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An Auto-Stabilized Weak Galerkin Method for Elasticity Interface Problems on Nonconvex Meshes

Chunmei Wang, Shangyou Zhang

TL;DR

This work develops an auto-stabilized weak Galerkin method for elasticity interface problems on general polytopal meshes without convexity assumptions, leveraging bubble functions to remove the need for stabilizers. The formulation yields a symmetric, positive-definite scheme and, under standard regularity, proves optimal-order convergence in the discrete $\,H^1$-norm. Theoretical results are supported by comprehensive numerical experiments on nonconvex meshes that demonstrate robustness to interface discontinuities and mesh geometry, with occasional superconvergence observed on certain grids. Overall, the approach offers a simple, stable, and efficient WG framework suitable for complex elasticity interfaces on general meshes.

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for elasticity interface problems on general polygonal and polyhedral meshes, without requiring convexity constraints. The method utilizes bubble functions as key analytical tools, eliminating the need for stabilizers typically used in traditional WG methods and leading to a more streamlined formulation. The proposed method is symmetric, positive definite, and easy to implement. Optimal-order error estimates are derived for the WG approximations in the discrete $H^1$-norm, assuming the exact solution has sufficient smoothness. Numerical experiments validate the accuracy and efficiency of the auto-stabilized WG method.

An Auto-Stabilized Weak Galerkin Method for Elasticity Interface Problems on Nonconvex Meshes

TL;DR

This work develops an auto-stabilized weak Galerkin method for elasticity interface problems on general polytopal meshes without convexity assumptions, leveraging bubble functions to remove the need for stabilizers. The formulation yields a symmetric, positive-definite scheme and, under standard regularity, proves optimal-order convergence in the discrete -norm. Theoretical results are supported by comprehensive numerical experiments on nonconvex meshes that demonstrate robustness to interface discontinuities and mesh geometry, with occasional superconvergence observed on certain grids. Overall, the approach offers a simple, stable, and efficient WG framework suitable for complex elasticity interfaces on general meshes.

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for elasticity interface problems on general polygonal and polyhedral meshes, without requiring convexity constraints. The method utilizes bubble functions as key analytical tools, eliminating the need for stabilizers typically used in traditional WG methods and leading to a more streamlined formulation. The proposed method is symmetric, positive definite, and easy to implement. Optimal-order error estimates are derived for the WG approximations in the discrete -norm, assuming the exact solution has sufficient smoothness. Numerical experiments validate the accuracy and efficiency of the auto-stabilized WG method.
Paper Structure (13 sections, 12 theorems, 69 equations, 7 figures, 8 tables, 1 algorithm)

This paper contains 13 sections, 12 theorems, 69 equations, 7 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discrete Weak Strain Tensor and Discrete Weak Divergence
  3. Auto-Stabilized Weak Galerkin Algorithm
  4. Solution Existence and Uniqueness
  5. Error Equations
  6. Error Estimates
  7. Numerical test
  8. Example \ref{['ex-1']}
  9. Example \ref{['ex-2']}
  10. Example \ref{['ex-3']}
  11. Example \ref{['ex-4']}
  12. Example \ref{['ex-5']}
  13. Example \ref{['ex-6']}

Key Result

Lemma 4.2

wang1wang2wangelas \newlabelnorm1 For ${\mathbf{v}}=\{{\mathbf{v}}_0, {\mathbf{v}}_b\}\in V_h$, there exists a constant $C$ such that

Figures (7)

  • Figure 7.1: Type-1 non-convex polygonal grids: first four grids $G_1$--$G_4$.
  • Figure 7.2: The $({\mathbf{u}}_h)_1$ and $({\mathbf{u}}_h)_2$ of the $P_3$ WG solution for \ref{['model']} with \ref{['l-1']} and \ref{['f-1']} on Grid 3 in Figure \ref{['g2-1']}.
  • Figure 7.3: Type-2 non-convex polygonal grids: first four grids $G_1$--$G_4$.
  • Figure 7.4: The $({\mathbf{u}}_h)_1$ and $({\mathbf{u}}_h)_2$ of the $P_1$ WG solution for \ref{['2-2']}--\ref{['2-s2']} on Grid 5 in Figure \ref{['g2-1']}.
  • Figure 7.5: The $({\mathbf{u}}_h)_1$ and $({\mathbf{u}}_h)_2$ of the $P_1$ WG solution for \ref{['3-1']}--\ref{['3-s1']} on Grid 6 in Figure \ref{['g2-1']}.
  • ...and 2 more figures

Theorems & Definitions (21)

  • Definition 4.1
  • Lemma 4.2
  • Lemma 4.3
  • Remark 4.1
  • Definition 4.4
  • Lemma 4.5
  • Lemma 4.6
  • Lemma 4.7
  • Remark 4.2
  • Remark 4.3
  • ...and 11 more