Variational difference method of solving the quasisingular classes of elliptic problems with rapidly changing coefficients in the equations

Javadian, A. D.

doi:10.1515/rnam.1996.11.4.303

Cited by 3 publications

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“…The following model problem has been studied in detail in [3]. The following model problem has been studied in detail in [3].…”

Section: Appendix Bmentioning

confidence: 99%

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Fast multigrid algorithm for quasisingular classes of diffusion problems

Javadian

2004

Russian Journal of Numerical Analysis and Mathematical Modelling

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In this paper it is shown that the fast multigrid algorithm (FMA) first proposed by the author in [1] is applicable to the solution of a sufficiently broad family of quasisingulsar classes of diffusion problems. The optimal accuracy in the sense of the Kolmogorov N-width is attained using O(| ln ε| 3 N) operations, where ε is the minimum value of the coefficient in the problem. This paper is a continuation of [1] studying a rather broad family in the quasisingular classes of elliptic problems (QCEP). The latter class contains diffusion problems of a sufficiently general form.Unlike [1], some facts in this paper are proved without cumbersome analytic representations of functions. Note that typically such representations are not available.When describing the studied class of problems, we formulate conditions sufficient for the FMA to converge. In Appendix B we give examples of model problems in which these conditions are satisfied.

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“…The following model problem has been studied in detail in [3]. The following model problem has been studied in detail in [3].…”

Section: Appendix Bmentioning

confidence: 99%

“…From [3] we have (G). We partition the side of the rectangle Ω along the 0ϕ-axis into segments of length of the order √ εh.…”

Section: Appendix Bmentioning

confidence: 99%

Fast multigrid algorithm for quasisingular classes of diffusion problems

Javadian

2004

Russian Journal of Numerical Analysis and Mathematical Modelling

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“…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].…”

Section: Posing the Problemmentioning

confidence: 99%

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

Javadian¹

1997

Russian Journal of Numerical Analysis and Mathematical Modelling

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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 g , which is constructed on the grid with W nodes, is the solution of this kind if

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“…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.…”

Section: Introductionmentioning

confidence: 99%

Fast multigrid algorithm for quasisingular classes of elliptic problems

Javadian

2001

Russian Journal of Numerical Analysis and Mathematical Modelling

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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 .

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Variational difference method of solving the quasisingular classes of elliptic problems with rapidly changing coefficients in the equations

Abstract: 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.

Cited by 3 publications

References 3 publications

Fast multigrid algorithm for quasisingular classes of diffusion problems

Fast multigrid algorithm for quasisingular classes of diffusion problems

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

Fast multigrid algorithm for quasisingular classes of elliptic problems

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