Variational difference method for elliptic problems with highly quasi-singular quadratic forms

Abstract

The paper considers the problem of construction of the best approximate solution (using the variational difference method) to two model elliptic boundary value problems with essentially different types of high quasi-singularity of the corresponding quadratic form.To solve approximately elliptic problems [9] by the variational difference method, sufficient conditions for localization of singularities and quasi-singularities [7] of these problems were obtained in [4,5,6]. The original domain was assumed to be composed of subdomains in each of which the behaviour of the solution was determined by one of the model type behaviours studied in advance.The present paper is devoted to the construction of the best approximate solution (using the variational difference method) to two model elliptic boundary value problems with essentially different types of high quasi-singularity of the corresponding quadratic form.By the best approximate solution to the elliptic boundary value problem Lu=/, u = 0, /€Z 7 (fl) 3Ω obtained by the variational difference method we mean a function v [belonging to an Af-dimensional subspace of the space W^( )] such that where |[ · ]| t Ω is the energy norm, C is a constant independent of singularities and quasi-singularities of the operator L and also of the original domain Ω, and d N (L,Q) is the Kolmogorov TV-diameter:{u\ Ι Here, H N denotes an arbitrary TV-dimensional subspace of the space To compute the TV-diameter associated with the elliptic operators of the form ~$xQ~ix ~ Oy^ily 9 one can ut^ze the technique from [6] based on the application of the Weyl theorem on the asymptotics of the TV-th eigenvalue of the operator [1]. (I) Let the boundary value problem (Q = ε + χ?) 3 du · dy z (1) u =0 3Ω Brought to you by | University of Queensland -UQ Library Authenticated Download Date | 7/16/15 2:07 AM

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

“…In 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].…”

Optimal methods of approximate solution of elliptic problems with weakened boundary conditions

“…The universal constants are numerated within every section. Note that in [1,[3][4][5][6][7][11][12][13] it has been shown, in fact, that the variation-difference method is optimal for solving a wide class of both regular and irregular problems & f ( in the traditional formulation) and Sy. Special condensing grids, and, sometimes, special finite elements constructed on the basis of the predetermined differential properties of solutions of these problems have been used therein for solving irregular classes of the problems.…”

“…Javadian [7] constructed an optimal method (a variation-difference one) for solving this problem (see Definition 1.1). He also showed that the following relation held for sufficiently large N:…”

Fast multigrid algorithm for quasisingular classes of elliptic problems

“…For all quasisingular classes of elliptic problems (QCEP) studied by the author in [3][4][5][6][7][8] condensed finite element grids were constructed to get an optimal approximate solution in the sense of (d N N)-width of Kolmogorov. It means that we chose N from the relation d N H. Then using a priori differential properties of QCEP under investigation, we constructed a certain condensed N-nodal finite element grid to get an accuracy H in energy norm.…”