An Efficient Numerical Scheme for the Biharmonic Equation by Weak Galerkin Finite Element Methods on Polygonal or Polyhedral Meshes

TL;DR

This paper introduces a new weak Galerkin finite element method for solving the biharmonic equation on polygonal or polyhedral meshes, providing an efficient, symmetric, and parameter-free scheme with optimal error estimates.

Contribution

The paper develops a novel weak Galerkin finite element scheme for the biharmonic equation using discrete weak derivatives on general meshes, with proven optimal convergence and efficiency.

Findings

Matrix is symmetric and positive definite.

Achieves optimal order error estimates in $H^2$-equivalent norm.

Numerical experiments confirm efficiency and accuracy.

Abstract

This paper presents a new and efficient numerical algorithm for the biharmonic equation by using weak Galerkin (WG) finite element methods. The WG finite element scheme is based on a variational form of the biharmonic equation that is equivalent to the usual $H^2$-semi norm. Weak partial derivatives and their approximations, called discrete weak partial derivatives, are introduced for a class of discontinuous functions defined on a finite element partition of the domain consisting of general polygons or polyhedra. The discrete weak partial derivatives serve as building blocks for the WG finite element method. The resulting matrix from the WG method is symmetric, positive definite, and parameter free. An error estimate of optimal order is derived in an $H^2$-equivalent norm for the WG finite element solutions. Error estimates in the usual $L^2$ norm are established, yielding optimal order of convergence for all the WG finite element algorithms except the one corresponding to the lowest order (i.e., piecewise quadratic elements). Some numerical experiments are presented to illustrate the efficiency and accuracy of the numerical scheme.