diff --git "a/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt" "b/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/59FJT4oBgHgl3EQflCzy/content/tmp_files/load_file.txt" @@ -0,0 +1,866 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf,len=865 +page_content='Adaptive Least-Squares Methods for Convection-Dominated Diffusion-Reaction Problems Zhiqiang Cai∗ Binghe Chen† Jing Yang‡ Abstract This paper studies adaptive least-squares finite element methods for convection- dominated diffusion-reaction problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares methods are based on the first-order system of the primal and dual variables with various ways of imposing outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The coercivity of the homogeneous least-squares func- tionals are established, and the a priori error estimates of the least-squares methods are obtained in a norm that incorporates the streamline derivative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' All methods have the same convergence rate provided that meshes in the layer regions are fine enough.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To increase computational accuracy and reduce computational cost, adaptive least- squares methods are implemented and numerical results are presented for some test problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ADAPTIVE FOSLS FOR THE CONVECTION-DOMINATED PROBLEMS 1 Introduction Due to the small diffusion coefficient, the solution of the convection-dominated diffusion- reaction problem develops the boundary or interior layers, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', narrow regions where derivatives of the solution change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It is well known that the conventional numerical methods do not work well on either stability or accuracy for such problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, the standard Galerkin method with continuous linear elements exhibits large spurious oscillation in the boundary layer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Over the decades, many successful numerical methods have been studied and may be roughly grouped into three categories: the mesh-fitted approach, the operator-fitted approach, and the stabilization approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The mesh-fitted approach utilizes the a priori information of the solution including the location and the width of the layer to construct a layer-fitted mesh, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the Shishkin mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The operator-fitted approach applies the layer-alike functions as the bases of the approximation space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The stabilization approach adds some stabilization term to the ∗Department of Mathematics, Purdue University, 150 N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' University Street, West Lafayette, IN 47907- 2067, zcai@math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='purdue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This work was supported in part by the National Science Foundation under grants DMS-1217081 and DMS-1522707.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' †Wells Fargo Corporate & Investment Banking, Charlotte, NC 28202-4200, binghe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='chen@wellsfargo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='com.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ‡School of Mathematical Science, Peking University, No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5 Yiheyuan Road Haidian District, Beijing, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='China 100871, yangjingmath@pku.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='cn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11582v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='NA] 27 Jan 2023 2 bilinear form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, the well-known streamline upwind Petrov-Galerkin (SUPG) method [21] adds the original equation tested by the convection term as the stabilization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For a comprehensive collection of the methods, see [23] and the references therein.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Recently, least-squares methods have been intensively studied for fluid flow and elas- ticity problems (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [5, 7, 8, 9, 12, 14, 15, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares methods minimize certain norms of the residual of the first-order system over appropriate finite element spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The method always leads to a symmetric positive definite problem, and choices of finite element spaces for the primal and dual variables are not subject to the LBB condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, one striking feature of the least-squares method is that the value of the least-squares functional at the current approximation provides an accurate estimates of the true error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The application of the least-squares methods to the convection-dominated diffusion- reaction problems is still in its infancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Reported in [17] is a new least-squares formulation with inflow boundary conditions weakly imposed and outflow boundary conditions ultra- weakly imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This formulation works well on regions away from the boundary layer, even on coarse meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it does not resolve the boundary layer, which is the primary interest of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This phenomena is also observed in the DG method [4], where the boundary conditions are weakly imposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' These works motivate us to treat outflow boundary conditions in different fashions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In particular, we study least-squares method for the convection-dominated diffusion-reaction problem with three different ways to handle the outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The a priori error estimates of finite element approximations based on these formulations are established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The solution of the convection-dominated diffusion-reaction problem usually consists of two parts: the solution of a transport problem (ϵ = 0) and the correction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the boundary layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To compute the first part, it is sufficient to use a coarse mesh, while it requires a very fine mesh to resolve the boundary layer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Without the a priori information on locations of the layers, this observation motivates the use of adaptive mesh refinement algorithm, which has been vastly studied (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [2, 3, 6, 13, 19, 24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, many a posteriori error estimators are not suitable for the convection-dominated diffusion-reaction problems, since they depend on the small diffusion parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To design a robust a posteriori error estimator is non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Nevertheless, for a least-squares formulation, the a posteriori error estimator is handy, which is simply the value of the least-squares functional at the current approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the least-squares functional has been computed when solving the algebraic equation, there is no additional cost.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Besides, the reliability and the efficiency stem easily from the coercivity and the continuity of the bilinear form, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this paper, we present numerical results of adaptive mesh refinement algorithms using the least-squares estimator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The rest of this paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In section 2, we present the convection- dominated diffusion-reaction problem and its first-order linear system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Based on the first- order system, three least-squares formulations are introduced and their coercivity are established in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Section 4 is a computable counterpart of the previous section, which introduces the computable mesh dependent norms to replace the fractional norms in the least-squares functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The main objective of section 5 is to establish the a priori error estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The adaptive mesh refinement algorithm and the numerical tests 3 are exhibited in section 6 and section 7, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 Notation We use the standard notation and definitions for the Sobolev spaces Hs(Ω)d and Hs(∂Ω)d for s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The standard associated inner products are denoted by (·, ·)s,Ω and (·, ·)s,∂Ω, and their respective norms are denoted by ∥·∥s,Ω and ∥·∥s,∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (We suppress the superscript d because the dependence on dimension will be clear by context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We also omit the subscript Ω from the inner product and norm designation when there is no risk of confusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=') For s = 0, Hs(Ω)d coincides with L2(Ω)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this case, the inner product and norm will be denoted by (·, ·) and ∥ · ∥, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Finally, we define some spaces H1 D(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD}, H1 D±(Ω) := {q ∈ H1(Ω) : q = 0 on ΓD±}, and H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) = {v ∈ L2(Ω)2 : ∇ · v ∈ L2(Ω)}, which is a Hilbert space under the norm ∥v∥H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) = � ∥v∥2 + ∥∇ · v∥2� 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 2 The convection-diffusion-reaction problem Let Ω be a bounded, open, connected subset in Rd (d = 2, 3) with a Lipschitz continu- ous boundary ∂Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by n = (n1, · · · , nd)t the outward unit vector normal to the boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For a given vector-valued function β, denote by Γ+ = {x ∈ ∂Ω : β · n(x) > 0} and Γ− = {x ∈ ∂Ω : β · n(x) < 0} the outflow and inflow boundaries, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Consider the following stationary convection-dominated diffusion-reaction problem: −ϵ ∆u + β · ∇u + c u = f in Ω, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) where the diffusion coefficient ϵ is a given small constant, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', 0 < ϵ ≪ 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' and c and f are given scalar-valued functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For simplicity, we consider homogeneous Dirichlet boundary condition: u|∂Ω = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) For the convection and reaction coefficients, we assume that: (1) β ∈ W 1 ∞(Ω)d and c ∈ L∞(Ω) with ∥c∥∞ ≤ γ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2) there exists a positive constant α0 such that 0 < α0 ≤ c − 1 2∇ · β a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) 4 Introducing the dual variable σ = −ϵ1/2∇u, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) may be rewritten as the following first-order system: � σ + ϵ1/2∇u = 0 in Ω, ϵ1/2∇ · σ + β · ∇u + c u = f in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 3 Least-squares formulations In this section, we study three least-squares formulations based on the first-order system in (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) with the inflow boundary conditions imposed strongly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' These formulations differ in how to handle the outflow boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' More specifically, the outflow boundary conditions are treated strongly for the first one and weakly for the other two through weighted boundary functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, introduce the following least-squares functionals: G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥τ + ϵ1/2 ∇v∥2 + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) G2(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + ∥ϵ−1/2 v∥2 1/2,Γ+, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and G3(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + ∥v∥2 1/2,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) Since ϵ is very small, the outflow boundary conditions are enforced stronger in G2 than in G3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let U1 = H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) × H1 0(Ω) and U2 = U3 = H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) × H1 Γ−(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the least-squares formulations are to find (σ, u) ∈ Ui such that Gi(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = min (τ , v)∈ Ui Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any (τ, v) ∈ Ui, define the following norms: M1(τ, v) = ∥τ∥2 + ∥v∥2 + ∥ϵ1/2 ∇v∥2, M2(τ, v) = M1(τ, v) + ∥ϵ−1/2 v∥2 1/2,Γ+, and M3(τ, v) = M1(τ, v) + ∥v∥2 1/2,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Below we show that the homogeneous least-squares functionals are coercive with respect to the corresponding norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In particular, the coercivity of the functionals G1 and G2 are independent of the ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 (Coercivity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Ui with i = 1, 2, 3, there exist positive constants Ci such that Mi(τ, v) ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) where C1 and C2 are independent of the ϵ and C3 is proportional to ϵ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 5 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We provide proofs for i = 2 and 3 in detail with an emphasis on how the weight in G2 leads to the coercivity constant independent of the ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The case of i = 1 may be proved in a similar fashion as the case of i = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Ui with i = 1, 2, 3, the triangle inequality gives ∥τ∥ ≤ ∥τ + ϵ1/2 ∇v∥ + ∥ϵ1/2 ∇v∥ ≤ G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) Hence, to show the validity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), it suffices to prove that ∥v∥2 + ∥ϵ1/2 ∇v∥2 ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) ∀ (τ, v) ∈ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) To this end, let I = − � ϵ1/2 ∇v, τ � + � v, (c − 1 2 ∇ · β) v � + 1 2 ∥(β · n)1/2 v∥2 0,Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) It follows from the definition of the outflow boundary condition and the Cauchy-Schwarz inequality that ∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ (ϵ1/2 ∇v, ϵ1/2∇v + τ) + I ≤ ∥ϵ1/2 ∇v∥ G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I, which implies ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C (G1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) To bound I, first note that integration by parts and the boundary conditions imply that (ϵ1/2 ∇v, τ) = (v, ϵ1/2 τ · n)∂Ω − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ − (ϵ1/2 v, ∇ · τ) = (v, ϵ1/2 τ · n)Γ+ + (v, c v) − (v, ϵ1/2 ∇ · τ + β · ∇v + c v) + (v β, ∇v) and that (∇v, v β) = 1 2 ∥(β · n)1/2 v∥2 0,Γ+ − 1 2 (v, v ∇ · β) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Combining the above two equalities yields I = � v, ϵ1/2 ∇ · τ + β · ∇v + c v � − (v, ϵ1/2 τ · n)Γ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) By the trace theorem and the Cauchy-Schwarz inequality, we have ∥τ · n∥−1/2,Γ+ ≤ C � ∥τ∥ + ∥∇ · τ∥ � ≤ C � G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥ + ϵ−1/2 ∥β · ∇v∥ + ϵ−1/2 ∥c v∥ � ≤ C ϵ−1/2� G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥∇v∥ + ∥v∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) 6 Let αi = 1 for i = 2 or 1/2 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then it follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), the Cauchy-Schwarz inequality, the definition of the dual norm, and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) that for i = 2 and 3 I ≤ ∥v∥ ∥ϵ1/2 ∇ · τ + β · ∇v + c v∥ + ∥ϵ1/2−αi v∥1/2,Γ+ ∥ϵαi τ · n∥−1/2,Γ+ (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) ≤ C � ∥v∥ + ∥ϵαi τ · n∥−1/2,Γ+ � G1/2 i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) ≤ C Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + C � ∥ϵαi−1/2 ∇v∥ + ∥v∥ � G1/2 i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9), implies ∥ϵ1/2 ∇v∥2 + α0 ∥v∥2 ≤ Ci Gi(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) with C2 independent of ϵ and C3 proportional to ϵ−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) and, hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for i = 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The validity of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for i = 1 may be established in a similar fashion by noticing that the boundary term of I in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) vanishes due to the boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 4 Mesh-dependent least-squares functionals For computational feasibility, in this section, we replace the 1 2-norm in the least-squares functionals defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) by mesh-dependent L2-norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the simplicity of presentation, assume that the domain Ω is a convex polygon in the two dimensional plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (The extension to the higher dimension is straightforward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=') Let Th = {K} be a triangulation of Ω with triangular elements K of diameter less than or equal to h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Assume that the triangulation Th is regular and quasi-uniform (see [18]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by Eh the set of all edges of the triangulation Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares functionals G2 and G3 defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) are modified by the following computable least-squares functionals: Gh 2(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈Eh∩Γ+ h−1 e ∥ϵ−1/2 v∥2 0,e (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and Gh 3(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = G1(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈Eh∩Γ+ h−1 e ∥v∥2 0,e, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) where he denotes the diameter of the edge e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any triangle K ∈ Th, let Pk(K) be the space of polynomials of degree less than or equal to k on K and denote the local Raviart–Thomas space of index k on K by RTk(K) = Pk(K)2 + � x1 x2 � Pk(K).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the standard H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) conforming Raviart–Thomas space of index k [22] and the standard (conforming) continuous piecewise polynomials of degree k + 1 are defined, re- spectively, by Σk h = {τ ∈ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) : τ|K ∈ RTk(K), ∀ K ∈ Th}, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) V k+1 h = {v ∈ H1(Ω) : v ∈ Pk+1(K), ∀ K ∈ Th}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 7 These spaces have the following approximation properties: let k ≥ 0 be an integer, and let l ∈ (0, k + 1]: inf τ ∈ Σk h ∥σ − τ∥H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) ≤ C hl (∥σ∥l + ∥∇ · σ∥l) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for σ ∈ Hl(Ω)2 ∩ H(div;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ω) with ∇ · σ ∈ Hl(Ω) and inf v∈V k+1 h ∥u − v∥1 ≤ C hl ∥u∥l+1 (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) for u ∈ Hl+1(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the subsequent sections, based on the smoothness of σ and u, we will choose k + 1 to be the smallest integer greater than or equal to l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the triangulation Th is regular, the following inverse inequalities hold for all K ∈ Th: ∥τ∥1,K ≤ C h−1 K ∥τ∥K, ∀ τ ∈ RTk(K) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) ∥v∥1,K ≤ C h−1 K ∥v∥K, ∀ v ∈ Pk(K) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) with positive constant C independent of hK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by Uh i the finite dimensional subspaces of Ui: Uh i = � Σk h × V k+1 h � ∩ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) For any (τ, v) ∈ Uh i , define the following norms: Mh 2 (τ, v) = M1(τ, v) + � e∈Eh∩Γ+ h−1 e ∥ϵ−1/2 v∥2 0,e and Mh 3 (τ, v) = M1(τ, v) + � e∈Eh∩Γ+ h−1 e ∥v∥2 0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Below we establish the discrete version of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the coercivity of the discrete functionals (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) with respect to the norms defined above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the consistence of notation, we also let Gh 1 = G1 and Mh 1 = M1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all (τ, v) ∈ Uh i with i = 2 and 3, there exist positive constants Ci independent of ϵ such that Mh i (τ, v) ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Similar to the argument in the proof of Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, in order to establish (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), it suffices to show that ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) for all (τ, v) ∈ Uh i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, we have ∥ϵ1/2 ∇v∥2 + ∥v∥2 ≤ C � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + I � (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) 8 with I defined in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any e ∈ Eh ∩ Γ+, let e be an edge of element K ∈ Th.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from the trace theorem and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) that he ∥τ · n∥2 0,e ≤ C he ∥τ∥2 0,e ≤ C he ∥τ∥0,K∥τ∥1,K ≤ C ∥τ∥2 0,K, which, together with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6), implies � � � e∈Eh∩Γ+ he ∥τ · n∥2 0,e � � 1/2 ≤ C ∥τ∥ ≤ C � G1/2 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇v∥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) Let αi = 1 for i = 2 or 1/2 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), the Cauchy-Schwarz inequality, and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) that I = � v, ϵ1/2 ∇ · τ + β · ∇v + c v � − (v, ϵ1/2 τ · n)Γ+ ≤ C � �∥v∥ + ϵαi � � e∈Eh∩Γ+ he ∥τ · n∥2 0,e �1/2 � � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0)1/2 ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + C � ∥v∥ + ∥ϵ1/2∇v∥ � Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0)1/2 which, together with (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12), implies the validity of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) and, hence, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This com- pletes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that the coercivity constant C3 in the discrete version is no longer depending on ϵ, that is better than the continuous version (see Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 5 Finite element approximations The least-squares problems are to find (σ, u) ∈ Ui (i = 1, 2, 3) such that Gh i (σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = min (τ , v)∈ Ui Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) The corresponding variational problems are to find (σ, u) ∈ Ui such that ai(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = Fi(τ, v), ∀ (τ, v) ∈ Ui, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) where the bilinear forms ai(· ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' ·) are symmetric and given by a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = (σ + ϵ1/2 ∇u, τ + ϵ1/2 ∇v) +(ϵ1/2 ∇ · σ + β · ∇u + c u, ϵ1/2 ∇ · τ + β · ∇v + c v), a2(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) + � e ∈Eh∩Γ+ h−1 e ϵ−1 (u, v)0,e, a3(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = a1(σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) + � e ∈Eh∩Γ+ h−1 e (u, v)0,e, 9 and the linear forms Fi(·) are given by Fi(τ, v) = (f, ϵ1/2 ∇ · τ + β · ∇v + c v) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares finite element approximations to the variational problems in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) are to find (σi h, ui h) ∈ Uh i such that ai(σi h, ui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = Fi(τ, v), ∀ (τ, v) ∈ Uh i , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Taking the difference between (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) implies the following orthog- onality: ai(σ − σi h, u − ui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' τ, v) = 0, ∀ (τ, v) ∈ Uh i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) In the rest of this section, we consider a stronger norm which incorporates the norm of the streamline derivative: |||(τ, v)|||2 i = Mh i (τ, v) + � K∈Th δK ∥β · ∇v∥2 K, where δK is a positive constant to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the following lemma, we show that Gh i (σ, u;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) are also elliptic with respect to these norms if the δK is appropriately chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For all K ∈ Th, assume that 0 < δK ≤ min{h2 K/ϵ, C}, then there exist positive constants Ci independent of ϵ such that |||(τ, v)|||2 i ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), ∀ (τ, v) ∈ Uh i , i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' By Theorems 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, to prove the validity of the lemma, it suffices to show that � K∈Th δK ∥β · ∇v∥2 K ≤ Ci Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) To this end, note the facts that δK ≤ C and δK ϵ h2 K ≤ min � 1, C ϵ h2 K � ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now it follows from the Cauchy-Schwarz inequality and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) that � K∈Th δK ∥β · ∇v∥2 K ≤ C � K∈Th δK � Gh 1,K (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥ϵ1/2 ∇ · τ∥2 K + ∥c v∥2 K � ≤ C � K∈Th � Gh 1,K (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + δK ϵ h2 K ∥τ∥2 K + ∥v∥2 K � ≤ C � Gh 1 (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) + ∥τ∥2 + ∥v∥2� ≤ C Gh i (τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), which establishes (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) and hence completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 10 To choose δK properly, first define the local mesh P´eclet number by PeK = ∥β∥0,∞,K hK 2 ϵ , then partition the triangulation Th into two subsets: T c h = {K ∈ Th : PeK > 1} and T d h = {K ∈ Th : PeK ≤ 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) The elements in T c h are referred to the convection-dominated elements, while the elements in T d h the diffusion-dominated elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, the δK is chosen to be δK = � � � � � � � � � � � 2 hK ∥β∥0,∞,K , if K ∈ T c h , h2 K ϵ , if K ∈ T d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The δK defined in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7) satisfies the assumption in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', δK ≤ min{h2 K/ϵ, C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since ∥β∥0,∞,K is large comparing to hK, we have 2 hK ∥β∥0,∞,K ≤ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9) For any K ∈ T c h , the fact that PeK > 1 implies 2 hK ∥β∥0,∞,K < h2 K ϵ , which, together with (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9), yields (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For any K ∈ T d h , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8) is again a consequence of the definition of δK in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), the fact that PeK ≤ 1, and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Denote by T ∂ h the set of elements that intersect the outflow boundary nontrivially, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', T ∂ h = {K ∈ Th : meas( ¯K ∩ Γ+) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this paper, we assume that T ∂ h ⊂ T d h .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) For any K ∈ T d h , the fact that PeK ≤ 1 implies hK < 2 ϵ ∥β∥0,∞,K .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Hence, assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10) means that the mesh size in the boundary layer region is com- parable to the perturbation parameter ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 11 Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σ, u) be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Assume that (σ, u) ∈ Hl(Ω)2×Hl+1(Ω) and that ∇ · σ ∈ Hl(Ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σi h, ui h), i = 1, 2, 3, be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) with k = l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Under the assumption in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), we have the following a priori error estimation: Ci ������(σ − σi h, u − ui h) ������2 i ≤ � K∈T c h h2l−1 K � ϵ ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ∥u∥2 l+1,K � + � K∈T d h h2l−1 K � ϵ2 hK ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ϵ hK ∥u∥2 l+1,K � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) where constants Ci > 0 are independent of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' We provide proof of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) only for i = 2 and 3 since (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) may be obtained in a similar fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, let σI and uI be the interpolants of σ and u, respectively, such that the approximation properties in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) hold and that (∇ · (σ − σI), v) = 0, ∀ v ∈ Dh k, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12) where Dh k = {v ∈ L2(Ω) : v|K ∈ Pk(K) ∀ K ∈ Th} is the space of discontinuous piecewise polynomials of degree less than or equal to k ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let EI = σ − σI, Ei h = σI − σi h, eI = u − uI, and ei h = uI − ui h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since Ei = σ − σi h = EI + Ei h and ei = u − ui h = eI + ei h, the triangle inequality gives ������(Ei, ei) ������ i ≤ |||(EI, eI)|||i + ������(Ei h, ei h) ������ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) Let αi = −1 or 0 for i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' By approximation property (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) and assumption (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10), we have � e ∈Eh∩Γ+ h−1 e ϵαi ∥eI∥2 0,e ≤ C � K∈T ∂ h h2l K ϵαi ∥u∥2 l+1,K ≤ C � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, it follows from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6), the trace theorem, and the fact δK ≤ C that |||(EI, eI)|||2 i ≤ C � �∥EI∥2 + ∥eI∥2 + ∥ϵ1/2 ∇eI∥2 + � e∈Γ+ h−1 e ϵαi ∥eI∥2 e + � K∈Th ∥β · ∇eI∥2 K � � ≤ C � � � K∈Th h2l K ∥σ∥2 l,K + � K∈Th h2l K ∥u∥2 l+1,K + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='14) 12 To bound the second term of the right-hand side in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13), by Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 and orthog- onality (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have Ci ������(Ei h, ei h) ������2 i ≤ ai(Ei h, ei h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Ei h, ei h) = ai(Ei h, ei h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' −EI, −eI) ≡ Ii 1 + Ii 2 + Ii 3 + Ii 4, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='15) where Ii 1 = (c ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI) + (Ei h + ϵ1/2 ∇ei h, −EI − ϵ1/2 ∇eI), Ii 2 = (ϵ1/2 ∇ · Ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), Ii 3 = (β · ∇ei h, −ϵ1/2 ∇ · EI − β · ∇eI − c eI), and Ii 4 = � e ∈Eh∩Γ+ h−1 e ϵαi (ei h, −eI)0,e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' It follows from the triangle and Cauchy-Schwarz inequalities, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5), and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) that Ii 1 ≤ C ∥ei h∥ � ∥ϵ1/2∇ · EI∥ + ∥∇eI∥ + ∥eI∥ � + C � ∥Ei h∥ + ∥ϵ1/2∇ei h∥ � � ∥EI∥ + ∥ϵ1/2∇eI)∥ � ≤C � ∥ei h∥ + ∥Ei h∥ + ∥ϵ1/2∇ei h∥ � � � � K∈Th h2l K � ϵ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + ∥u∥2 l+1,K � � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='16) By (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='12), the Cauchy-Schwarz and triangle inequalities, and the inverse inequality in (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), we have Ii 2 = −(ϵ1/2 ∇ · Ei h, β · ∇eI + c eI), ≤ C � K∈Th ϵ1/2 hK ∥Ei h∥K � ∥∇eI∥K + ∥eI∥K � ≤ C ∥Ei h∥ � � � K∈Th ϵ h2l−2 K ∥u∥2 l+1,K � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='17) By the Cauchy-Schwarz and the triangle inequalities, I3 is bounded by Ii 3 ≤ C � K∈Th ∥β · ∇ei h∥K � ϵ1/2 ∥∇ · EI∥K + ∥∇eI∥K + ∥eI∥K � ≤ C � K∈Th ∥β · ∇ei h∥K � ϵ1/2 hl K ∥∇ · σ∥l,K + hl K ∥u∥l+1,K � ≤ C � � � K∈Th δK∥β · ∇ei h∥2 K � � 1/2� � � K∈Th δ−1 K � ϵ h2l K ∥∇ · σ∥2 l,K + h2l K ∥u∥2 l+1,K � � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='18) 13 For Ii 4, it follows from the Cauchy-Schwarz inequality and the trace theorem that Ii 4 ≤ C � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥ei h∥2 0,e � � 1/2 � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥eI∥2 0,e � � 1/2 ≤ C � � � e ∈Eh∩Γ+ h−1 e ϵαi ∥ei h∥2 0,e � � 1/2 � � � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K � � 1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='19) Combining (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='15), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='16), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='17), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='18), (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='19), and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8), we have Ci ������(Ei h, ei h) ������2 i ≤ � K∈Th h2l K∥σ∥2 l,K + � K∈Th � 1 + δ−1 K � ϵ h2l K ∥∇ · σ∥2 l,K + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K + � K∈Th � 1 + ϵ h−2 K + δ−1 K � h2l K∥u∥2 l+1,K ≤ � K∈Th �ϵ h2l K δK ∥∇ · σ∥2 l,K + h2l K ∥σ∥2 l,K + h2l K δK ∥u∥2 l+1,K � + � K∈T ∂ h h2l+αi K ∥u∥2 l+1,K, which, together with the definition of δK in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7), implies Ci ������(Ei h, ei h) ������2 i ≤ � K∈T c h h2l−1 K � ϵ ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ∥u∥2 l+1,K � + � K∈T d h h2l−1 K � ϵ2 hK ∥∇ · σ∥2 l,K + hK ∥σ∥2 l,K + ϵ hK ∥u∥2 l+1,K � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Now, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='11) is a consequence of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='13) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that the a priori error estimate in Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3 is not optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This is because the coercivity of the homogeneous least-squares functionals in Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 are established in a norm that is weaker than the norm used for the continuity of the functionals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To restore the full order of convergence, one may use piecewise polynomials of degree l + 1 to approximate u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σi h, ui h), i = 1, 2, 3, be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) with Uh i = (Σl h×V l+1 h )∩ Ui.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 14 Under the assumption of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3, we have the following a priori error estimation: Ci ������(σ − σi h, u − ui h) ������2 i ≤ � K∈T c h h2l K � ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + hK ∥u∥2 l+2,K � + � K∈T d h h2l K � ϵ2 h2 K ∥∇ · σ∥2 l,K + ∥σ∥2 l,K + ϵ ∥u∥2 l+2,K � , (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='20) where constants Ci > 0 are independent of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The a priori error estimate in (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='20) may be obtained in a similar fashion by noting that ∥u − uI∥1 ≤ C hl+1∥u∥l+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 6 Adaptive algorithm Asymptotic analysis (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [20]) shows that the solution of a convection-dominated diffusion-reaction problem consists of two parts: the solution of the reduced equation (ϵ = 0) and the correction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', the boundary or interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The boundary and interior layers are narrow regions where derivatives of the solution change dramatically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For example, for the following problem [20]: � � � � � −ϵ ∆u + ∂u ∂y = f in Ω = (0, 1)2, u = 0 on ∂Ω, the exponential layer is of width O(ϵ) at y = 1, and the width of the parabolic boundary layers is O(ϵ1/2) at both x = 0 and x = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Therefore, two sets of largely different scales exist simultaneously in the convection-dominated diffusion problem, and hence it is difficult computationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the one hand, one can apply the small scale over the entire domain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', to use uniform fine meshes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' With such a fine mesh, the standard Galerkin finite element method can also produce a good approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it is computationally inefficient due to the small region of the boundary and/or interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the other hand, one can use the large scale over the entire domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' If the outflow boundary conditions are imposed strongly, the numerical solution (away from the boundary layers) will be polluted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In contrast, if the outflow boundary conditions are imposed weakly, the boundary layers can not be resolved (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', numerical results in [4, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Neither of the above two approaches leads to a satisfactory numerical scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The fail- ure is due to the fact that these approaches ignore this intrinsic property of the convection- dominated diffusion problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In contrast, the Shishkin mesh is aware of and respect it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 15 Basically, the Shishkin mesh is a piecewise uniform mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the diffusion-dominated re- gion where the layers stand, it is a fine mesh suitable to the layer and in the convective region, it turns to be a coarse mesh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The disadvantage of the Shishkin mesh is that it needs the a priori information of the solution, such as the location and the width of the layer, in order to construct a mesh of high quality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, this information is not always available in advance, especially, for a complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Based on the above considerations, we employ adaptive least-squares finite element methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares estimators are simply defined as the value of the least-squares functionals at the current approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' To this end, for each element K ∈ Th, denote the local least-squares functionals by Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥τ + ϵ1/2 ∇v∥2 K + ∥ϵ1/2 ∇ · τ + β · ∇v + c v − f∥2 K, Gh 2,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = � � � � � Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f), if K ∩ Γ+ = ∅, Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈K∩Γ+ h−1 e ∥ϵ−1/2v∥2 0, e, otherwise, and Gh 3,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = � � � � � Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f), if K ∩ Γ+ = ∅, Gh 1,K(τ, v;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) + � e∈K∩Γ+ h−1 e ∥v∥2 0, e, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (ˆσh i , ˆuh i ) be the current approximations to the solutions of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Then the least-squares indicators are simply the square root of the value of the local least-squares functionals at the current approximation: ηi K = Gh i,K (ˆσi h, ˆui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f)1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) for all K ∈ Th and for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The least-squares estimators are ηi = � � � K∈Th � ηi K �2 � � 1/2 = Gh i (ˆσi h, ˆui h;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f)1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let (σ, u) be the solution of (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) and denote the true errors by ˆEi = σ − ˆσi h and ˆei = u − ˆu1 h for i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' There exist positive constants Ce,1 and Cr,1 independent of ϵ such that η1 K ≤ Ce,1 � M1,K(ˆE1, ˆe1) + ∥β · ∇ ˆe1∥2 K + ϵ ∥∇ · ˆE1∥2 K �1/2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) for all K ∈ T and that M1(ˆE1, ˆe1)1/2 ≤ Cr,1 η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4) 16 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Since the exact solution (σ, u) satisfies (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have � η1 K �2 = Gh 1,K(ˆE1, ˆe1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0) and � η1�2 = Gh 1(��E1, ˆe1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' which, together with the triangle inequality and Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1, imply the efficiency and the reliability bounds, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' There exist positive constants Ce,i independent of ϵ such that Ce, i � ηi K �2 ≤ Mh i,K(ˆEi, ˆei) + ∥β · ∇ˆei∥2 K + ϵ ∥∇ · ˆEi∥2 (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5) for all K ∈ T and i = 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Let αi = −1 for i = 2 or 0 for i = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' With the fact that (σ, u) is the exact solution satisfying (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4), we have ηi(ˆσh i , ˆuh i )2 = Gh i (ˆσh i , ˆuh i ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' f) = ∥ˆσh i + ϵ1/2 ∇ˆuh i ∥2 + ∥ϵ1/2 ∇ · ˆσh i + β · ∇ˆuh i + c ˆuh i − f∥2 + � e∈Eh∩Γ+ ϵαi h−1 e ∥ˆuh i ∥2 0,e = ∥ˆEi + ϵ1/2 ∇ˆei∥2 + ∥ϵ1/2 ∇ · ˆEi + β · ∇ˆei + c ˆei∥2 + � e∈Eh∩Γ+ ϵαi h−1 e ∥ˆei∥ = Gh i (ˆEi, ˆei;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0), (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) with which, the efficiency bound simply follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6) and the Cauchy-Schwarz in- equality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In the remainder of this section, we describe the standard adaptive mesh refinement algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Starting with an initial triangulation T0, a sequence of nested triangulations {Tl} is generated through the well known AFEM-Loop: SOLVE −→ ESTIMATE −→ MARK −→ REFINE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The SOLVE step solves (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='3) in the finite element space corresponding to the mesh Tl for a numerical approximation (σi h(l), ui h(l)) ∈ Uh i (l), where Uh i (l) is the finite element space defined on Tl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Hereafter, we shall explicitly express the dependence of a quantity on the level l by either the subscript like Tl or the variable like Uh i (l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The ESTIMATE step computes the indicators {ηi K(l)} and the estimator ηi(l) defined in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The way to choose elements for refinement influences the efficiency of the adaptive algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' If most of elements are marked for refinement, then it is comparable to uniform refinement, which does not take full advantage of the adaptive algorithm and results in redundant degrees of freedom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' On the other hand, if few elements are refined, then it requires many iterations, which undermines the efficiency of the adaptive algorithm, since each iteration is costly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' For the singularly perturbed problems, it is well known that the indicators associated with the elements in the layer region are much larger than others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 17 Therefore, we MARK by the maximum algorithm, which defines the set ˆTl of marked elements such that for all K ∈ ˆTl ηi K(l) ≥ θ max K∈Tl ηi K(l).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The REFINE step is to bisect all the triangles in ˆTl into two sub-triangles to generate a new triangulation Tl+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Note that some triangles in Tl \\ ˆTl adjacent to triangles in ˆTl are also refined in order to avoid hanging nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In summary, the adaptive least-squares finite element algorithm can be cast as follows: with the initial mesh T0, marking parameter θ ∈ (0, 1), and the maximal number of iteration maxIt, for l = 0, 1, · · · , maxIt, do (1) (σi h(l), ui h(l)) = SOLVE(Tl);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (2) {ηi K(l)} = ESTIMATE(Tl, σi h(l), ui h(l));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (3) ˆTl = MARK(Tl, {ηi K(l)});' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (4) Tl+1 = REFINE(Tl, ˆTl).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 7 Numerical experiments In this section, we conduct several numerical experiments on two model problems used by many authors (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=', [4, 17]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Both the model problems are defined in the unit square and all numerical experiments are started with the same initial mesh, which consists of sixteen isosceles right triangles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The marking parameter θ is chosen to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 Boundary layer In this example, β = [1, 1]T , and c = 0, and the external force f is chosen such that the exact solution is u(x, y) = sin πx 2 + sin πy 2 � 1 − sin πx 2 � + e−1/ϵ − e−(1−x)(1−y)/ϵ 1 − e−1/ϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This solution is smooth, but develops boundary layers at x = 1 and y = 1 with width O(ϵ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This example is suitable for testing capability of the numerical approximations on resolving the boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' In this numerical experiment, ϵ = 10−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Given the tolerance tol = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5, computation is terminated if ηi(l) ≤ tol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) Since the exact solution is available, the true error is computed and the effectivity index is defined as follows: eff-index := ηi(σi h, ui h) ������(σ − σi h, u − ui h) ������ i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) 18 Figure 1: The final meshes and the numerical solutions are, respectively, displayed in the first and the second columns and the rows are corresponding to i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The final meshes are displayed in the first column of Figure 1 when the stopping criterion (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1) is satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' They clearly show that the refinements cluster around the boundary layer area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The numerical solutions on the final meshes are depicted in the second column of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' All the three methods successfully capture the sharp boundary layers, and no visible oscillation appears in the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Reported in Figure 2 is the 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='9 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='. A : 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6 : -?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='..' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='4 c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 019 convergence rates of the numerical solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The errors with the norm |||·|||i that are used in the a priori error estimate are tracked, which converge in the order of (DoF)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, the convergence rate is independent of the value of ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' This is also verified by the test problem with ϵ = 10−4, where the convergence rate does not deteriorate (see the second column of Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 2: The convergence rates corresponding to ϵ = 10−3 and 10−4 are displayed in the first and the second columns, respectively, and the rows are corresponding to i = 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' 10° 10° 10 10° 10 10* 10° errEne3 estimator DoF-1 effindex 10 102 103 104 105 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='10° ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='Degree of Freedom20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 Interior layer In the second example, β = [1/2, √ 3/2]T , c = 0, f = 0, and the boundary condition is u = � � � � � � � 1, on {(x, y) : y = 0, 0 ≤ x ≤ 1}, 1, on {(x, y) : x = 0, 0 ≤ y ≤ 1/5}, 0, otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The exact solution of the problem remains unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' However, it is known that, additional to the boundary layers, the solution develops an interior layer along the line y = √ 3 x+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2 due to the discontinuity at (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content='2) of the boundary condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' The problem is chosen to test whether the formulations can capture the interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 3 shows that all the three methods capture both the boundary and the interior layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Moreover, the numerical solutions do not exhibit any visible oscillation, which is much better than the results reported in [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Figure 3: Numerical solutions corresponding to i = 1, 2, 3 from left to right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Acknowledgements We thank Dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59FJT4oBgHgl3EQflCzy/content/2301.11582v1.pdf'} +page_content=' Shuhao Cao for the discussion and helpful suggestions on the computation of the test problems.' metadata={'source': 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