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Penalty-Free Any-Order Weak Galerkin FEMs for Linear Elasticity on Quadrilateral Meshes

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Abstract

This paper develops a family of new weak Galerkin (WG) finite element methods (FEMs) for solving linear elasticity in the primal formulation. For a convex quadrilateral mesh, degree \( k \ge 0 \) vector-valued polynomials are used independently in element interiors and on edges for approximating the displacement. No penalty or stabilizer is needed for these new methods. The methods are free of Poisson-locking and have optimal order \( (k+1) \) convergence rates in displacement, stress, and dilation (divergence of displacement). Numerical experiments on popular test cases are presented to illustrate the theoretical estimates and demonstrate efficiency of these new solvers. Extension to cuboidal hexahedral meshes is briefly discussed.

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Data Availability

The code used in this work will be made available upon request to the authors.

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Funding

J. Liu was supported in part by US National Science Foundation under grant DMS-2208590. R. Wang was partially supported by the National Natural Science Foundation of China (grant No. 12001230, 11971198), the National Key Research and Development Program of China (grant No. 2020YFA0713602), and the Key Laboratory of Symbolic Computation and Knowledge Engineering of Ministry of Education of China housed at Jilin University. Z. Wang was partially supported by the National Natural Science Foundation of China (grant No.12101626).

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Authors and Affiliations

  1. School of Mathematics, Jilin University, Changchun, 130012, Jilin, China

    Ruishu Wang

  2. School of Mathematics (Zhuhai), Sun Yat-sen University, Zhuhai, 519082, Guangdong, China

    Zhuoran Wang

  3. Department of Mathematics, Colorado State University, Fort Collins, CO, 80523-1874, USA

    Jiangguo Liu

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  1. Ruishu Wang
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  2. Zhuoran Wang
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Correspondence to Zhuoran Wang.

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Wang, R., Wang, Z. & Liu, J. Penalty-Free Any-Order Weak Galerkin FEMs for Linear Elasticity on Quadrilateral Meshes. J Sci Comput 95, 20 (2023). https://doi.org/10.1007/s10915-023-02151-3

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  • DOI: https://doi.org/10.1007/s10915-023-02151-3

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