Augmented Lagrangian finite element methods for contact problems

TL;DR

This paper introduces two augmented Lagrangian finite element methods for contact problems, addressing obstacle and Signorini cases, with stability analysis and optimal convergence guarantees.

Contribution

It presents novel Lagrange multiplier methods for contact problems, including stability analysis and convergence proofs for both continuous and discontinuous approximations.

Findings

Methods are proven to have existence and uniqueness of solutions.

Optimal approximation and convergence estimates are established.

Discontinuous approximation requires a penalty for stability.

Abstract

We propose two different Lagrange multiplier methods for contact problems derived from the augmented Lagrangian variational formulation. Both the obstacle problem, where a constraint on the solution is imposed in the bulk domain and the Signorini problem, where a lateral contact condition is imposed are considered. We consider both continuous and discontinuous approximation spaces for the Lagrange multiplier. In the latter case the method is unstable and a penalty on the jump of the multiplier must be applied for stability. We prove the existence and uniqueness of discrete solutions, best approximation estimates and convergence estimates that are optimal compared to the regularity of the solution.