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

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 L2norm 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 H2norm. Superconvergence of four orders in the L2norm is also derived for k⩾3, where kis the degree of the approximation polynomial. The postprocessing is proved to lift a PkSFWG solution to a Pk+4solution elementwise which converges at the optimal order. Numerical examples are tested to verify the theories.