A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

TL;DR

This paper presents a novel weak Galerkin mixed finite element method for biharmonic equations, enabling flexible discretization on arbitrary polygonal meshes with proven error estimates and computational validation.

Contribution

It extends the weak Galerkin method to biharmonic problems using a mixed formulation, providing error analysis and demonstrating computational efficiency.

Findings

Error estimates are established for the finite element approximations.

The method performs efficiently on arbitrary polygonal meshes.

Computational results confirm the theoretical analysis.

Abstract

This article introduces and analyzes a weak Galerkin mixed finite element method for solving the biharmonic equation. The weak Galerkin method, first introduced by two of the authors (J. Wang and X. Ye) in an earlier publication for second order elliptic problems, is based on the concept of discrete weak gradients. The method allows the use of completely discrete finite element functions on partitions of arbitrary polygon or polyhedron. In this article, the idea of weak Galerkin method is applied to discretize the Ciarlet-Raviart mixed formulation for the biharmonic equation. In particular, an a priori error estimation is given for the corresponding finite element approximations. The error analysis essentially follows the framework of Babuska, Osborn, and Pitkaranta and uses specially designed mesh-dependent norms. The proof is technically tedious due to the discontinuous nature of the weak Galerkin finite element functions. Some computational results are presented to demonstrate the efficiency of the method.