Combined Regularization and Discretization of Equilibrium Problems and Primal-Dual Gap Estimators

TL;DR

This paper develops a finite element approach combining regularization and discretization for equilibrium problems with moving constraints, providing convergence results and adaptive error estimation techniques validated through numerical experiments.

Contribution

It introduces a novel combined regularization and discretization framework with convergence analysis and adaptive error estimators for equilibrium problems.

Findings

Convergence of regularized and discretized solutions to the original problem.

Effective error estimators for adaptive mesh refinement.

Numerical validation on obstacle and quasi-variational inequality problems.

Abstract

The present work aims at the application of finite element discretizations to a class of equilibrium problems involving moving constraints. Therefore, a Moreau--Yosida based regularization technique, controlled by a parameter, is discussed and, using a generalized $\Gamma$-convergence concept, a priori convergence results are derived. The latter technique is applied to the discretization of the regularized problems and is used to prove the convergence to the orginal equilibrium problem, when both -- regularization and discretization -- are imposed simultaneously. In addition, a primal-dual gap technique is used for the derivation of error estimators suitable for adaptive mesh refinement. A strategy for balancing between a refinement of the mesh and an update of the regularization parameter is established, too. The theoretical findings are illustrated for the obstacle problem as well as numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.