Residual-based a posteriori error analysis for symmetric mixed Arnold–Winther FEM

Abstract

This paper introduces an explicit residual-based a posteriori error analysis for the symmetric mixed finite element method in linear elasticity after Arnold–Winther with pointwise symmetric and H(div) -conforming stress approximation. The residual-based a posteriori error estimator of this paper is reliable and efficient and truly explicit in that it solely depends on the symmetric stress and does neither need any additional information of some skew symmetric part of the gradient nor any efficient approximation thereof. Hence, it is straightforward to implement an adaptive mesh-refining algorithm. Numerical experiments verify the proven reliability and efficiency of the new a posteriori error estimator and illustrate the improved convergence rate in comparison to uniform mesh-refining. A higher convergence rate for piecewise affine data is observed in the L 2 stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.