An approximation scheme for variational inequalities with convex and coercive Hamiltonians

TL;DR

This paper introduces a new approximation scheme for a class of semilinear variational inequalities with convex, coercive Hamiltonians, extending previous methods and establishing convergence with explicit error bounds.

Contribution

The paper develops a novel approximation scheme for variational inequalities with convex, coercive Hamiltonians, including convergence proof and error rate analysis using advanced techniques.

Findings

The scheme converges to the true solution.

Error bounds are explicitly derived.

The method extends previous splitting schemes.

Abstract

We propose an approximation scheme for a class of semilinear variational inequalities whose Hamiltonian is convex and coercive. The proposed scheme is a natural extension of a previous splitting scheme proposed by Liang, Zariphopoulou and the author for semilinear parabolic PDEs. We establish the convergence of the scheme and determine the convergence rate by obtaining its error bounds. The bounds are obtained by Krylov's shaking coefficients technique and Barles-Jakobsen's optimal switching approximation, in which a key step is to introduce a variant switching system.