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.

翻译：过去十年来,对完全非线性汉密尔顿-Jacobi-Bellman(HJB)方程式的限定要素方法进行了许多研究,线性系统的条件数量很大,不仅取决于网状尺寸,而且取决于Cordes条件的参数,本文涉及设计和分析HJB方程式的线性离散(Calcolo,58, 20211美元)的有限要素的辅助空间先决条件。根据辅助空间的稳定分解,我们提议了添加和多倍化前孔器,它们一致地组合在一起,因为由此产生的条件数量与Cordes条件下的自由度数量和参数$\lambda$独立。进行了数值实验,以说明拟议先决条件的效率。