Fast multigrid algorithm for quasisingular classes of elliptic problems

Abstract

For certain quasisingular classes of elliptic problems (QCEP) we have obtained condensed finite element grids to get an optimal approximate solution in the sense of the Kolmogorov N-width. But the computational cost of any method of solving the system of finite element equations is high.That is why in this paper we undertook to invent a new fast multigrid algorithm (FMA) in order to get the optimal approximate solution for QCEP. We show that the computational cost of the FMA is asymptotically proportional to N´ln Nµ 3 .

A fast solving method for elliptic problems in domains with re-entrant corners

“…This paper is the continuation of the series of papers by the author devoted to the construction of approximate solutions for singular and quasisingular classes of elliptic problems optimal in accuracy (see [8] and the references therein) and in the number of arithmetic operations [6,7].…”

“…For the case n = 0 this problem may be solved with the optimal computation cost and the accuracy of the order O(| ln h|h) by the FMA method proposed by the author in [6,7]. In the case n 1 this method is not applicable.…”

Fast multigrid algorithm for quasisingular classes of diffusion problems

“…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. 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.…”

“…Unlike [1], some facts in this paper are proved without cumbersome analytic representations of functions. Note that typically such representations are not available.…”