Proximal Discontinuous Galerkin Methods for Variational Inequalities

TL;DR

This paper develops a family of proximal discontinuous Galerkin methods for variational inequalities, providing a unified analysis and establishing the first higher-order convergence result for such methods.

Contribution

It introduces a new family of proximal DG methods for variational inequalities, with a unified analysis and the first higher-order convergence proof.

Findings

Proved existence and uniqueness of solutions.

Established energy dissipation and error estimates.

Achieved the first higher-order convergence result for proximal Galerkin methods.

Abstract

We introduce a family of proximal discontinuous Galerkin methods for variational inequalities, focusing on the obstacle problem as a didactic example. Each member of this family is born from applying a different well-known nonconforming finite element discretization to the Bregman proximal point method. We explicitly treat four examples: the symmetric interior penalty discontinuous Galerkin, the enriched Galerkin, the hybridizable interior penalty and the hybrid high-order methods. We formulate a unified analysis framework for this family of methods and prove the existence and uniqueness of solutions, energy dissipation, and error estimates for both the primal and dual variables. Remarkably, the proximal hybrid high-order method with piecewise constant cell unknowns and piecewise affine facet unknowns leads to the first higher-order convergence result for any proximal Galerkin method.