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

• Published in: Communications on Applied Mathematics and Computation (Online) 2023-12, Vol.5 (4), p.1323-1338
• Main Authors: Ye, Xiu, Zhang, Shangyou
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Computational Science and Engineering; Mathematics; Mathematics and Statistics; Original Paper
• ISSN: 2096-6385; 2661-8893
• DOI: 10.1007/s42967-022-00201-5

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 ⩾ 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.