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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
additivemultiplicative separation of singularity. We prove that the convergence rate of the proposed technique is optimal
in a given class of smoothness of righthand sides.
Keywords and phrasestwopoint boundary value problem–finite element schemes–weight function spaces–multiplicative and additivemultiplicative separation of singularity Russian Mathematics 01/2011; 55(5):7477.

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ABSTRACT: Twopoint boundary value problem for a differential equation of fourthorder with degeneration is considered. This problem is solved by the finite element method of highorder accuracy with a multiplicative separation of singularity. The optimal convergence rate of the presented method for a given class of smoothness of the righthand sides is proved. Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta. Seriya FizikiMatematicheskie Nauki. 01/2010; 1(1).

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ABSTRACT: In this paper for the finite elementmethod for systems of degenerate elliptic equations we develop highaccuracy 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. Russian Mathematics 01/2009; 53(7):1727.

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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. Differential Equations 01/2008; 44(7):9991005. · 0.42 Impact Factor

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ABSTRACT: The purpose of the paper is construction of a finite element schemes with a high approximation order for twopointed heterogeneous boundaryvalue problem with degenerated coefficients, based on a multiplicative allocation of singularities. It is proved, that this method has optimal order of convergence rate. Uchenye Zapiski Kazanskogo Gosudarstvennogo Universiteta. Seriya FizikiMatematicheskie Nauki. 01/2006; 4(4).

M. R. Timerbaev
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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. Differential Equations 06/2000; 36(7):10861093. · 0.42 Impact Factor

Differential Equations 06/2000; 36(7):10501057. · 0.42 Impact Factor

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ABSTRACT: From the introduction: The article is devoted to investigation of existence and uniqueness of generalized solution of a linear degenerate elliptic equation of order 2m for two cases. First, for the case where the equation degenerates on a part of the domain’s boundary (in particular, on the whole boundary). Second, when the degeneration occurs inside the domain. We analyze the problem’s statement in dependence on equation coefficients’ degeneration degree by application of the embedding theorem of the weight Sobolev spaces. For approximate solution we suggest the finite element method. Russian Mathematics 01/1999; 43(5).

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ABSTRACT: The authors propose a finite element method in order to solve some fourthorder quasilinear elliptic boundary value problems which degenerate on the boundary of the domain. They follow the classical lines of analysis in the finite element method (existence and uniqueness of a generalized solution, interpolation results in weighted Sobolev spaces, etc.). Consequently, they obtain estimations relating the accuracy of the method with the smoothness of the solution, the degeneration degree and the roughness of the mesh nearby the degeneration line. Differential Equations 35(2). · 0.42 Impact Factor