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

TL;DR

This work develops a stabilized, symmetric, and positive-definite weak Galerkin method for the biharmonic equation $\Delta^2 u=f$ on domains that may include non-convex polytopal meshes, incorporating Dirichlet/Neumann boundary data. The key idea is to eliminate stabilization terms by using higher-degree polynomials to compute the weak second-order derivatives and to employ bubble functions, enabling accurate discretization on general polytopal elements in arbitrary dimensions. The authors prove existence and uniqueness, derive an error equation, and establish optimal-order convergence in the discrete $H^2$-norm and in the $L^2$-norm, with rates $|||u-u_h||| \lesssim h^{k-1}\|u\|_{k+1}$ and, under regularity, $\|u-u_h\| \lesssim h^{k+1}\|u\|_{k+1}$. The approach preserves sparsity and extends stabilizer-free WG methods to non-convex meshes, offering a robust and practically efficient framework for high-order biharmonic problems.

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.

Theorems & Definitions (21)

• Lemma 4.1
• proof
• Remark 4.1
• Lemma 4.2
• Lemma 4.3
• proof
• Lemma 4.4
• proof
• Theorem 4.5
• proof