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 Sib. Zh. Vychisl. Mat., 1998, Volume 1, Number 2, Pages 153–170 (Mi sjvm299)

On the $h$-$p$ version of the finite element method for one-dimensional boundary value problem with singularity of a solution

V. A. Rukavishnikov, A. Yu. Bespalov

Computing Center, Far-Eastern Branch, Russian Academy of Sciences, Khabarovsk

Abstract: The paper analyzes the $h$-$p$ version of the finite element method for a one-dimensional model boundary value problem with coordinated degeneration of initial data and with strong singularity of a solution. The scheme of the finite element method is constructed on the basis of the definition of $R_\nu$-generalized solution to the problem, and the finite element space contains singular power functions. By using meshes with concentration at a singular point and by constructing the linear degree vector of approximating functions in a special way, a nearly optimal two-sided exponential estimate is obtained for the residual of the finite element method.

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UDC: 519.632
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Citation: V. A. Rukavishnikov, A. Yu. Bespalov, “On the $h$-$p$ version of the finite element method for one-dimensional boundary value problem with singularity of a solution”, Sib. Zh. Vychisl. Mat., 1:2 (1998), 153–170

Citation in format AMSBIB
\Bibitem{RukBes98} \by V.~A.~Rukavishnikov, A.~Yu.~Bespalov \paper On the $h$-$p$ version of the finite element method for one-dimensional boundary value problem with singularity of a~solution \jour Sib. Zh. Vychisl. Mat. \yr 1998 \vol 1 \issue 2 \pages 153--170 \mathnet{http://mi.mathnet.ru/sjvm299} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1700959} \zmath{https://zbmath.org/?q=an:0906.65084} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. Bespalov A., Rukavishnikov V., “The Exponential Rate of Convergence of the Finite-Element Method for the Dirichlet Problem with Singularity of the Solution”, Dokl. Math., 62:2 (2000), 266–270
2. Rukavishnikov V., “On the Uniqueness of an R-V-Generalized Solution to Boundary Value Problems with Inconsistently Degenerate Initial Data”, Dokl. Math., 63:1 (2001), 68–70
3. A. Yu. Bespalov, V. A. Rukavishnikov, “The use of singular functions in the $h$-$p$ version of the finite element method for a Dirichlet problem with degeneration of the input data”, Sib. zhurn. vychisl. matem., 4:3 (2001), 201–228
4. E. V. Kashuba, V. A. Rukavishnikov, “On the $p$-version of the finite element method for the boundary value problem with singularity”, Sib. zhurn. vychisl. matem., 8:1 (2005), 31–42
5. Rukavishnikov V.A., “Methods of Numerical Analysis for Boundary Value Problems with Strong Singularity”, Russ. J. Numer. Anal. Math. Model, 24:6 (2009), 565–590
6. Rukavishnikov V.A., “O kraevykh zadachakh s silnoi singulyarnostyu”, Vestnik Tikhookeanskogo gosudarstvennogo universiteta, 2011, no. 2, 033–042
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