Numerical analysis of Krylov multigrid methods for stationary advection-diffusion equation

Abstract

This paper outlines the problem of applying multigrid methods for the stationary scalar advection-diffusion equation for the given advection filed. We assume that all functions are smooth enough to be represented by discretization methods. Two main discretization are considered: finite difference method for Dirichlet boundary conditions and pseudo-spectral method for the periodic boundary. We test the following smoothers for multigrid methods: Jacobi smoother, Gauss-Seidel smoother, Krylov subspace smoother (GMRES method with and without preconditioners). The analysis is performed in the space formed by the cross product of discretization parameters, diffusion coefficient values, multigrid levels and smoothers.We demonstrate that the most efficient strategy depends on parameter value and given velocity field. Best variants include Gauss-Seidel smoothers which is optimal for advection-dominated problem while multigrid method is used as a preconditioner for a Krylov method. Such methods can be used for spectral or pseudo-spectral methods where explicit dense matrix storage is impossible.