A Weak Galerkin Mixed Finite Element Method for Biharmonic Equations

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 (Wang et al., Comput. Appl. Math. 241:103–115, 2013) for second-order elliptic problems, is based on the concept of discrete weak gradients. The method uses completely discrete finite element functions, and, using certain discrete spaces and with stabilization, it works on partitions of arbitrary polygon or polyhedron. In this article, the 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 Babus̆ka, Osborn, and Pitkäranta (Math. Comp. 35:1039–1062, 1980) 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.