Discontinuous Galerkin finite element methods for time-dependent Hamilton--Jacobi--Bellman equations with Cordes coefficients

TL;DR

This paper introduces a fully-discrete discontinuous Galerkin method for solving parabolic Hamilton--Jacobi--Bellman equations with Cordes coefficients, achieving high accuracy, stability, and optimal convergence rates on complex meshes.

Contribution

It develops a novel unconditionally stable DG time-stepping scheme for HJB equations with Cordes coefficients, providing error bounds and demonstrating high-order convergence.

Findings

Method is unconditionally stable on unstructured meshes.

Achieves arbitrarily high-order accuracy for smooth solutions.

Numerical experiments show exponential convergence with $hp$- and $ au q$-refinement.

Abstract

We propose and analyse a fully-discrete discontinuous Galerkin time-stepping method for parabolic Hamilton--Jacobi--Bellman equations with Cordes coefficients. The method is consistent and unconditionally stable on rather general unstructured meshes and time-partitions. Error bounds are obtained for both rough and regular solutions, and it is shown that for sufficiently smooth solutions, the method is arbitrarily high-order with optimal convergence rates with respect to the mesh size, time-interval length and temporal polynomial degree, and possibly suboptimal by an order and a half in the spatial polynomial degree. Numerical experiments on problems with strongly anisotropic diffusion coefficients and early-time singularities demonstrate the accuracy and computational efficiency of the method, with exponential convergence rates under combined $hp$- and $\tau q$-refinement.