Auxiliary Space Preconditioners for $C^{0}$ Finite Element Approximation of Hamilton--Jacobi--Bellman Equations with Cordes Coefficients

TL;DR

This paper develops auxiliary space preconditioners for linear systems from finite element discretizations of Hamilton--Jacobi--Bellman equations with Cordes coefficients, achieving uniform convergence independent of mesh size and parameters.

Contribution

It introduces novel additive and multiplicative preconditioners based on stable auxiliary space decompositions for HJB equations with Cordes conditions.

Findings

Preconditioners achieve uniform convergence regardless of degrees of freedom.

Condition number is independent of the parameter in Cordes condition.

Numerical experiments confirm the efficiency of the proposed methods.

Abstract

In the past decade, there are many works on the finite element methods for the fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordes condition. The linearised systems have large condition numbers, which depend not only on the mesh size, but also on the parameters in the Cordes condition. This paper is concerned with the design and analysis of auxiliary space preconditioners for the linearised systems of $C^0$ finite element discretization of HJB equations [Calcolo, 58, 2021]. Based on the stable decomposition on the auxiliary spaces, we propose both the additive and multiplicative preconditoners which converge uniformly in the sense that the resulting condition number is independent of both the number of degrees of freedom and the parameter $\lambda$ in Cordes condition. Numerical experiments are carried out to illustrate the efficiency of the proposed preconditioners.