Augmented Lagrangian finite element methods for contact problems

¹ Department of Mathematics, University College London, Gower Street, UK–WC1E 6BT London, UK
² Department of Mechanical Engineering, Jönköping University, 55111 Jönköping, Sweden
³ Department of Mathematics and Mathematical Statistics, Umeå University, 901 87 Umeå, Sweden

^* Corresponding author: e.burman@ucl.ac.uk

Received: 21 July 2017
Accepted: 18 July 2018

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.