Multigrid methods for convergent mixed finite difference scheme for Monge–Ampère equation

Abstract

We propose multigrid methods for convergent mixed finite difference discretization for the two dimensional Monge–Ampere equation. We apply mixed standard 7-point stencil and semi-Lagrangian wide stencil discretization, such that the numerical solution is guaranteed to converge to the viscosity solution of the Monge–Ampere equation. We investigate multigrid methods for two scenarios. The first scenario considers applying standard 7-point stencil discretization on the entire computational domain. We use full approximation scheme with four-directional alternating line smoothers. The second scenario considers the more general mixed stencil discretization and is used for the linearized problem. We propose a coarsening strategy where wide stencil points are set as coarse grid points. Linear interpolation is applied on the entire computational domain. At wide stencil points, injection as the restriction yields a good coarse grid correction. Numerical experiments show that the convergence rates of the proposed multigrid methods are mesh-independent.