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A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes

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Abstract

A modified weak Galerkin (MWG) finite element method is developed for solving the biharmonic equation. This method uses the same finite element space as that of the discontinuous Galerkin method, the space of discontinuous polynomials on polytopal meshes. But its formulation is simple, symmetric, positive definite, and parameter independent, without any of six inter-element face-integral terms in the formulation of the discontinuous Galerkin method. Optimal order error estimates in a discrete \(H^2\) norm are established for the corresponding finite element solutions. Error estimates in the \(L^2\) norm are also derived with a sub-optimal order of convergence for the lowest-order element and an optimal order of convergence for all high-order of elements. The numerical results are presented to confirm the theory of convergence.

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Authors and Affiliations

  1. College of Applied Science, Beijing University of Technology, Beijing, China

    Ming Cui

  2. Department of Mathematics, University of Arkansas at Little Rock, Little Rock, AR, 72204, USA

    Xiu Ye

  3. Department of Mathematical Sciences, University of Delaware, Newark, DE, 19716, USA

    Shangyou Zhang

Authors
  1. Ming Cui
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  2. Xiu Ye
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  3. Shangyou Zhang
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Corresponding author

Correspondence to Shangyou Zhang.

Additional information

M. Cui was supported in part by the National Natural Science Foundation of China (Grant No. 11571026) and the Beijing Municipal Natural Science Foundation of China (Grant No. 1192003). Xiu Ye was supported in part by the National Science Foundation Grant DMS-1620016.

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Cui, M., Ye, X. & Zhang, S. A Modified Weak Galerkin Finite Element Method for the Biharmonic Equation on Polytopal Meshes. Commun. Appl. Math. Comput. 3, 91–105 (2021). https://doi.org/10.1007/s42967-020-00071-9

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  • DOI: https://doi.org/10.1007/s42967-020-00071-9

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