ∥p − ph∥2 L2(Ω) + ∥u − ˜uh∥2 cur l,Ω + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f = ∥(u − ˜uh, p − ph)∥U×Q + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≲ ∥˜ℓ1∥2 Q∗ + ∥˜ℓ2∥2 U∗ + ˜ζ2 + � f ∈�h αh−1 f < [[uh]],[[uh]] >f ≤ C1η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The error reduces on two successive meshes For convenience, for any v ∈ U and vh ∈ Uh, we denote ∥|v − vh|∥2 h = ∥κ(v − vh)∥2 L2(Ω) + � τ∈�h ∥∇ × µ(v − vh)∥2 L2(τ) + � f ∈�h αh−1 f < [[vh]],[[vh]] >f .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='20) Let Uc h be the H(cur l) conforming subspace of Uh given by Uc h := Uh ∩ H0(curl;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, there is a subspace U⊥ h which can orthogonally decompose Uh under L2 inner product such that Uh := Uc h ⊕ U⊥ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Especially, if (uh, ph) ∈ Uh × Qh is the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), then we have ∥|u⊥ h |∥2 h ≲ α � f ∈∂ τ ∥h−1/2 f [[uh]]∥2 L2(f ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21) In fact, from the Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, notice that uh satisfies the IPDG scheme of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15), and ac- cording to Lemma 2 in [20], we can obtain (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 14 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In order to easily estimate the jump term of face �h, we need to introduce the lifting operators and the corresponding stability estimates, more details are referenced to Propo- sition 12 in [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let �h : H1(Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) → Uh be the lifting operators, which satisfies the following equality � Ω �h(v) · wdx =< [[v]],{{w}} >�h, ∀w ∈ Uh, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='22) and ∥�h(v)∥L2(Ω) ≤ C� ∥h−1/2[[v]]∥L2(�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23) where the constant C� depending on the shape regularity of mesh �h and the degree of polynomial l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q and (uh, ph) ∈ Uh × Qh be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively, we have ∥p − ph∥L2(Ω) ≲ ∥∇ × (u − uh)∥L2(Ω) + η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='24) ∥ph − pH∥L2(Ω) ≲ ∥∇ × (uh − uH)∥L2(Ω) + � η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='25) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that Qh ⊆ Q, and using (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1), the definition of R1(uh, ph) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='16), we have ∥p − ph∥L2(�h) ≤ sup ∀q∈Q (p − ph,q)�h ∥q∥L2(�h) = sup ∀q∈Q (µ∇ × u,q)�h − � R1(uh, ph) + µ∇ × uh,q � �h ∥q∥L2(�h) ≤ sup ∀q∈Q (µ∇ × (u − uh),q)�h − � R1(uh, ph),q � �h ∥q∥L2(�h) ≲ ∥∇ × (u − uh)∥L2(�h) + η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of R1(uh, ph), (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' and the fact [[uh]] = Convergence of AMIPDG methods for H(cur l)-elliptic problems 15 [[uc h + u⊥ h ]] = [[u⊥ h ]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we have ∥ph − pH∥L2(�h) ≤ sup ∀qh∈Qh (ph − pH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h ∥qh∥L2(�h) ≤ sup ∀qh∈Qh (ph,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH) + µ∇ × uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) ≤ sup ∀qh∈Qh (µ∇ × uh,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h+ < {{qh}},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[µuh]] >�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH) + µ∇ × uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) = sup ∀qh∈Qh (µ∇ × (uh − uH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh)�h+ < {{qh}},' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[µuh]] >�h − � R1(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='qh � �h ∥qh∥L2(�h) ≲ ∥∇ × (uh − uH)∥L2(�h) + ∥h−1/2 τ [[uh]]∥L2(�h) + η(uH,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) ≲ ∥∇ × (uh − uH)∥L2(�h) + C� ∥h−1/2 τ [[u⊥ h ]]∥L2(�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) ≲ ∥∇ × (uh − uH)∥L2(τ) + � η(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that ∥(u, p)−(uh, ph)∥2 DG+η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) and ∥|u−uh|∥2 h+η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In fact, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='24), we first know that ∥(u, p) − (uh, ph)∥2 DG + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = ∥|u − uh|∥2 h + ∥p − ph∥2 L2(�h) + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≲ ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Secondly, it is shown by the definition of ∥ · ∥DG ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ ∥(u, p) − (uh, ph)∥2 DG + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, we next only need to consider the convergence of ∥|u − uh|∥2 h + η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We first show that the error plus some quantity reduces with a fixed factor on two successive meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given f ∈ L2(Ω) and two tetrahedral mesh �h and �H, where �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (u, p) ∈ U × Q be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), and (uh, ph) ∈ Uh × Qh, (uH, pH) ∈ UH × QH be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exit two constants δ1,δ2 ∈ (0,1), such that ∥|u − uh|∥2 h ≤ (1 + δ1)∥|u − uH|∥2 H − 1 − δ2 2 ∥|uh − uH|∥2 h + C3 δ1δ2α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='26) where C3 depending on the C� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 16 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Choosing that q = ∇ × v, and subtracting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1) from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2), we obtain (κu, v) + (µ∇ × u,∇ × v) = ( f , v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='27) Subtracting (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='15) from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='27) with v = vh = uc h − uc H, and using [[uc h − uc H]] = 0, we have (κ(u − uh), uc h − uc H)0,�h + (µ∇ × (u − uh),∇ × (uc h − uc H))0,�h + < [[uh]],{{µ∇ × (uc h − uc H)}} >�h= 0, which leads to (κ(u − uh), uc h − uc H)0,�h + (µ∇ × (u − uh),∇ × (uc h − uc H))0,�h = − < [[uh]],{{µuc h − uc H}} >�h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='28) Using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='22) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='23), we have < [[uh]],{{∇ × (uc h − uc H)}} >�h = (�h(uh),∇ × (uc h − uc H))0,�h ≤ C� ∥h−1/2[[uh]]∥0,�h∥∇ × (uc h − uc H)∥0,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='29) Let uh = uc h + u⊥ h and uH = uc H + u⊥ H, we have uh + uc H − uc h = uH − u⊥ H + u⊥ h , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='30) where uc H ∈ Uc H, uc h ∈ Uc h, u⊥ H ∈ U⊥ H, u⊥ h ∈ U⊥ h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='30), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='28), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='29) and Young’s inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' we get ∥|u − uh|∥2 h = ∥κ(u − uh)∥2 L2(Ω) + ∥∇ × µ(u − uh)∥2 L2(Ω) + � f ∈�h αh−1 f < [[(u − uh)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[u − uh]] >�h = ∥|u − uh − uc H + uc h|∥2 h − ∥|uc h − uc H|∥2 h − 2(κ(u − uh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' uc h − uc H)0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h −2(µ∇ × (u − uh),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='∇ × (uc h − uc H))0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h −2 � f ∈�h αh−1 f < [[(u − uh)]],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='[[uc h − uc H]] > ≲ ∥|u − uH|∥2 H + 2∥|u − uH|∥H∥|u⊥ h − u⊥ H|∥h + ∥|u⊥ h − u⊥ H|∥2 h − ∥|uc h − uc H|∥2 h +2∥h−1/2[[uh]]∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h∥∇ × (uc h − uc H)∥0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h ≤ (1 + δ1)∥|u − uH|∥2 H + (1 + 1 δ1 )∥|u⊥ h − u⊥ H|∥2 h − (1 − ˆδ2C� )∥|uc h − uc H|∥2 h +C� ˆδ2 ∥h−1/2[[uh]]∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h = (1 + δ1)∥|u − uH|∥2 H + (1 + 1 δ1 )∥|u⊥ h − u⊥ H|∥2 h − (1 − δ2)∥|uc h − uc H|∥2 h + C2 � δ2 ∥h−1/2[[uh]]∥2 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 17 where δ2 = ˆδ2C� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using uc H = uH − u⊥ H, uc h = uh − u⊥ h , triangle inequality and average inequality, we have ∥|uc h − uc H|∥2 h ≥ 1 2∥|uh − uH|∥2 h − ∥|u⊥ h − u⊥ H|∥2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' By triangle inequality and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='21), we obtain ∥|u⊥ h − u⊥ H|∥2 h ≤ 2(∥|u⊥ h |∥2 h + ∥|u⊥ H|∥2 H) ≤ 2α∥h−1/2[[u⊥ h ]]∥2 0,�h + 2α∥h−1/2[[u⊥ H]]∥2 0,�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Combining [[uH]] = [[u⊥ H + uc H]] = [[u⊥ H]] and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='19), we have ∥|u − uh|∥2 h ≤ (1 + δ1)∥|u − uH|∥2 H − 1 − δ2 2 ∥|uh − uH|∥2 h + C3 δ1δ2α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Contraction of the error estimator In this subsection, we prove the reduction of error indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let us first consider the effect of changing the finite element function used in the estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given f ∈ L2(Ω) and two tetrahedral mesh �h, �H with �H ≤ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (vh,qh) ∈ Uh × Qh and (v H,q H) ∈ UH × QH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any ε > 0, we have η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε)η2(v H,q H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + Cε∥(vh,qh) − (v H,q H)∥2 DG, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='31) where Cε depending on the ε, and the mesh size h < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any τ∗ ∈ �h, we will discuss each of the five components of the mark η2(vh,qh;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Firstly, using the definition of R1(vh,qh) and triangle inequality, we have ∥R1(vh,qh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='32) = ∥qh − µ∇ × vh∥L2(τ∗) = ∥qh − q H + µ∇ × (v H − vh) + q H − µ∇ × v H∥L2(τ∗) ≲ ∥q H − ∇ × v H∥L2(τ∗) + ∥qh − q H∥L2(τ∗) + ∥∇ × (vh − v H)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Secondly, using the definition of R2(vh,qh), triangle inequality and inverse inequality, we get hτ∗∥R2(vh,qh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='33) = hτ∗(∥ f − ∇ × qh − κvh∥L2(τ∗)) = hτ∗(∥ f − ∇ × (qh − q H) − κ(vh − v H) − ∇ × q H − κv H∥L2(τ∗)) ≤ hτ∗(∥ f − ∇ × q H − κv H∥L2(τ∗) + ∥∇ × (qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) ≲ hτ∗(∥R2(v H,q H)∥L2(τ∗) + h−1 τ∗ ∥(qh − q H)∥L2(τ∗) + ∥κ(vh − v H)∥L2(τ∗)) ≲ hτ∗∥R2(v H,q H)∥L2(τ∗) + ∥(qh − q H)∥L2(τ∗) + hτ∗∥κ(vh − v H)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 18 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of R3(vh), triangle inequality and inverse inequality, we get hτ∗∥R3(vh)∥L2(τ∗) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='34) = hτ∗∥∇ · ( f − κvh)∥L2(τ∗) = hτ∗∥∇ · ( f − κv H + κv H − κvh)∥L2(τ∗) ≤ hτ∗(∥∇ · ( f − κv H)∥L2(τ∗) + ∥∇ · κ(v H − vh)∥L2(τ∗)) ≲ hτ∗(∥R3(v H)∥L2(τ∗) + h−1 τ∗ ∥κ(v H − vh)∥L2(τ∗)) ≲ hτ∗∥R3(v H)∥L2(τ∗) + ∥κ(v H − vh)∥L2(τ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we discuss the jump J1(qh) and J2(vh).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any f ∈ �(�h), we let f = τ1 ∗ � τ2 ∗ with τ1 ∗,τ2 ∗ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, using the definition of J1(qh), triangle inequality and trace inequality, we have h1/2 f ∥J1(qh)∥L2(f ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='35) = h1/2 f ∥[[qh]]∥L2(f ) = h1/2 f ∥[[q H + qh − q H]]∥L2(f ) ≤ h1/2 f (∥[[q H]]∥L2(f ) + ∥[[qh − q H]]∥L2(f )) ≤ h1/2 f ∥[[q H]]∥L2(f ) + h1/2 f ∥(qh − q H)|τ1 ∗∥L2(f ) + h1/2 f ∥(qh − q H)|τ2 ∗∥L2(f ) ≲ h1/2 f ∥J1(q H)∥L2(f ) + ∥(qh − q H)∥L2(τ1 ∗∪τ2 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Similarly, using the definition of J2(vh), triangle inequality and trace inequality, we have h1/2 f ∥J2(vh)∥L2(f ) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='36) = h1/2 f ∥[[( f − κvh)]]∥L2(f ) = h1/2 f ∥[[( f − κv H + κv H − κvh)]]∥L2(f ) ≤ h1/2 f (∥[[(f − κv H)]]∥L2(f ) + ∥[[κ(v H − vh)]]∥L2(f )) ≤ h1/2 f ∥J2(v H)∥L2(f ) + h1/2 f (∥κ(v H − vh)|τ1 ∗∥L2(f ) + ∥κ(v H − vh)|τ2 ∗∥L2(f )) ≲ h1/2 f ∥J2(v H)∥L2(f ) + ∥κv H − κvh∥L2(τ1 ∗∪τ2 ∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, the desired result (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='31) is obtained by combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='32)-(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='36), Young’s in- equality and the shape regularity of mesh �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We then prove the contraction of the error estimator under the assumptions on the problem of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uH, pH) ∈ UH × QH be the solution of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), and ��H−→�h = �H \\ (�h ∩ �H) be the Convergence of AMIPDG methods for H(cur l)-elliptic problems 19 set of all element refined into �h on �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, there is a constant λ ∈ (0,1) independent of mesh size, such that η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='37) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Assume that the tetrahedral mesh τ ∈ �H is divided into two new tetrahedral mesh τ1 ∗ and τ2 ∗ with equal volumes, where τ1 ∗,τ2 ∗ ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus, h3 τ1 ∗ = |τ1 ∗| = |τ2 ∗| = h3 τ2 ∗ = 2−1h3 τ by the shape regularity of mesh, which implies hτ1 ∗ = hτ2 ∗ = 2−1/3hτ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we have ∥R1(uH, pH)∥2 L2(τ1 ∗) + ∥R1(uH, pH)∥2 L2(τ2 ∗) ≤ ∥R1(uH, pH)∥2 L2(τ), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='38) and h2 τ1 ∗(∥R2(uH, pH)∥2 L2(τ1 ∗) + ∥R3(uH)∥2 L2(τ1 ∗)) + h2 τ2 ∗(∥R2(uH, pH)∥2 L2(τ2 ∗) + ∥R3(uH)∥2 L2(τ2 ∗)) ≤ 2−2/3h2 τ(∥R2(uH, pH)∥2 L2(τ) + ∥R3(uH)∥2 L2(τ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='39) For any f ∈ ∂ (τ1 ∗ ∪ τ2 ∗), which can be divided into three parts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (1) For the first part, there are two of the faces are constant and belong to τ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (2) For the second part, there are two new faces that overlap and are used to divide the mesh τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Since (uH, ph) ∈ UH × QH is a continuous polynomial in the region τ, it follows that the value of [[ph]] and [[( f − κuH)]] on this surface is equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3) For the third part, there are four faces that are obtained by dividing the two faces in the τ into two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, we obtain η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ1 ∗) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ2 ∗) ≤ γη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='τ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='40) where constant γ ∈ (0,1) independent of mesh τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, since ��H→�h represents the part of the set in the tetrahedral set �H that will be used to be refined, it follows that ��H→�h ⊂ �H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let ��H→�h denote the part of the cell set that has been refined in the tetrahedral set �H, we have ��h→�H ∈ �h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Obviously, �H \\��H→�h = �h \\��H→�h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then combining the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='40), and the marking strategy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18), we have η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) = η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h \\ ��H→�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H \\ ��H→�h) + γη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) + (γ − 1)η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) ≤ η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h), where λ = 1 − γ ∈ (0,1) independent of mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Now, we combine the Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9 to prove the reduction of error indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 20 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Given a constant θ ∈ (0,1) and two tetrahedral mesh �h, �H(�H ≤ �h).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let (uh, ph) ∈ Uh × Qh and (uH, pH) ∈ UH × QH be the solutions of (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For any ε > 0 and λ ∈ (0,1), we have (1 − Cε α )η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε + Cε α )η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − (1 + ε)λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) + Cε∥|uh − uH|∥2 h, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='41) where constant Cε depending on the ε and mesh size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using the Lemmas 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='6, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='8 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='9, we have η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥(uh, ph) − (uH, pH)∥2 DG ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥|uh − uH|∥2 h + ∥ph − pH∥2 L2(Ω) ≤ (1 + ε) � η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) − λη2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��H→�h) � +Cε∥|uh − uH|∥2 h + Cε α � η2(uh, ph;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�h) + η2(uH, pH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�H) � , which completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence result Now, we proved that the sum of the norm of the error and the scaled error indicator is attenuated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For a given θ ∈ (0,1),let {�k,Uk,Qk, uk, pk,η(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k)}k≥0 be the se- quence of meshes, Mixed discrete solution (defined by (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='13)-(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='14)), and the estimate in- dicator produced by the AMIPDG algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then there exist constants ρ > 0, δ ∈ (0,1), which depend on marking parameter θ and the shape regularity of the initial mesh �0, such that ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Setting �ρ = 1−δ2 2Cε , then multiply the both sides of the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='41) inequality by �ρ, we get �ρ(1 − Cε α )η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ �ρ(1 + ε + Cε α )η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) − �ρ(1 + ε)λη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1) +1 − δ2 2 ∥|uk+1 − uk|∥2 h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='42) Convergence of AMIPDG methods for H(cur l)-elliptic problems 21 Next, by the (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='26) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='42), we have ∥|u − uk+1|∥2 k+1 + �ρ(1 − Cε α )η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ (1 + δ1)∥|u − uk|∥2 k + C3 δ1δ2α � η2(v k+1,q k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) + η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � +�ρ(1 + ε + Cε α )η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) − �ρ(1 + ε)λη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='43) First move the term and then according to Dörfler marking strategy (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='18), the Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 and ∥| · |∥h ≤ ∥ · ∥DG, we know −η2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='��k→�k+1) ≤ −θη2(v k,q k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k), then ∥|u − uk+1|∥2 k+1 + �ρ(1 − Cε α − C3 �ρδ1δ2α)η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ (1 + δ1)∥|u − uk|∥2 k − �ρ(1 + ε)λθ 2 η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) +�ρ � 1 + ε + Cε α + C3 �ρδ1δ2α − (1 + ε)λθ 2 � η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ (1 + δ1 − �ρ(1 + ε)λθC−1 1 2 )∥|u − uk|∥2 k +�ρ � 1 + ε + Cε α + C3 �ρδ1δ2α − (1 + ε)λθ 2 � η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' For convenience, denote β1 = 1 − Cε α − C3 �ρδ1δ2α, β2 = 1 + δ1 − �ρ(1 + ε)λθC−1 1 2 , β3 = (1 + ε)(1 − λθ 2 ) + Cε α + C3 �ρδ1δ2α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Thus ∥|u − uk+1|∥2 k+1 + �ρβ1η2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ β2∥|u − uk|∥2 k + �ρβ3η2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next, we firstly choose δ1 = �ρ(1+ε)λθC−1 1 4 , then select the appropriate δ2 to make �ρ = 1−δ2 2Cε smaller to ensure 0 < δ1 < 1, Secondly, we let ε > 0 and (1 + ε)(1 − λθ 2 ) = 1 − λθ 4 ( λθ ∈ (0,1)), therefore β2 = 1 − δ1 ∈ (0,1), (1 + ε)(1 − λθ 2 ) < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Furthermore, we choose a sufficiently large penalty parameter α such that β1 > β3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 22 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Finally, there is a constant δ = max{β2, β1 β3 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Then, we let ρ = �ρβ1, and obtain ∥|u − uk+1|∥2 k+1 + ρη2(uk+1, pk+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k+1) ≤ δ � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Corollary 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Under the conditions of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we have ∥(u, p) − (uk, pk)∥2 DG + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ δk �Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' where �Cδ = C � ∥(u, p) − (u0, p0)∥2 DG + ρη2(u0, p0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�0) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Therefore, for a given precision, the AMIPDG method will terminate after a finite number of operations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Using the Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2 and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we have ∥(u, p) − (uk, pk)∥2 DG + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) ≤ C � ∥|u − uk|∥2 k + ρη2(uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k) � ≤ δk �Cδ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numerical experiments In this section, we test some numerical experiments to show the efficiency and the robustness of AMIPDG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We carry out these numerical experiments by using the MATLAB software package iFEM [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Experiments 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2, we take p = ∇ × u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1, we discuss the influence of the penalty parameter α on the error in ∥ · ∥DG norm, and observe the dependency of the condition number of stiffness matrix on α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2): u = � � x(x − 1)y(y − 1)z(z − 1) sin(πx)sin(πy)sin(πz) (1 − ex)(1 − ex−1)(1 − e y)(1 − e y−1)(1 − ez)(1 − ez−1) � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' It is easy to see that the solution u satisfies the boundary condition u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' In this example, we get a uniform mesh by partitioning the x−, y− and z−axes into equally distributed M(M ≥ 2) subintervals, and then dividing one cube into six tetrahe- drons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let h = 1/M be mesh sizes for different tetrahedrons meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We fixed mesh with h = 1/4 and report the error estimates in ∥ · ∥DG norm and condition number of stiffness matrices for different penalty parameters α = 1,10,100,500 and 1000 in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' We note that ∥u − uh∥0 increases at first and then decreases as the penalty parameter α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Convergence of AMIPDG methods for H(cur l)-elliptic problems 23 Table 1: The error in ∥ · ∥DG norms and condition number of stiffness matrices with h = 1/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' α 1 10 100 500 1000 ∥ � p − ph, u − uh � ∥DG 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='949e+00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='133e-00 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='614e-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='649e-01 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='659e-01 Cond 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='235e+04 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='021e+04 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='959e+05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='995e+06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='150e+06 The condition numbers of stiffness matrices increase with the increase of penalty parame- ters α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' As a way to balance, in the following numerical tests, we always choose α = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Noting that we only consider uniform meshes in Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Next we test adaptive meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Let Ω := [0,1] × [0,1] × [0,1], we construct the following analytical solution of the model (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='1)-(1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='2) u = � � � x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 − x(x−1)y(y−1)z(z−1) x2+y2+z2+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='001 � � �.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Note that the solution u satisfies the condition u × n = 0 on ∂ Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The right of Figure 1 shows an adaptively refined mesh with marking parameter- θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7 after k = 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The grid is locally refined near the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Figure 1: Left: the initial mesh with 1152 DoFs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Right: the adaptive mesh(θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7) with 181104 DoFs after 18 refinements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The Figure 2 shows the curves of log N−logη � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � for parameters θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='3,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='5,0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The curves indicate the convergence and the quasi-optimality of the adaptive algorithm AMIPDG of η � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Acknowledgment The first author is supported by the East China University of Technology (DHBK2019209) and Jiangxi Province Education Department (GJJ200755).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The second, third and fourth authors are supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 12071160).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' The third author is also supported by the National Natural Science Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 11901212).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 24 K Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Figure 2: Quasi optimality of the AMIPDG of the error η � uk, pk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='�k � with different marking parameters θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' References [1] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' AYUSO DE DIOS, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' HIPTMAIR AND C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content='L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' PAGLIANTINI, Auxiliary space preconditioners for SIP-DG discretizations of H(curl)-elliptic problems with discontinuous coefficients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' IMA J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Numer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' Anal.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'} +page_content=' 18–21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0dAzT4oBgHgl3EQfefzh/content/2301.01439v1.pdf'}