A posteriori error estimates via equilibrated stress reconstructions for contact problems approximated by Nitsche's method

TL;DR

This paper develops an a posteriori error estimator for contact problems using equilibrated stress reconstructions, enabling adaptive algorithms and improved accuracy in finite element approximations.

Contribution

It introduces a guaranteed upper bound for the residual norm, refines error estimates, and proposes an algorithm with adaptive stopping and parameter tuning for Nitsche's method.

Findings

Estimator effectively guides adaptive refinement.

Algorithm improves solver efficiency and accuracy.

Numerical tests confirm estimator reliability.

Abstract

We present an a posteriori error estimate based on equilibrated stress reconstructions for the finite element approximation of a unilateral contact problem with weak enforcement of the contact conditions. We start by proving a guaranteed upper bound for the dual norm of the residual. This norm is shown to control the natural energy norm up to a boundary term, which can be removed under a saturation assumption. The basic estimate is then refined to distinguish the different components of the error, and is used as a starting point to design an algorithm including adaptive stopping criteria for the nonlinear solver and automatic tuning of a regularization parameter. We then discuss an actual way of computing the stress reconstruction based on the Arnold-Falk-Winther finite elements. Finally, after briefly discussing the efficiency of our estimators, we showcase their performance on a panel of numerical tests.