Fast multigrid algorithm for quasisingular classes of diffusion problems

Abstract

In this paper it is shown that the fast multigrid algorithm (FMA) first proposed by the author in [1] is applicable to the solution of a sufficiently broad family of quasisingulsar classes of diffusion problems. The optimal accuracy in the sense of the Kolmogorov N-width is attained using O(| ln ε| 3 N) operations, where ε is the minimum value of the coefficient in the problem. This paper is a continuation of [1] studying a rather broad family in the quasisingular classes of elliptic problems (QCEP). The latter class contains diffusion problems of a sufficiently general form.Unlike [1], some facts in this paper are proved without cumbersome analytic representations of functions. Note that typically such representations are not available.When describing the studied class of problems, we formulate conditions sufficient for the FMA to converge. In Appendix B we give examples of model problems in which these conditions are satisfied.