On the Convergence of Finite Element Methods for Hamilton-Jacobi-Bellman Equations

TL;DR

This paper proves the strong uniform convergence of monotone P1 finite element methods on unstructured meshes for fully non-linear Hamilton-Jacobi-Bellman equations, including convergence of gradients under certain conditions.

Contribution

It extends convergence results to discretizations that violate traditional consistency conditions using elliptic projection operators.

Findings

Strong uniform convergence of numerical solutions

L2 convergence of gradients under non-degeneracy

Applicability to degenerate, isotropic diffusions

Abstract

In this note we study the convergence of monotone P1 finite element methods on unstructured meshes for fully non-linear Hamilton-Jacobi-Bellman equations arising from stochastic optimal control problems with possibly degenerate, isotropic diffusions. Using elliptic projection operators we treat discretisations which violate the consistency conditions of the framework by Barles and Souganidis. We obtain strong uniform convergence of the numerical solutions and, under non-degeneracy assumptions, strong L2 convergence of the gradients.