Inhomogeneous Dirichlet Boundary Condition in the A Posteriori Error Control of the Obstacle Problem

TL;DR

This paper introduces a simplified residual-based a posteriori error estimator for the elliptic obstacle problem, effectively addressing inhomogeneous Dirichlet boundary conditions and providing more reliable error bounds.

Contribution

It presents a new approach to handle inhomogeneous boundary conditions in a posteriori error estimation, with simplified bounds and post-processing methods for finite element solutions.

Findings

The proposed estimator effectively accounts for inhomogeneous Dirichlet conditions.

New post-processing methods improve boundary condition satisfaction.

Results demonstrate enhanced reliability over existing estimators.

Abstract

We propose a new and simpler residual based a posteriori error estimator for finite element approximation of the elliptic obstacle problem. The results in the article are two fold. Firstly, we address the influence of the inhomogeneous Dirichlet boundary condition in {\em a posteriori} error control of the elliptic obstacle problem. Secondly by rewriting the obstacle problem in an equivalent form, we derive simpler {\em a posteriori} error bounds which are free from min/max functions. To accomplish this, we construct a post-processed solution $\tilde u_h$ of the discrete solution $u_h$ which satisfies the exact boundary conditions although the discrete solution $u_h$ may not satisfy. We propose two post processing methods and analyze them. We remark that the results known in the literature are either for the homogeneous Dirichlet boundary condition or that the estimator is only weakly reliable in the case of inhomogeneous Dirichlet boundary condition.