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

TL;DR

This paper presents an auto-stabilized weak Galerkin finite element method for Stokes equations that works on non-convex polytopal meshes, offering a stable, symmetric, and positive definite approach with optimal error estimates.

Contribution

It introduces a novel auto-stabilized weak Galerkin method that handles non-convex meshes without traditional stabilizers, enhancing stability and accuracy.

Findings

Method is symmetric and positive definite.

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

Applicable to convex and 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 $H^1$ norm and the $L^2$ norm.