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 Sib. Zh. Vychisl. Mat., 2005, Volume 8, Number 1, Pages 31–42 (Mi sjvm208)

On the $p$-version of the finite element method for the boundary value problem with singularity

E. V. Kashuba, V. A. Rukavishnikov

Computer Centre Far-Eastern Branch of RAS

Abstract: The one-dimensional first-type boundary value problem for the second order differential equation with strong singularity of a solution caused by coordinated degeneration of input data at the origin is considered. For this problem we define the solution as $R_{\nu}$-generalized one. It has been proved that solution belongs to the weighted Sobolev space $H^3_{2,\nu+\beta/2+1}$ under proper assumptions for coefficients and the right-hand side of the differential equation. The scheme of the finite element method is constructed on a fixed mesh using polynomials of an arbitrary degree $p$ (the $p$-version of the finite element method). The finite element space contains singular polynomials. Using the regularity of $R_{\nu}$-generalized solution, the estimate for the rate of convergence of the second order with respect to the degree $p$ of polynomials is proved in the norm of the weighted Sobolev space.

Key words: the $p$-version of the finite element method, boundary value problems with singularity, the weighted Sobolev spaces, an orthonormalized singular polynomials set.

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UDC: 519.6
Revised: 31.05.2004
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Citation: E. V. Kashuba, V. A. Rukavishnikov, “On the $p$-version of the finite element method for the boundary value problem with singularity”, Sib. Zh. Vychisl. Mat., 8:1 (2005), 31–42

Citation in format AMSBIB
\Bibitem{KasRuk05} \by E.~V.~Kashuba, V.~A.~Rukavishnikov \paper On the $p$-version of the finite element method for the boundary value problem with singularity \jour Sib. Zh. Vychisl. Mat. \yr 2005 \vol 8 \issue 1 \pages 31--42 \mathnet{http://mi.mathnet.ru/sjvm208} \zmath{https://zbmath.org/?q=an:1078.65104} 

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This publication is cited in the following articles:
1. Rukavishnikov V.A., Kuznetsova E.V., “Coercive estimate for a boundary value problem with noncoordinated degeneration of the data”, Differ. Equ., 43:4 (2007), 550–560
2. V. A. Rukavishnikov, E. V. Kuznetsova, “A scheme of a finite element method for boundary value problems with non-coordinated degeneration of input data”, Num. Anal. Appl., 2:3 (2009), 250–259
3. Rukavishnikov V.A., “Methods of numerical analysis for boundary value problems with strong singularity”, Russian J. Numer. Anal. Math. Modelling, 24:6 (2009), 565–590
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