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Auto-Stabilized Weak Galerkin Finite Element Methods for Stokes Equations on Non-Convex Polytopal Meshes

Chunmei Wang, Shangyou Zhang

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for solving Stokes equations without relying on traditional stabilizers. The proposed WG method accommodates both convex and non-convex polytopal elements in finite element partitions, leveraging bubble functions as a key analytical tool. The simplified WG method is symmetric and positive definite, and optimal-order error estimates are derived for WG approximations in both the discrete $H^1$ norm and the $L^2$ norm.

Auto-Stabilized Weak Galerkin Finite Element Methods for Stokes Equations on Non-Convex Polytopal Meshes

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method for solving Stokes equations without relying on traditional stabilizers. The proposed WG method accommodates both convex and non-convex polytopal elements in finite element partitions, leveraging bubble functions as a key analytical tool. The simplified WG method is symmetric and positive definite, and optimal-order error estimates are derived for WG approximations in both the discrete norm and the norm.
Paper Structure (8 sections, 13 theorems, 123 equations, 8 figures, 8 tables, 1 algorithm)

This paper contains 8 sections, 13 theorems, 123 equations, 8 figures, 8 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Discrete Weak Gradient and Discrete Weak Divergence
  3. Auto-Stabilized Weak Galerkin Algorithms
  4. Solution Existence and Uniqueness
  5. Error Equations
  6. Error Estimates in $H^1$
  7. $L^2$ Error Estimates
  8. Numerical tests

Key Result

Lemma 4.1

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

Figures (8)

  • Figure 1: The first three meshes $G_1$--$G_3$ for Table \ref{['t2-1']}
  • Figure 2: The first three meshes $G_1$--$G_3$ for Table \ref{['t2-2']}
  • Figure 3: The first three meshes $G_1$--$G_3$ for Table \ref{['t2-3']}
  • Figure 4: The first three meshes $G_1$--$G_3$ for Table \ref{['t2-4']}
  • Figure 5: The first three meshes $G_1$--$G_3$ for Table \ref{['t3-1']}
  • ...and 3 more figures

Theorems & Definitions (28)

  • Lemma 4.1
  • proof
  • Remark 4.1
  • Lemma 4.2
  • proof
  • Remark 4.2
  • Lemma 4.3
  • proof
  • Remark 4.3
  • Remark 4.4
  • ...and 18 more