Multigrid analysis for higher order finite difference scheme

Abstract

We introduce and analyze a multigrid algorithm for higher order finite difference schemes for elliptic problems on a nonuniform rectangular mesh. These schemes are presented by 9-point stencils. We prove the V-cycle convergence adopting the theory developed for finite element methods to these schemes. To be more precise, we show that the energy norm of the prolongation operator is less than one and hence obtain the conclusion using the approximation and regularity property as in [2]. In the numerical experiment section, we report contraction numbers, eigenvalues and condition numbers of the multigrid algorithm. The numerical test shows that for higher order schemes the multigrid algorithm converges much faster than for low order schemes. We also test the case of a nonuniform grid with a line smoother which also shows good convergence behavior.