A C^0-Weak Galerkin Finite Element Method for the Biharmonic Equation

TL;DR

This paper introduces a symmetric, positive definite C^0-weak Galerkin finite element method for solving the biharmonic equation in 2D and 3D, with optimal error estimates and confirmed by numerical results.

Contribution

It develops a novel C^0-weak Galerkin formulation for the biharmonic equation, including a refined interpolation operator for improved error analysis.

Findings

Optimal error estimates in discrete H^2 and L^2 norms.

Numerical results confirm theoretical convergence rates.

The method is symmetric, positive definite, and parameter free.

Abstract

A C^0-weak Galerkin (WG) method is introduced and analyzed for solving the biharmonic equation in 2D and 3D. A weak Laplacian is defined for C^0 functions in the new weak formulation. This WG finite element formulation is symmetric, positive definite and parameter free. Optimal order error estimates are established in both a discrete H^2 norm and the L^2 norm, for the weak Galerkin finite element solution. Numerical results are presented to confirm the theory. As a technical tool, a refined Scott-Zhang interpolation operator is constructed to assist the corresponding error estimate. This refined interpolation preserves the volume mass of order (k+1-d) and the surface mass of order (k+2-d) for the P_{k+2} finite element functions in d-dimensional space.