Weakly symmetric stress equilibration and a posteriori error estimation for linear elasticity. Issue 4 (13th January 2021)

• Record Type: Journal Article
• Authors: Bertrand, Fleurianne; Kober, Bernhard; Moldenhauer, Marcel; Starke, Gerhard
• Is Part Of: Numerical methods for partial differential equations. Volume 37:Issue 4(2021)
• Page Start: 2783
• Page End: 2802
• Publication Date: 2021-01-13
• Subjects: a posteriori error estimation -- incompressible linear elasticity -- Raviart‐Thomas elements -- Taylor‐Hood elements -- weakly symmetric stress equilibration; Differential equations, Partial -- Numerical solutions -- Periodicals; 515.353
• DOI: 10.1002/num.22741
• Languages: English
• ISSNs: 0749-159X

Abstract

Abstract: This paper proposes and analyzes a posteriori error estimator based on stress equilibration for linear elasticity with emphasis on the behavior for (nearly) incompressible materials. It is based on an H (div)‐conforming, weakly symmetric stress reconstruction from the displacement‐pressure approximation computed with a stable finite element pair. Our focus is on the Taylor‐Hood combination of continuous finite element spaces of polynomial degrees k + 1 and k for the displacement and the pressure, respectively. This weak symmetry allows us to prove that the resulting error estimator constitutes a guaranteed upper bound for the error with a constant that depends only on local constants associated with the patches and thus on the shape regularity of the triangulation. It does not involve global constants like those from Korn's in equality which may become very large depending on the location and type of the boundary conditions. Local efficiency, also uniformly in the incompressible limit, is deduced from the upper bound by the residual error estimator. Numerical results for the popular Cook's membrane test problem confirm the theoretical predictions.