Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes

Abstract

A stabilizer-free weak Galerkin (SFWG) finite element method was introduced and analyzed in Ye and Zhang (SIAM J. Numer. Anal. 58: 2572–2588, 2020) for the biharmonic equation, which has an ultra simple finite element formulation. This work is a continuation of our investigation of the SFWG method for the biharmonic equation. The new SFWG method is highly accurate with a convergence rate of four orders higher than the optimal order of convergence in both the energy norm and the $L^2$ norm on triangular grids. This new method also keeps the formulation that is symmetric, positive definite, and stabilizer-free. Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete $H^2$ norm. Superconvergence of four orders in the $L^2$ norm is also derived for $k\geqslant 3$, where k is the degree of the approximation polynomial. The postprocessing is proved to lift a $P_k$ SFWG solution to a $P_{k+4}$ solution elementwise which converges at the optimal order. Numerical examples are tested to verify the theories.