Nitsche's method for unilateral contact problems

TL;DR

This paper develops optimal error estimates for Nitsche's method applied to unilateral contact problems, providing theoretical analysis and numerical validation for its effectiveness in approximating Signorini problems.

Contribution

It offers a new interpretation of Nitsche's method as a stabilized finite element approach for the mixed Lagrange multiplier formulation, with comprehensive error analysis.

Findings

Optimal a priori error estimates derived

Reliable a posteriori error estimators developed

Numerical results confirm estimator efficiency

Abstract

We derive optimal a priori and a posteriori error estimates for Nitsche's method applied to unilateral contact problems. Our analysis is based on the interpretation of Nitsche's method as a stabilised finite element method for the mixed Lagrange multiplier formulation of the contact problem wherein the Lagrange multiplier has been eliminated elementwise. To simplify the presentation, we focus on the scalar Signorini problem and outline only the proofs of the main results since most of the auxiliary results can be traced to our previous works on the numerical approximation of variational inequalities. We end the paper by presenting results of our numerical computations which corroborate the efficiency and reliability of the a posteriori estimators.