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Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes

Abstract

We consider spectral mixed discontinuous Galerkin finite element discretizations of the Lamé system of linear elasticity in polyhedral domains in \({\mathbb {R}}^3\). In order to resolve possible corner, edge, and corner-edge singularities, anisotropic geometric edge meshes consisting of hexahedral elements are applied. We perform a computational study on the discrete inf-sup stability of these methods, and especially focus on the robustness with respect to the Poisson ratio close to the incompressible limit (i.e. the Stokes system). Furthermore, under certain realistic assumptions (for analytic data) on the regularity of the exact solution, we illustrate numerically that the proposed mixed DG schemes converge exponentially in a natural DG norm.

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Correspondence to Thomas P. Wihler.

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Thomas P. Wihler acknowledges the financial support of the Swiss National Science Foundation under Grant no. 200021–182524.

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Wihler, T.P., Wirz, M. Stability and Convergence of Spectral Mixed Discontinuous Galerkin Methods for 3D Linear Elasticity on Anisotropic Geometric Meshes. J Sci Comput 82, 49 (2020). https://doi.org/10.1007/s10915-020-01153-9

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Keywords

  • Linear elasticity in polyhedra
  • Anisotropic geometric meshes
  • Spectral methods
  • Discontinuous Galerkin methods
  • Inf-sup stability
  • Exponential convergence

Mathematics Subject Classification

  • 65N30