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Least-Squares Methods for Elasticity and Stokes Equations with Weakly Imposed Symmetry

  • Fleurianne Bertrand EMAIL logo , Zhiqiang Cai and Eun Young Park

Abstract

This paper develops and analyzes two least-squares methods for the numerical solution of linear elasticity and Stokes equations in both two and three dimensions. Both approaches use the L2 norm to define least-squares functionals. One is based on the stress-displacement/velocity-rotation/vorticity-pressure (SDRP/SVVP) formulation, and the other is based on the stress-displacement/velocity-rotation/vorticity (SDR/SVV) formulation. The introduction of the rotation/vorticity variable enables us to weakly enforce the symmetry of the stress. It is shown that the homogeneous least-squares functionals are elliptic and continuous in the norm of H(div;Ω) for the stress, of H1(Ω) for the displacement/velocity, and of L2(Ω) for the rotation/vorticity and the pressure. This immediately implies optimal error estimates in the energy norm for conforming finite element approximations. As well, it admits optimal multigrid solution methods if Raviart–Thomas finite element spaces are used to approximate the stress tensor. Through a refined duality argument, an optimal L2 norm error estimates for the displacement/velocity are also established. Finally, numerical results for a Cook’s membrane problem of planar elasticity are included in order to illustrate the robustness of our method in the incompressible limit.

MSC 2010: 65N30

Award Identifier / Grant number: DMS-1217081

Award Identifier / Grant number: DMS-1522707

Funding statement: This work was supported in part by the National Science Foundation under grants DMS-1217081 and DMS-1522707.

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Received: 2018-10-05
Revised: 2019-02-17
Accepted: 2019-03-21
Published Online: 2019-06-13
Published in Print: 2019-07-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

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