Geodesic
Application of SDE's to estimating the solution of heat equations with discontinuous coefficients
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 147-161.

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This paper proposes the use of the numerical solution to stochastic differential equations (SDE's) to find estimates of the solutions to boundary value problems for linear parabolic equations with discontinuous coefficients. The solution of the problem with smoothed coefficients is taken as an approximation of the generalized solution to the considered boundary value problem. The results of calculations for a thermal barrier coating comprising a composite cellular material are presented. 
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S. A. Gusev. Application of SDE's to estimating the solution of heat equations with discontinuous coefficients. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 18 (2015) no. 2, pp. 147-161. http://geodesic.mathdoc.fr/item/SJVM_2015_18_2_a3/

[1] Seepersad C. C., Dempsey B. M., Allen J. K., Mistree F., McDowell D. L., “Design of multifunctional honeycomb materials”, J. AIAA, 42:5 (2004), 1025–1032 | DOI

[2] Diwan G. C., Mohamed M. S., Seaid M., Trevelyan J., Laghrouche O., “Mixed enrichment for the finite element method in heterogeneous media”, Int. J. Numer. Meth. Engng., 101:1 (2015), 54–78 | DOI | MR

[3] Zhang X., Zhang P., “Heterogeneous heat conduction problems by an improved element-free Galerkin method”, Numerical Heat Transfer. Part B: Fundamentals. An Int. J. of Comput. and Methodology, 65:4 (2014), 359–375 | DOI

[4] Xiao-qi Liu, “Multiscale finite element methods for heat equation in three dimension honeycomb structure”, Proceedings of the Third international conference on Artificial intelligence and computational intelligence, Lecture Notes in Computer Science, 7004, 2011, 186–194

[5] Zhikov V. V., Kozlov S. M., Olejnik O. A., “Usrednenie parabolicheskikh operatorov”, Tr. Moskovskogo matematicheskogo obshchestva, 45, 1982, 182–236 | MR | Zbl

[6] Zhikov V. V., Kozlov S. M., Olejnik O. A., Usrednenie differentsial'nykh operatorov, Fizmatlit, M., 1993 | MR

[7] Vlasov A. N., Savatorova V. L., Talonov A. V., “Asimptoticheskoe usrednenie dlya resheniya zadach teploprovodnosti s fazovymi perekhodami v sloistykh sredakh”, Prikladnaya mekhanika i tekhnicheskaya fizika, 36:5 (1995), 155–163 | MR | Zbl

[8] Savatorova V. L., Talonov A. V., Volkov-Bogorodskij D. B., Vlasov A. N., “Matematicheskoe modelirovanie protsessa teploprovodnosti v periodicheskoj srede s tsilindricheskimi vklyucheniyami, otdelennymi ot matritsy tonkim sloem”, Vestnik Nizhegorodskogo universiteta im. N. I. Lobachevskogo, 2010, no. 6, 168–179

[9] Volkov-Bogorodskij D. B., Sushko G. B., Kharchenko S. A., “Kombinirovannaya MPI+THREADS parallel'naya realizatsiya metoda blokov dlya modelirovaniya teplovykh protsessov v strukturno-neodnorodnykh sredakh”, Vychislitel'nye metody i programmirovanie, 11:1 (2010), 127–136

[10] Misnar A., Teploprovodnost' tverdykh tel, zhidkostej, gazov i ikh sostavlyayushchikh, Mir, M., 1968

[11] Olejnik O. A., “Ob uravneniyakh ellipticheskogo i parabolicheskogo tipa s razryvnymi koeffitsientami”, UMN, 14:5(89) (1959), 164–166

[12] Ladyzhenskaya O. A., Rivkind V. Ya., Ural'tseva N. N., “O klassicheskoj razreshimosti zadach difraktsii”, Kraevye zadachi matematicheskoj fiziki. 4, Tr. MIAN SSSR, 92, 1966, 116–146 | MR | Zbl

[13] Ladyzhenskaya O. A., Solonnikov V. A., Ural'tseva N. N., Linejnye i kvazilinejnye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR

[14] Sobolev S. L., Nekotorye primeneniya funktsional'nogo analiza v matematicheskoj fizike, Izdanie tret'e, Nauka, M., 1988 | MR

[15] Nikol'skij S. M., Kurs matematicheskogo analiza, v. II, izdanie tret'e, pererab. i dop., Nauka, M., 1983

[16] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsij i funktsional'nogo analiza, Nauka, M., 1976 | MR

[17] Gikhman I. I., Skorokhod A. V., Vvedenie v teoriyu sluchajnykh protsessov, Nauka, M., 1977 | MR

[18] Gobet E., “Weak approximation of killed diffusion using Euler schemes”, Stochastic Processes and their Applications, 87 (2000), 167–197 | DOI | MR | Zbl

[19] Kloeden P. E., Platen E., Numerical Solution of Stochastic Differential Equations, Springer-Verlag, Berlin, 1992 | MR | Zbl

[20] Lépingle D., “Un schéma d'Euler pouréquations différentiells stochastiques réfléchies”, C. R. Acad. Sci. Paris Série I Math., 316 (1993), 601–605 | MR | Zbl

[21] Gobet E., “Euler schemes and half-space approximation for the simulation of diffusion in a domain”, ESAIM Probability and Statistics, 5 (2001), 261–297 | DOI | MR | Zbl

[22] Gusev S. A., Nikolaev V. N., “Calculation of heat transfer in heterogeneous structures such as honeycomb by using numerical solution of stochastic differential equations”, Advanced Materials Research, 1016 (2014), 758–763 | DOI