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 Zh. Vychisl. Mat. Mat. Fiz., 2005, Volume 45, Number 9, Pages 1594–1605 (Mi zvmmf597)

Averaging algorithms and the support-operator method in elliptic problems with discontinuous coefficients

M. Yu. Zaslavsky, A. Kh. Pergament

Keldysh Institute of Applied Mathematics, Russian Academy of Sciences, Miusskaya pl. 4, Moscow, 125047, Russia

Abstract: A method is proposed for solving elliptic boundary value problems with discontinuous coefficients. The method is based on an approximation of the energy integral followed by the construction of a finite-difference scheme by varying the corresponding functionals. It is shown that the solution to the original problem can be approximated by an element of the linear span spanned by basis vectors reflecting the features of the solution: for span elements, the flux component normal to the boundary and the tangent component of the gradient are both continuous across the discontinuity. The expression for the energy functional is exact for span elements and approximates the energy integral for arbitrary solutions. Numerical grids can be structure-fitted (as in the support-operator method) or not structure-fitted (e.g., rectangular, as in the averaging method). The weak convergence of the algorithms is proved. A method is discussed for choosing the control volume associated with a mesh point so as to satisfy the approximation conditions on the faces of the volume. It is shown that such a volume can be constructed for two-dimensional problems, and strong convergence is proved for them.

Key words: elliptic boundary value problems, discontinuous coefficients, approximation of energy integral, weak convergence.

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English version:
Computational Mathematics and Mathematical Physics, 2005, 45:9, 1538–1548

Bibliographic databases:
UDC: 519.632
Revised: 09.03.2005

Citation: M. Yu. Zaslavsky, A. Kh. Pergament, “Averaging algorithms and the support-operator method in elliptic problems with discontinuous coefficients”, Zh. Vychisl. Mat. Mat. Fiz., 45:9 (2005), 1594–1605; Comput. Math. Math. Phys., 45:9 (2005), 1538–1548

Citation in format AMSBIB
\Bibitem{ZasPer05} \by M.~Yu.~Zaslavsky, A.~Kh.~Pergament \paper Averaging algorithms and the support-operator method in elliptic problems with discontinuous coefficients \jour Zh. Vychisl. Mat. Mat. Fiz. \yr 2005 \vol 45 \issue 9 \pages 1594--1605 \mathnet{http://mi.mathnet.ru/zvmmf597} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2216071} \zmath{https://zbmath.org/?q=an:1117.65366} \transl \jour Comput. Math. Math. Phys. \yr 2005 \vol 45 \issue 9 \pages 1538--1548 

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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. A. Kh. Pergament, V. A. Semiletov, “Metod opornykh operatorov dlya ellipticheskikh i parabolicheskikh kraevykh zadach s razryvnymi koeffitsientami v anizotropnykh sredakh”, Matem. modelirovanie, 19:5 (2007), 105–115
2. Zaslavsky M.Yu., “An Averaging Algorithm for Solving Elliptic Problems with Discontinuous Coefficients”, Dokl. Phys., 53:3 (2008), 160–165
3. A. Kh. Pergament, V. A. Semiletov, P. Yu. Tomin, “On some multiscale algorithms for sector modeling in multiphase flow problems”, Math. Models Comput. Simul., 3:3 (2011), 365–374
4. M. Yu. Zaslavskii, P. Yu. Tomin, “O modelirovanii protsessov mnogofaznoi filtratsii v treschinovatykh sredakh v primenenii k zadacham adaptatsii modeli mestorozhdeniya”, Preprinty IPM im. M. V. Keldysha, 2010, 045, 20 pp.
5. A. Kh. Pergament, P. Yu. Tomin, “On study of relative phase permeabilities for anisotropic media”, Math. Models Comput. Simul., 4:1 (2012), 1–9
6. Zaslavsky M., Druskin V., Davydycheva S., Knizhnerman L., Abubakar A., Habashy T., “Hybrid finite-difference integral equation solver for 3D frequency domain anisotropic electromagnetic problems”, Geophysics, 76:2 (2011), F123–F137
7. Druskin V., Mamonov A.V., Zaslavsky M., “Multiscale S-Fraction Reduced-Order Models for Massive Wavefield Simulations”, Multiscale Model. Simul., 15:1 (2017), 445–475
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