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

TL;DR

This paper presents an auto-stabilized weak Galerkin finite element method that effectively handles Poisson equations on polytopal meshes without convexity constraints, extending applicability and maintaining optimal accuracy.

Contribution

It introduces a novel auto-stabilized WG method using bubble functions, applicable to non-convex elements and any dimension, overcoming previous limitations.

Findings

Achieves optimal error estimates in discrete H^1 and L^2 norms.

Applicable to convex and non-convex polytopal meshes in any dimension.

Maintains a simple, symmetric, and positive definite structure.

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.