Far Eastern Mathematical Journal

To content of the issue


Inverse extremum problems for stationary equations of convection-diffusion-reaction


O. V. Soboleva

2010, issue 2, Ñ. 170–184


Abstract
We study the coefficient inverse extremum problem for stationary equations of convection-diffusion-reaction in a bounded domain with mixed boundary conditions. We prove the stability of the solution of this problem with respect to small perturbations of both the cost functional and of the given function entering into the initial boundary value problem. The numerical algorithm is developed for solution of this extremum problem. It is based on Newton method for solving nonlinear equations and discretization of the linear boundary value problem by finite difference method or finite element method. Some results of numerical experiments are discussed.

Keywords:
elliptic equation, third boundary value problem, mass transfer, coefficient inverse problem, solvability, stability, numerical algorithm

Download the article (PDF-file)

References

[1] G. I. Marchuk, Matematicheskoe modelirovanie v probleme okruzhayushhej sredy, Nauka, M., 1982, 319 s.
[2] A. N. Tixonov, V. Ya. Arsenin, Metody resheniya nekorrektnyx zadach, Nauka, M., 1979.
[3] A. A. Samarskij, P. N. Vabishhevich, Chislennye metody resheniya obratnyx zadach matematicheskoj fiziki, Editorial URSS, Moskva, 2004, 480 s.
[4] K. Ito, K. Kunisch, “Estimation of the convection coefficient in elliptic equations”, Inverse Problems, 1997, no. 14, 995–1013.
[5] B. Lowe, W. Rundell, “The determination of a coefficient in an elliptic equation from average flux data”, J. Comput. Math., 70 (1996), 173–187.
[6] D. A. Tereshko, “Chislennoe reshenie zadach identifikacii parametrov primesi dlya stacionarnyx uravnenij massoperenosa”, Vych. texn., 9:4 (2004), 92–98.
[7] E. A. Kalinina, “Chislennoe issledovanie obratnoj e'kstremal'noj zadachi identifikacii mladshego koe'fficienta dvumernogo e'llipticheskogo uravneniya”, Dal'nevost. matem. zhurn., 6:1-2 (2005), 57–70.
[8] G. V. Alekseev, E. A. Kalinina, “Identifikaciya mladshego koe'fficienta dlya stacionarnogo uravneniya konvekcii – diffuzii – reakcii”, Sib. zhurn. industr. matem., 10:1 (2007), 3–16.
[9] G. V. Alekseev, “Koe'fficientnye obratnye e'kstremal'nye zadachi dlya stacionarnyx uravnenij teplomassoperenosa”, Zhurn. vychisl. matem. i matem. fiz., 47:6 (2007), 1055–1076.
[10] G. V. Alekseev, O. V. Soboleva, D. A. Tereshko, “Zadachi identifikacii dlya stacionarnoj modeli massoperenosa”, Prikl. mex. texn. fiz., 49:4 (2008), 24–35.
[11] G. V. Alekseev, O. V. Soboleva, “Ob ustojchivosti reshenij e'kstremal'nyx zadach dlya stacionarnyx uravnenij massoperenosa”, Dal'nevost. matem. zhurn., 9:1-2 (2009), 5–14.
[12] G. V. Alekseev, D. A. Tereshko, Analiz i optimizaciya v gidrodinamike vyazkoj zhidkosti, Dal'nauka, Vladivostok, 2008, 365 s.
[13] V. G. Maz'ya, Prostranstva Soboleva, Izd-vo LGU, L., 1985, 416 s.
[14] P. Grisvard, Elliptic problems in nonsmooth domains. Monograph and studies in mathematics, Pitman, London, 1985.
[15] V. A. Trenogin, Funkcional'nyj analiz, Nauka, M., 1980, 496 s.
[16] G. V. Alekseev, O. V. Soboleva, D. A. Tereshko, Koe'fficientnye obratnye e'kstremal'nye zadachi dlya stacionarnogo uravneniya konvekcii – diffuzii – reakcii, Preprint ¹5 In-ta prikl. matem. DVO RAN, Dal'nauka, Vladivostok, 2008, 40 s.
[17] A. D. Ioffe, V. M. Tixomirov, Teoriya e'kstremal'nyx zadach, Nauka, M., 1974, 480 s.
[18] Olivier Pironneau, Frederic Hecht, Antoine Le Hyaric, Jacques Morice, FreeFem++ version 3.9-0 (2d and 3d), http://www.freefem.org/ff++/index.htm.
[19] Home–Scilab WebSite, http://www.scilab.org.

To content of the issue