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

doi:10.1007/s42967-022-00201-5

Communications on Applied Mathematics and Computation ›› 2023, Vol. 5 ›› Issue (4): 1323-1338.doi: 10.1007/s42967-022-00201-5

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

Xiu Ye¹, Shangyou Zhang²

1 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

收稿日期:2022-02-16 修回日期:2022-05-09 发布日期:2023-12-16
通讯作者: Shangyou Zhang,E-mail:szhang@udel.edu;Xiu Ye,E-mail:xxye@ualr.edu E-mail:szhang@udel.edu;xxye@ualr.edu

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

Xiu Ye¹, Shangyou Zhang²

1 Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR 72204, USA;
2 Department of Mathematical Sciences, University of Delaware, Newark, DE 19716, USA

Received:2022-02-16 Revised:2022-05-09 Published:2023-12-16
Contact: Shangyou Zhang,E-mail:szhang@udel.edu;Xiu Ye,E-mail:xxye@ualr.edu E-mail:szhang@udel.edu;xxye@ualr.edu

摘要/Abstract

摘要： 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.

关键词: Finite element, Weak Hessian, Weak Galerkin(WG), Biharmonic equation, Triangular mesh

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

中图分类号:

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.

参考文献

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

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

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