The fictitious domain method and homotopy as a new alternative for multidimensional partial differential equations in domains of arbitrary shape

Gavrilyuk, IP
1Makarov, VL
1Institute of Mathematics of the NAS of Ukraine, Kyiv
Dopov. Nac. akad. nauk Ukr. 2019, 11:8-16
https://doi.org/10.15407/dopovidi2019.11.008
Section: Mathematics
Language: Ukrainian
Abstract: 

The ideas of the method of fictitious domains and of homotopy are united with the aim to reduce the solution of multidimensional partial differential equations in an arbitrary domain to an exponentially convergent sequence of problems in a parallelepiped (rectangle in the 2D case). This makes it possible to reduce the amount of computational work due to the absence of the need for the triangulation of a domain.

Keywords: boundary-value problem for a partial differential equation, domain of arbitrary shape, exponential convergence rate, fictitious domain method, homotopy, parallelepiped
References: 

1. Sauljev, V. K. (1963). On the solution of certain boundary value problems on high-speed computers by the fictitious domain method. Sib. mat. zhurn., 4, No. 4, pp. 912-925 (in Russian).
2. Marchuk, G. I. (1989). Methods of numerical mathematics. Moscow: Nauka (in Russian).
3. Kopchenov, V. D. (1968). The approximation of the solution of the Dirichlet problem by the method of fictitious domains. Differents. uravneniya, 4, No. 1, pp. 151-164 (in Russian).
4. Kobel'kov, G. M. (1987). Fictitious domain method and the solution of elliptic equations with highly varying coefficients. Sov. J. Numer. Anal. Math. Modelling, 2, Iss. 6, pp. 407-419. doi: https://doi.org/10.1515/rnam.1987.2.6.407
5. Brusnikin, M. B. (2002). On effective algorithms for solving problems of the fictitious domain method in the multiply connected case. Dokl. AN, 387, No. 2, pp. 151-155 (in Russian).
6. Bakhvalov, N. S., Bogachev, K. Ju. & Metr, J. F. (1999). An efficient algorithm for stiff elliptic problems with applications to the method of fictitious domains. Comput. Math. Math. Phys., 39, No. 6, pp. 884-896.
7. Lebedev, V. I. (1964). Difference analogues of orthogonal decompositions, basic differential operators and some boundary problems of mathematical physics. I. USSR Comput. Math. Math. Phys., 4, No. 3, pp. 69-92. doi: https://doi.org/10.1016/0041-5553(64)90240-X
8. Konovalov, A. N. (1973). The method of fictitious domains in torsion problems. Chislennyie metody mehaniki sploshnoy sredy, 4, No. 2, pp. 109-115 (in Russian).
9. Bogachev, K. Ju. (1996). Justification of the method of fictitious domains for solving mixed boundary-value problems for quasilinear elliptic equations. Vestn. Mosk. Un-ta, Ser. 1, Matematica. Mehanika, No. 3, pp.16-23 (in Russian).
10. Rukhovets, L. A. (1967). A remark on the method of fictitious regions. Differents. uravneniya, 3, No. 4, pp. 698-701 (in Russian).
11. Glowinski, R., Pan, T. W. & Periaux, J. (1994). A fictitious domain method for Dirichlet problem and applications. Comput. Methods Appl. Mech. Eng., 111, Iss. 3-4, pp. 283-303. doi: https://doi.org/10.1016/0045-7825(94)90135-X
12. Vojtsekhovskij, S. A., Gavrilyuk, I. P. & Makarov, V. L. (1982). Convergence of difference solutions to generalized solutions of the Dirichlet problem for the Helmholtz equation in an arbitrary domain. Dokl. AN SSSR, 267, No. 1, pp. 34-37 (in Russian).
13. Kopchenov, V. D. (1974). A method of fictitious domains for the second and third boundary value problems. Proc. Steklov Inst. Math., 131, pp. 125-134. 14. Vabishchevich, P. N. (2017). The method of fictitious domains in problems of mathematical physics. Moscow: LENAND (in Russian).
15. Makarov, V. L. (1991). About functional-discrete method of arbitrary accuracy order for solving Sturm—Liouville problem with piecewise smooth coefficients. Dokl. AN SSSR, Ser. math., 320, No. 1, pp. 34-39 (in Russian).