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Convergent finite elements on arbitrary meshes, the WG method

Ran Zhang, Shangyou Zhang

TL;DR

This paper tackles the convergence failure of classical finite element methods on meshes that violate the maximum angle condition by introducing a weak Galerkin finite element method that avoids inter-element penalties and relies on a weak gradient. The authors prove convergence on arbitrary triangular and tetrahedral meshes, achieving two-order superconvergence, and develop a gradient-preserving Zhang-Zhang transformation to support the analysis. Numerical experiments in 2D and 3D Poisson problems validate robustness on degenerate meshes and illustrate the superior convergence of the WG method compared to standard P1 elements. The work broadens the applicability of finite element methods to poorly shaped meshes with guaranteed convergence and improved accuracy.

Abstract

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e., on arbitrary meshes. Numerical tests confirm the theory.

Convergent finite elements on arbitrary meshes, the WG method

TL;DR

This paper tackles the convergence failure of classical finite element methods on meshes that violate the maximum angle condition by introducing a weak Galerkin finite element method that avoids inter-element penalties and relies on a weak gradient. The authors prove convergence on arbitrary triangular and tetrahedral meshes, achieving two-order superconvergence, and develop a gradient-preserving Zhang-Zhang transformation to support the analysis. Numerical experiments in 2D and 3D Poisson problems validate robustness on degenerate meshes and illustrate the superior convergence of the WG method compared to standard P1 elements. The work broadens the applicability of finite element methods to poorly shaped meshes with guaranteed convergence and improved accuracy.

Abstract

On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e., on arbitrary meshes. Numerical tests confirm the theory.
Paper Structure (5 sections, 3 theorems, 25 equations, 3 figures, 4 tables)

This paper contains 5 sections, 3 theorems, 25 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. The weak Galerkin finite element method
  3. Convergence
  4. Numerical test
  5. Ethical Statement

Key Result

Lemma 2.1

For all $v_h=\{v_0,v_b\} \in V_h$ of V-h, it holds that where $C>0$ is independent of the shape regularity and the size of $\mathcal{T}_h$, $\|\cdot\|_0$ is the $L^2$ norm, and $\nabla_w$ is defined in w-g.

Figures (3)

  • Figure 1: (A) A triangular mesh violates the maximum angle condition, when $h\to 0$. (B) The area of triangles (interpolating at three vertices) converges to different a number. (C) The area of triangles (interpolating at three mid-edge points) converges to different a number. (D) The area of triangles ($L^2$-projection) converges to that of the cylinder. (E) The arc-length of line segments converges to that of the circle when mesh size goes to zero.
  • Figure 2: (A) Quasi-uniform but non-nested meshes. (B) Degenerate (violates the maximum angle condition) meshes.
  • Figure 3: (A) Quasi-uniform but non-nested 3D meshes. (B) Degenerate (violates the maximum angle condition) 3D meshes.

Theorems & Definitions (6)

  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • proof