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

Abstract

Abstract. Recently, the technique of defect correction for the refinement of discrete solutions to elliptic boundary value problems has gained new acceptance in connection with the multigrid approach. In the present paper we give an analysis of a specific application, namely to finite-difference analogues of the Dirichlet problem for Helmholtz's equation, emphasizing the case of nonrectangular domains. A quantitative convergence proof is presented for a class of convex polygonal domains.

Defect corrections for multigrid solutions of the Dirichlet problem in general domains

“…In this section we consider Helmholtz's equation in the unit square. This example has already been discussed in Auzinger and Stetter [4] and, in more detail, in Auzinger [1]. Let ß = (0,1) X (0,1), and let Helmholtz's equation (1.1) be given.…”

“…The quoted results apply to the case of c = const > 0. (See [1] for the handling of variable c(x, y) by partial summation.) On a uniform grid with mesh spacing h = 2~m, m e N, the basic discretization Lhuh = fh is defined by the usual five-point stencil…”

“…Proof. (4.2) is the reformulation of a "discrete Green's function estimate" in Bramble and Hubbard [6] (see also [1]). Q Proposition 4.1 shows that the full order of consistency is not required near the boundary.…”

“…We have found an example where \L~hlALh\0fl > 1, even though ß is convex (cf. [1]). (In contrast to this, |ALALA"1|00 < 1 can easily be derived from Theorem 4.6.)…”

“…After some general remarks in Section 2, we present in Section 3 an account of the model problem analysis given in Auzinger [1]. In Section 4, which is the heart of the paper, we derive explicit bounds for the contraction number of the defect correction iteration for a class of convex polygonal domains.…”