A priori error analysis of the proximal Galerkin method

TL;DR

This paper develops a general theoretical framework for analyzing the error of the proximal Galerkin method, a finite element approach for variational problems with inequality constraints, demonstrating optimal convergence in key applications.

Contribution

It provides the first abstract a priori error analysis for PG methods, establishing convergence and error estimates for obstacle and Signorini problems.

Findings

Optimal convergence rates achieved for obstacle problems

Framework applicable to various finite element subspaces

First abstract error analysis for PG methods

Abstract

The proximal Galerkin (PG) method is a finite element method for solving variational problems with inequality constraints. It has several advantages, including constraint-preserving approximations and mesh independence. This paper presents the first abstract a priori error analysis of PG methods, providing a general framework to establish convergence and error estimates. As applications of the framework, we demonstrate optimal convergence rates for both the obstacle and Signorini problems using various finite element subspaces.