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

doi:10.1007/s42967-022-00201-5

Abstract

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 L² norm on triangular grids. This new method also keeps the formulation that is symmetric, positive definite, and stabilizerfree. Four-order superconvergence error estimates are proved for the corresponding SFWG finite element solutions in a discrete H² norm. Superconvergence of four orders in the L² 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.

Key words: Finite element, Weak Hessian, Weak Galerkin(WG), Biharmonic equation, Triangular mesh

CLC Number:

Xiu Ye, Shangyou Zhang. Four-Order Superconvergent Weak Galerkin Methods for the Biharmonic Equation on Triangular Meshes[J]. Communications on Applied Mathematics and Computation, 2023, 5(4): 1323-1338.

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References

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