[Show abstract][Hide abstract] ABSTRACT: We obtain sharp estimates for the accuracy of the best approximation of functions by algebraic polynomials on an interval, the half-line, and the entire line in weighted Sobolev spaces with Jacobi, Laguerre, and Hermite weights, respectively. We show that the orthogonal polynomials associated with these weights form orthogonal bases in the respective weighted Sobolev spaces. We obtain sharp estimates of Markov–Bernstein type.
[Show abstract][Hide abstract] ABSTRACT: We consider a class of elliptic boundary value problems with degenerating coefficients for which we construct FEM schemes with optimal convergence on the basis of multiplicative extraction of the singularity. For a scale of weighted Sobolev norms including the energy norm of the differential operator, we prove a posteriori estimates for the error of the discrete solutions.
[Show abstract][Hide abstract] ABSTRACT: Sobolev spaces with weights taking infinite values at some interior points of a two-dimensional domain are considered. For functions from these spaces, a Hardy inequality is obtained. Embedding theorems for weighted Lebesgue spaces and equivalent renorming theorems are proved.
Lobachevskii Journal of Mathematics 10/2013; 34(4). DOI:10.1134/S1995080213040069
[Show abstract][Hide abstract] ABSTRACT: In this paper we construct a high accuracy variant of the finite element method for an ordinary differential equation of the
fourth order whose coefficients are degenerate on the boundary. The proposed technique is based on the multiplicative and
additive-multiplicative separation of singularity. We prove that the convergence rate of the proposed technique is optimal
in a given class of smoothness of right-hand sides.
Keywords and phrasestwo-point boundary value problem–finite element schemes–weight function spaces–multiplicative and additive-multiplicative separation of singularity
[Show abstract][Hide abstract] ABSTRACT: In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop high-accuracy schemes based
on multiplicative singularity extraction. We prove theorems about the smoothness of a solution. Based on these theorems we
estimate the error of the proposed method.
[Show abstract][Hide abstract] ABSTRACT: The paper deals with the numerical solution of a generalized spectral boundary value problem for an elliptic operator with degenerating coefficients. We suggest an approximate method based on the multiplicative separation of the singularity, whereby the eigenfunctions are approximated by piecewise linear functions multiplied by a weight specially chosen depending on the order of degeneration of the coefficients. For this method, we obtain error estimates justifying its optimality.
[Show abstract][Hide abstract] ABSTRACT: We suggest a numerical method for degenerate equations in which the solution u(x) is represented in the boundary layer region in the form of the product u(x) = a(x)~o(x), where the function a(x) is constructed with the use of the coefficients of the degenerate differential operator and takes account of the asymptotics of the solution in a neighborhood of the degeneration points and the function ~(x) is much more suitable for FE approximations than u(x). The approximate solution is sought in the form ~th(X ) ~- O'(X)~gh(X), where ~h(X) is an FE approximation to p(x) on a quasiuniform grid. The performance characteristics of the method (such as accuracy, programmability, and time and memory requirements) are the same as in the regular case.
[Show abstract][Hide abstract] ABSTRACT: In the present paper, we suggest and investigate a mixed finite-element scheme for quasilinear fourth-order elliptic equations degenerating on the domain boundary. For example, equations appearing in the description of elastoplastic deflections of thin plates of variable thickness (e.g., see [1, p. 195; 2, p. 69]) are special cases of such equations. The structure of our schemes is close to that studied in  (see also ). We use products of second-order derivatives of the desired function by a weight function determining the character of the equation degeneration as auxiliary unknowns in the mixed statement of the problem. This permits us to construct and investigate schemes of an arbitrary-order accuracy on the basis of Lagrangian elements in a unified way. A number of conformal schemes for quasilinear degenerating fourth-order elliptic equations based on classical affine-equivalent or almost affine-equivalent sets of finite elements of the class C-(1) [5, pp. 325-353 of the Russian translation] were investigated in .