Virtual Element Methods for HJB Equations with Cordes Coefficients

TL;DR

This paper introduces and analyzes virtual element methods for solving fully nonlinear HJB equations with Cordes coefficients, providing stability, error estimates, and numerical validation.

Contribution

It develops conforming and nonconforming VEMs for HJB equations with Cordes coefficients, including stabilization and error analysis, which was not previously established.

Findings

Optimal error estimates in discrete $H^2$ norm.

Stable methods without discrete Miranda-Talenti estimate.

Numerical experiments confirm theoretical results.

Abstract

In this paper, we propose and analyze both conforming and nonconforming virtual element methods (VEMs) for the fully nonlinear second-order elliptic Hamilton-Jacobi-Bellman (HJB) equations with Cordes coefficients. By incorporating stabilization terms, we establish the well-posedness of the proposed methods, thus avoiding the need to construct a discrete Miranda-Talenti estimate. We derive the optimal error estimate in the discrete $H^2$ norm for both numerical formulations. Furthermore, a semismooth Newton's method is employed to linearize the discrete problems. Several numerical experiments using the lowest-order VEMs are provided to demonstrate the efficacy of the proposed methods and to validate our theoretical results.