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

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 .

We prove the generalized Hyers-Ulam stability of the 2nd-order linear differential equation of the form , with condition that there exists a nonzero in such that and is an open interval. As a consequence of our main theorem, we prove the generalized Hyers-Ulam stability of several important well-known differential equations.

Abstract. Let A 1 , A 2 , ..., A n be normed ternary algebras over the complex field C and let B be a Banach ternary algebra over C. A mapping δ k from A 1 × · · · × A n into B is called a k-th partial ternary quadratic derivation if there exists a mapping g k : A k

This work deals with the construction of variational difference solutions of optimal order of accuracy for classes of elliptic problems with quasidegenerate quadratic form and rapidly changing coefficients in the equations.