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Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes without Convexity Constraints

Chunmei Wang

TL;DR

This work develops an auto-stabilized weak Galerkin method for the Poisson equation on general polytopal meshes, including non-convex elements, and in arbitrary dimension. Leveraging bubble functions as a core analytical tool, the approach yields a simple, symmetric, and positive definite discretization with built-in stabilization, while preserving sparsity. The authors prove existence and uniqueness, derive an error equation, and establish optimal-order convergence in the discrete $H^1$-norm and $L^2$-norm under standard regularity assumptions. The resulting framework broadens applicability of WG methods beyond stabilizer-free schemes, enables flexible polynomial degrees, and provides a solid theoretical foundation for practical finite element computations on complex polyhedral grids.

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method with a built-in stabilizer for Poisson equations. By utilizing bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, marking a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches and is applicable in any dimension $d$, offering substantial advantages. The proposed method maintains a simple, symmetric, and positive definite structure. These benefits are evidenced by optimal order error estimates in both discrete $H^1$ and $L^2$ norms, highlighting the effectiveness and accuracy of our WG method for practical applications.

Auto-Stabilized Weak Galerkin Finite Element Methods on Polytopal Meshes without Convexity Constraints

TL;DR

This work develops an auto-stabilized weak Galerkin method for the Poisson equation on general polytopal meshes, including non-convex elements, and in arbitrary dimension. Leveraging bubble functions as a core analytical tool, the approach yields a simple, symmetric, and positive definite discretization with built-in stabilization, while preserving sparsity. The authors prove existence and uniqueness, derive an error equation, and establish optimal-order convergence in the discrete -norm and -norm under standard regularity assumptions. The resulting framework broadens applicability of WG methods beyond stabilizer-free schemes, enables flexible polynomial degrees, and provides a solid theoretical foundation for practical finite element computations on complex polyhedral grids.

Abstract

This paper introduces an auto-stabilized weak Galerkin (WG) finite element method with a built-in stabilizer for Poisson equations. By utilizing bubble functions as a key analytical tool, our method extends to both convex and non-convex elements in finite element partitions, marking a significant advancement over existing stabilizer-free WG methods. It overcomes the restrictive conditions of previous approaches and is applicable in any dimension , offering substantial advantages. The proposed method maintains a simple, symmetric, and positive definite structure. These benefits are evidenced by optimal order error estimates in both discrete and norms, highlighting the effectiveness and accuracy of our WG method for practical applications.
Paper Structure (7 sections, 10 theorems, 89 equations, 1 algorithm)

This paper contains 7 sections, 10 theorems, 89 equations, 1 algorithm.

Table of Contents

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

Key Result

Lemma 4.1

For $v=\{v_0, v_b\}\in V_h$, there exists a constant $C$ such that

Theorems & Definitions (23)

  • 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 13 more