Mortaring for linear elasticity using mixed and stabilized finite elements

TL;DR

This paper investigates mortar methods for linear elasticity using low order finite elements, introducing a stabilized approach and comparing it with an unstabilized method, demonstrating stability and convergence in contact problems.

Contribution

It presents a stabilized mortar method for linear elasticity based on residual stabilization and compares it to the classical mixed mortar method, including stability analysis and numerical validation.

Findings

Stabilized mortar method is stable and convergent for contact problems.

Mixed $P_1-P_1$ approximation stability criteria derived.

Method successfully extended to three-dimensional problems.

Abstract

The purpose of this work is to study mortar methods for linear elasticity using standard low order finite element spaces. Based on residual stabilization, we introduce a stabilized mortar method for linear elasticity and compare it to the unstabilized mixed mortar method. For simplicity, both methods use a Lagrange multiplier defined on a trace mesh inherited from one side of the interface only. We derive a quasi-optimality estimate for the stabilized method and present the stability criteria of the mixed $P_1-P_1$ approximation. Our numerical results demonstrate the stability and the convergence of the methods for tie contact problems. Moreover, the results show that the mixed method can be successfully extended to three dimensional problems.