First-Order Least-Squares Method for the Obstacle Problem

TL;DR

This paper introduces a least-squares finite element method for the obstacle problem, providing theoretical analysis, error estimates, and numerical validation for improved solution accuracy and adaptive algorithms.

Contribution

It presents a novel least-squares approach for the obstacle problem, including error analysis and adaptive strategies, with proven optimal convergence rates.

Findings

Optimal convergence rates for the lowest-order case.

A priori and a posteriori error estimates derived.

Numerical studies confirm theoretical results.

Abstract

We define and analyse a least-squares finite element method for a first-order reformulation of the obstacle problem. Moreover, we derive variational inequalities that are based on similar but non-symmetric bilinear forms. A priori error estimates including the case of non-conforming convex sets are given and optimal convergence rates are shown for the lowest-order case. We provide also a posteriori bounds that can be be used as error indicators in an adaptive algorithm. Numerical studies are presented.