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In the past decade, there are many works on the finite element methods forthe fully nonlinear Hamilton--Jacobi--Bellman (HJB) equations with Cordescondition. The linearised systems have large condition numbers, which dependnot 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 spacepreconditioners for the linearised systems of $C^0$ finite elementdiscretization of HJB equations [Calcolo, 58, 2021]. Based on the stabledecomposition on the auxiliary spaces, we propose both the additive andmultiplicative preconditoners which converge uniformly in the sense that theresulting condition number is independent of both the number of degrees offreedom and the parameter $\lambda$ in Cordes condition. Numerical experimentsare carried out to illustrate the efficiency of the proposed preconditioners.

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