Geodesic
Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 6, pp. 989-998 
R. V. Zhalnin; M. E. Ladonkina; V. F. Masyagin; V. F. Tishkin. Solving the problem of non-stationary filtration of substance by the discontinuous Galerkin method on unstructured grids. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 6, pp. 989-998. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_6_a5/
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     title = {Solving the problem of non-stationary filtration of substance by the discontinuous {Galerkin} method on unstructured grids},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
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     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_6_a5/}
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Voir la notice de l'article provenant de la source Math-Net.Ru

A numerical algorithm is proposed for solving the problem of non-stationary filtration of substance in anisotropic media by the Galerkin method with discontinuous basis functions on unstructured triangular grids. A characteristic feature of this method is that the flux variables are considered on the dual grid. The dual grid comprises median control volumes around the nodes of the original triangular grid. The flux values of the quantities on the boundary of an element are calculated with the help of stabilizing additions. For averaging the permeability tensor over the cells of the dual grid, the method of support operators is applied. The method is studied on the example of a two-dimensional boundary value problem. The convergence and approximation of the numerical method are analyzed, and results of mathematical modeling are presented. The numerical results demonstrate the applicability of this approach for solving problems of non-stationary filtration of substance in anisotropic media by the discontinuous Galerkin method on unstructured triangular grids.

[1] Kondaurov V. I., Mekhanika i termodinamika nasyschennoi poristoi sredy, Uchebnoe posobie, MFTI, M., 2007, 310 pp.

[2] Masyagin V. F., Zhalnin R. V., Tishkin V. F., “Ob odnom sposobe approksimatsii uravnenii diffuzionnogo tipa s pomoschyu razryvnogo metoda Galerkina na nestrukturirovannykh setkakh”, Matem. i kompyuternoe modelirovanie estestvenno-nauchn. i sotsialnykh problem, Sb. st. VIII Mezhdunar. nauch.-tekhn. konf. molodykh spetsialistov, aspirantov i studentov (Rossiya, g. Penza, 26–30 maya 2014 g.), ed. I. V. Boikov, Izd-vo PGU, Penza, 2014, 49–56

[3] Samarskii A. A., Koldoba A. V., Poveschenko Yu. A., Tishkin V. F., Favorskii A. P., Raznostnye skhemy na neregulyarnykh setkakh, ZAO “Kriterii”, Minsk, 1996, 276 pp.

[4] Samarskii A. A., Tishkin V. F., Favorskii A. P., Shashkov M. Yu., “Ispolzovanie metoda opornykh operatorov dlya postroeniya raznostnykh analogov operatsii tenzornogo analiza”, Differents. ur-niya, 18:7 (1982), 1251–1256 | MR

[5] Pergament A. Kh., Semiletov V. A., “Metod opornykh operatorov dlya ellipticheskikh i parabolicheskikh kraevykh zadach s razryvnymi koeffitsientami v anizotropnykh sredakh”, Matem. modelirovanie, 19:5 (2007), 105–116 | Zbl

[6] Fletcher R., Chislennye metody na osnove metoda Galerkina, Per. s angl., Mir, M., 1988

[7] Khasanov M. M., Mukminov I. R., Bachin S. I., “K raschetu pritoka zhidkosti k skvazhinam, rabotayuschim v usloviyakh lokalnogo razgazirovaniya”, Neftepromyslovoe delo, 2000, no. 8–9, 2–6

[8] Charnyi I. A., Podzemnaya gidromekhanika, NITs “Regulyarnaya i khaoticheskaya dinamika”. In-t kompyuternykh issledovanii, M.–Izhevsk, 2006

[9] Schelkachev V. N., Osnovy i prilozheniya teorii neustanovivsheisya filtratsii, Monografiya: V 2 ch., Neft i gaz, M., 1995

[10] Erlager R. (ml.), Gidrodinamicheskie issledovaniya skvazhin, In-t kompyuternykh issledovanii, M.–Izhevsk, 2006

[11] Cockburn B., “An Introduction to the discontinuous Galerkin method for convection - dominated problems, advanced numerical approximation of nonlinear hyperbolic equations”, Lecture Notes in Math., 1697, 1998, 151–268 | MR | Zbl

[12] Arnold D. N., Brezzi F., Cockburn B., Marini L. D., “Unified analysis of discontinuous Galerkin methods for elliptic problems”, SIAM Journal on Numerical Analysis, 29 (2002), 1749–1779 | DOI | MR

[13] Bassi F., Rebay S., “A High-order accurate discontinuous finite element method for the numerical solution of the compressible Navier–Stokes equations”, J. Comput. Phys., 131 (1997), 267–279 | DOI | MR | Zbl

[14] Cockburn B., Shu C.-W., “Runge–Kutta discontinuous Galerkin methods for convection-dominated problems”, J. Sci. Comput., 3 (2001), 173–261 | DOI | MR | Zbl

[15] Belotserkovskii O. M., Davydov Yu. M., “Nestatsionarnyi metod “krupnykh chastits” dlya gazodinamicheskikh raschetov”, Zh. vychisl. matem. i matem. fiz., 11:1 (1971), 182–207 | MR | Zbl

[16] Belotserkovskii O. M., Guschin V. A., Schennikov V. V., “Metod rasschepleniya v primenenii k resheniyu zadach dinamiki vyazkoi neszhimaemoi zhidkosti”, Zh. vychisl. matem. i matem. fiz., 15:1 (1975), 197–207 | Zbl