The proximal Galerkin method for non-symmetric variational inequalities

TL;DR

The paper introduces the proximal Galerkin method for non-symmetric variational inequalities, providing a new approach that ensures mesh-independence and constraint preservation, with proven error estimates and diverse applications.

Contribution

It presents the novel proximal Galerkin method and its variants, with theoretical error bounds and practical applications in complex boundary problems.

Findings

Optimal a priori error estimates established and verified numerically.

Method is asymptotically mesh-independent and constraint-preserving.

Successfully applied to American options, porous media, and advection-diffusion problems.

Abstract

We introduce the proximal Galerkin (PG) method for non-symmetric variational inequalities. The proposed approach is asymptotically mesh-independent and yields constraint-preserving approximations. We present both a conforming PG formulation and a hybrid mixed first-order system variant (FOSPG). We establish optimal a priori error estimates for each variant, which are verified numerically. We conclude by applying the method to American option pricing, free boundary problems in porous media, advection-diffusion with a semipermeable boundary, and the enforcement of discrete maximum principles.