A priori error analysis of the proximal Galerkin method
Brendan Keith, Rami Masri, Marius Zeinhofer
TL;DR
We present the first general a priori error analysis of the proximal Galerkin (PG) method for quadratic variational problems with pointwise inequality constraints. The framework combines latent proximal point reformulations with Legendre/Bregman regularization to yield a discretized saddle-point system, energy dissipation, and mesh-independent iteration complexity. We prove existence/uniqueness of discrete subproblems, derive best-approximation error estimates, and establish convergence rates that are mesh-independent, then instantiate the theory for obstacle and Signorini problems to achieve optimal rates across several finite element pairs. The results justify the practical robustness of PG, enable analysis under low regularity, and point toward extensions to higher-order accuracy and broader classes of inequality-constrained problems.
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.