The convergence rate of a multigrid method with Gauss-Seidel relaxation for the Poisson equation

Abstract

Abstract. The convergence rate of a multigrid method for the numerical solution of the Poisson equation on a uniform grid is estimated. The results are independent of the shape of the domain as long as it is convex and polygonal. On the other hand, pollution effects become apparent when the domain contains reentrant corners. To estimate the smoothing of the Gauss-Seidel relaxation, the smoothness is measured by comparing the energy norm with a (weaker) discrete seminorm.

When does two‐grid optimality carry over to the V‐cycle?

“…Another possibility consists in checking the so-called smoothing and approximation properties [19][20][21][22][23][24][25]. Regarding the latter approach, the best results for SPD matrices have been obtained by Hackbusch [22,Theorem 7.2.2] and McCormick [24].…”

Multigrid with Matrix-dependent Transfer Operators for Convection-diffusion Problems

“…We note that for the Poisson equation the bound for the two-grid contraction number of our method (cf. Remark 3.6.1) is the same as for the two-grid methods in [11,3,4]. In these papers red-black coarsening and a matrix-dependent prolongation are used too.…”

“…(3.14) based on the Galerkin condition. The bound for the Poisson equation as in Remark 3.6.1 holds with damped Ja.cobi smoothing, whereas in [11,3,4] red-black Gauss-Seidel smoothing is used. Finally note that in [11,3,4] the main subject is an analysis of a very efficient multigrid solver for Poisson-like equations, whereas our purpose is to develop a robust and reasonably efficient muitigrid solver for convection-diffusion problems.…”

Defect Corrections for Multigrid Solutions of the Dirichlet Problem in General Domains

“…(In contrast to this, |ALALA"1|00 < 1 can easily be derived from Theorem 4.6.) 5. General Domains; Concluding Remarks.…”

“…Let Ü c R2 be a bounded polygonal domain such that, for some sequence of uniform grids ßA c ß (with mesh size h), its boundary consists of horizontal, vertical or diagonal grid lines (see Figure 4.1). For this type of domains, explicit bounds are known for multigrid convergence rates; see Braess [5]. We are going to establish bounds for the contraction rate of the defect correction method.…”