A stabilizer-free weak Galerkin mixed finite element method for the biharmonic equation

Abstract

In this paper, we present and study a stabilizer-free weak Galerkin (SFWG) finite element method for the Ciarlet-Raviart mixed form of the biharmonic equation on general polygonal meshes. We utilize the SFWG solutions of the second order elliptic problem to define projection operators and build error equations. Further, using weak functions formed by discontinuous k-th order polynomials, we derive the $O\left(h^k\right)$ convergence rate for the exact solution u in the $H^1$ norm and the $O\left(h^{k+1}\right)$ convergence rate in the $L^2$ norm. Finally, numerical examples support the results reached by the theory.