Simplified Weak Galerkin Finite Element Methods for Biharmonic Equations on Non-Convex Polytopal Meshes

TL;DR

This paper introduces a simplified, stabilizer-free weak Galerkin finite element method for biharmonic equations that works on non-convex polytopal meshes, providing optimal error estimates and maintaining symmetry and positive definiteness.

Contribution

It develops a novel simplified weak Galerkin method that avoids stabilizers and supports non-convex meshes, with proven optimal error bounds.

Findings

Method is symmetric and positive definite.

Achieves optimal-order error estimates in discrete $H^2$ and $L^2$ norms.

Supports both convex and non-convex polytopal meshes.

Abstract

This paper presents a simplified weak Galerkin (WG) finite element method for solving biharmonic equations avoiding the use of traditional stabilizers. The proposed WG method supports both convex and non-convex polytopal elements in finite element partitions, utilizing bubble functions as a critical analytical tool. The simplified WG method is symmetric and positive definite. Optimal-order error estimates are established for WG approximations in both the discrete $H^2$ norm and the $L^2$ norm.