Convergence Analysis of Penalty Based Numerical Methods for Constrained Inequality Problems

TL;DR

This paper develops a comprehensive convergence theory for penalty-based numerical methods applied to various constrained inequality problems, demonstrating their solutions approach the true solution as discretization and penalty parameters diminish.

Contribution

It provides the first unified convergence analysis for penalty methods across elliptic variational, hemivariational, and constrained inequality problems.

Findings

Proves convergence of penalty methods as mesh size and penalty parameter tend to zero independently.

Establishes convergence for general elliptic variational-hemivariational inequalities.

Includes applications to contact mechanics problems.

Abstract

This paper presents a general convergence theory of penalty based numerical methods for elliptic constrained inequality problems, including variational inequalities, hemivariational inequalities, and variational-hemivariational inequalities. The constraint is relaxed by a penalty formulation and is re-stored as the penalty parameter tends to zero. The main theoretical result of the paper is the convergence of the penalty based numerical solutions to the solution of the constrained inequality problem as the mesh-size and the penalty parameter approach zero simultaneously but independently. The convergence of the penalty based numerical methods is first established for a general elliptic variational-hemivariational inequality with constraints, and then for hemivariational inequalities and variational inequalities as special cases. Applications to problems in contact mechanics are described.