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Simplified Weak Galerkin Methods for Linear Elasticity on Nonconvex Domains

Chunmei Wang, Shangyou Zhang

Abstract

This paper presents a weak Galerkin (WG) finite element method for linear elasticity on general polygonal and polyhedral meshes, free from convexity constraints, by leveraging bubble functions as central analytical tools. The proposed method eliminates the need for stabilizers commonly used in traditional WG methods, resulting in a simplified formulation. The method is symmetric, positive definite, and straightforward to implement. Optimal-order error estimates are established for the WG approximations in the discrete $H^1$-norm, assuming sufficient smoothness of the exact solution, and in the standard $L^2$-norm under regularity assumptions for the dual problem. Numerical experiments confirm the efficiency and accuracy of the proposed stabilizer-free WG method.

Simplified Weak Galerkin Methods for Linear Elasticity on Nonconvex Domains

Abstract

This paper presents a weak Galerkin (WG) finite element method for linear elasticity on general polygonal and polyhedral meshes, free from convexity constraints, by leveraging bubble functions as central analytical tools. The proposed method eliminates the need for stabilizers commonly used in traditional WG methods, resulting in a simplified formulation. The method is symmetric, positive definite, and straightforward to implement. Optimal-order error estimates are established for the WG approximations in the discrete -norm, assuming sufficient smoothness of the exact solution, and in the standard -norm under regularity assumptions for the dual problem. Numerical experiments confirm the efficiency and accuracy of the proposed stabilizer-free WG method.

Paper Structure

This paper contains 8 sections, 14 theorems, 93 equations, 5 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discrete Weak Strain Tensor and Discrete Weak Divergence
  3. Weak Galerkin Algorithms without Stabilizers
  4. Solution Existence and Uniqueness
  5. Error Equations
  6. Error Estimates
  7. Error Estimates in $L^2$ Norm
  8. Numerical verification

Key Result

Lemma 4.1

wang1 For ${\mathbf{v}}=\{{\mathbf{v}}_0, {\mathbf{v}}_b\}\in V_h$, there exists a constant $C$ such that

Figures (5)

  • Figure 1: The first three grids for the computation in Tables \ref{['t-1']}--\ref{['t-2']}.
  • Figure 2: The first three non-convex polygonal grids for the computation in Tables \ref{['t-3']}--\ref{['t-4']}.
  • Figure 3: The first three grids for the computation in Tables \ref{['t-5']}-\ref{['t-6']}.
  • Figure 4: The first three grids for the computation in Table \ref{['t-7']}.
  • Figure 5: The first three grids for the computation in Table \ref{['t-8']}.

Theorems & Definitions (29)

  • Lemma 4.1
  • proof
  • Lemma 4.2
  • proof
  • Remark 4.1
  • Lemma 4.3
  • proof
  • Lemma 4.4
  • proof
  • Lemma 4.5
  • ...and 19 more