Finite Element Methods with Artificial Diffusion for Hamilton-Jacobi-Bellman Equations

TL;DR

This paper evaluates the numerical performance of artificial diffusion finite element methods for solving second-order fully nonlinear Hamilton-Jacobi-Bellman equations, focusing on degeneracy, local diffusion effects, and irregular geometries.

Contribution

It provides practical insights into the activation and impact of artificial diffusion in finite element methods for complex nonlinear PDEs, extending previous theoretical frameworks.

Findings

Artificial diffusion activates in degenerate regions.

Local diffusion parameters influence numerical dissipation.

Method effectively handles irregular geometries.

Abstract

In this short note we investigate the numerical performance of the method of artificial diffusion for second-order fully nonlinear Hamilton-Jacobi-Bellman equations. The method was proposed in (M. Jensen and I. Smears, arxiv:1111.5423); where a framework of finite element methods for Hamilton-Jacobi-Bellman equations was studied theoretically. The numerical examples in this note study how the artificial diffusion is activated in regions of degeneracy, the effect of a locally selected diffusion parameter on the observed numerical dissipation and the solution of second-order fully nonlinear equations on irregular geometries.