Penalty-Free Any-Order Weak Galerkin FEMs for Elliptic Problems on Quadrilateral Meshes

Abstract

This paper presents a family of weak Galerkin finite element methods for elliptic boundary value problems on convex quadrilateral meshes. These new methods use degree \(k \ge 0\) polynomials separately in element interiors and on edges for approximating the primal variable. The discrete weak gradients of these shape functions are established in the local Arbogast–Correa \(AC_k \) spaces. These discrete weak gradients are then used to approximate the classical gradient in the variational formulation. These new methods do not use any nonphysical penalty factor but produce optimal-order approximation to the primal variable, flux, normal flux, and divergence of flux. Moreover, these new solvers are locally conservative and offer continuous normal fluxes. Numerical experiments are presented to demonstrate the accuracy of this family of new methods.

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Acknowledgements

J. Liu and Z. Wang were supported in part by US National Science Foundation under Grant DMS-1819252. S. Tavener was supported in part by US National Science Foundation under Grant DMS-1720473. We sincerely thank the anonymous reviewers for their constructive comments, which have helped improve the quality of this paper.

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Correspondence to Jiangguo Liu.

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J. Liu and Z. Wang were partially supported by US NSF under Grant DMS-1819252. S. Tavener was partially supported by US NSF under Grant DMS-1720473.

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Liu, J., Tavener, S. & Wang, Z. Penalty-Free Any-Order Weak Galerkin FEMs for Elliptic Problems on Quadrilateral Meshes. J Sci Comput 83, 47 (2020). https://doi.org/10.1007/s10915-020-01239-4

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Keywords

  • Arbogast–Correa spaces
  • Elliptic boundary value problems
  • Penalty-free
  • Quadrilateral meshes
  • Weak Galerkin

Mathematics Subject Classification

  • 65N15
  • 65N30
  • 76S06