Galerkin least squares finite element method for the obstacle problem

TL;DR

This paper introduces a new multiplier-free finite element method for the obstacle problem, providing theoretical guarantees and demonstrating effective numerical performance with adaptive error control.

Contribution

It develops a consistent augmented Lagrangian-based Galerkin least squares method that eliminates the multiplier and offers optimal error estimates.

Findings

Proves existence and uniqueness of discrete solutions.

Establishes optimal a priori error estimates.

Demonstrates effectiveness through numerical examples.

Abstract

We construct a consistent multiplier free method for the finite element solution of the obstacle problem. The method is based on an augmented Lagrangian formulation in which we eliminate the multiplier by use of its definition in a discrete setting. We prove existence and uniqueness of discrete solutions and optimal order a priori error estimates for smooth exact solutions. Using a saturation assumption we also prove an a posteriori error estimate. Numerical examples show the performance of the method and of an adaptive algorithm for the control of the discretization error.