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

## TL;DR

This paper develops an explicit residual-based a posteriori error estimator for the symmetric mixed finite element method in linear elasticity, enabling reliable adaptive mesh refinement and improved convergence rates.

## Contribution

It introduces a new explicit residual-based a posteriori error estimator that is reliable, efficient, and easy to implement for Arnold-Winther FEM in linear elasticity.

## Key findings

- Estimator is reliable and efficient
- Adaptive refinement improves convergence rates
- Higher accuracy with adaptive mesh in non-smooth cases

## 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. Opposed to a previous publication, 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 obligatory in practical computations. 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 rates for piecewise affine data is observed in the L2 stress error and reproduced in non-smooth situations by the adaptive mesh-refining strategy.