Optimal variational difference method of solving the limiting problem for one quasisingular class of elliptic problems

Abstract

In this work we construct a quasisingular class of elliptic problems. We show that the problem is in a definite sense a limiting problem for this class. Finally, we prove that we can choose the optimal variational difference solution to one of the problems in the class constructed as the optimal (in the sense of the Kolmogorov TV-width) approximate solution of the original problem. POSING THE PROBLEMIn this paper we continue to study the model problems [1][2][3][4][5][6][7]. The method of dividing the arbitrary original problem (with certain properties) into subproblems that are reduced to the model problems has adequately been studied by the author in [1][2][3][4].We introduce the class of elliptic problems.In the square we consider the problem where > 0, /e L 2 (ß), ||/||^( ö) < 1. It is clear that when varying in (0,1] we obtain the quasisingular class of elliptic problems. The definition of the quasisingular classes of elliptic problems is formally given in [6] though these classes have been studied by the author since 1985.The problem D(L £ ,ß,/) for = 0 is the original problem of interest here. In Section 4 we show that it is a limiting problem for the class {D(L e ,i2,/)} 0<fiia .Thus, our aim is to construct the optimal (in sense of the Kolmogorov TV-width) approximate solution to the arbitrary problem D(L e ,ß,/), 0< < 1. Recall [1] that H> g , which is constructed on the grid with W nodes, is the solution of this kind if

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].…”