In this paper, we study numerical solutions for the Hamilton-Jacobi-Bellman (HJB) and Kolmogorov–Fokker–Planck (KFP) equations arising from mean field games. In order to solve the nonlinear discretized systems efficiently, we propose a multigrid method. Our proposed multigrid method is developed on the joint spacetime and is a full approximation scheme (FAS). We consider hybrid full-semi coarsening and kernel preserving biased restriction to address the anisotropy in time and convections in space. The main novelty of this paper is that we propose adding artificial viscosity to the direct discretization coarse grid operators, such that the coarse grid error estimations are more accurate. We use Fourier analysis to illustrate the efficiency of our proposed multigrid method. Numerical experiments show that the convergence rate of the proposed multigrid method is mesh-independent and faster than the existing methods in the literature.

在本文中，我们研究了均值场博弈产生的汉密尔顿-雅各比-贝尔曼（HJB）和Kolmogorov-Fokker-Planck（KFP）方程的数值解。为了有效地解决非线性离散系统，我们提出了一种多网格方法。我们提出的多重网格方法是在联合时空上开发的，是一种全近似方案（FAS）。我们考虑混合半完全粗化和保留核的偏向约束，以解决时间各向异性和空间对流问题。本文的主要新颖之处在于，我们建议将人工粘度添加到直接离散化粗网格算子中，以使粗网格误差估计更加准确。我们使用傅立叶分析来说明我们提出的多网格方法的效率。