Error estimates for perturbed variational inequalities of the first kind

TL;DR

This paper develops a unified approach to estimate errors in variational inequalities of the first kind, especially when perturbed by numerical approximations, with applications to finite element methods for obstacle problems.

Contribution

It introduces a general abstract framework combining Strang Lemma and Falk Theorem to derive a priori error estimates under various perturbations, including inexact quadrature.

Findings

Derived guaranteed error rates for finite element approximations of obstacle problems.

Analyzed the impact of quadrature point number on approximation and quadrature errors.

Validated theoretical estimates through numerical experiments.

Abstract

In this paper, we derive a priori error estimates for variational inequalities of the first kind in an abstract framework. This is done by combining the first Strang Lemma and the Falk Theorem. The main application consists in the derivation of a priori error estimates for Galerkin methods, in which "variational crimes" may perturb the underlying variational inequality. Different types of perturbations are incorporated into the abstract framework and discussed by various examples. For instance, the perturbation caused by an inexact quadrature is examined in detail for the Laplacian obstacle problem. For this problem, guaranteed rates for the approximation error resulting from the use of higher-order finite elements are derived. In numerical experiments, the influence of the number of quadrature points on the approximation error and on the quadrature-related error itself is studied for several discretization methods.