Multigrid analysis for higher order finite difference scheme. (English) Zbl 1085.65116

Kwak, D. Y.; Lee, J. S.

J. Numer. Math. 12, No. 4, 285-296 (2004).

Authors’ 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.
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.

Reviewer: Snezhana Gocheva-Ilieva (Plovdiv)

MSC:

65N55	Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N06	Finite difference methods for boundary value problems involving PDEs
35J25	Boundary value problems for second-order elliptic equations
65N12	Stability and convergence of numerical methods for boundary value problems involving PDEs
65N50	Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs

Keywords:

nonuniform grid; V-cycle convergence; higher order finite difference method; multigrid method; algorithm; elliptic problems; numerical experiment; condition numbers