Geodesic
On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 3, pp. 329-342.

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An iterative process for the grid problem of conjugation with iterations on the boundary of the discontinuity of the solution is considered. Similar grid problem arises in difference approximation of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions. The study of iterative processes for the states of such problems is of independent interest for theory and practice. The paper shows that the numerical solution of boundary problems of this type can be efficiently implemented using iterations on the inner boundary of the grid solution discontinuity in combination with other iterative methods for nonlinearities separately in each of the grid subregions. It can be noted that problems for states of controlled processes described by equations of mathematical physics with discontinuous coefficients and solutions arise in mathematical modeling and optimization of heat transfer, diffusion, filtration, elasticity theory, etc. The proposed iterative process reduces the solution of the initial grid boundary problem for a state with a discontinuous solution to a solution of two special boundary problems in two grid subdomains at every fixed iteration. The convergence of the iteration process in the Sobolev grid norms to the unique solution of the grid problem for each initial approximation is proved.
Keywords: iterative method, boundary value problem, discontinuous solution, difference approximation, summation identity, grid function.
Mots-clés : elliptic equation 
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F. V. Lubyshev; M. È. Fairuzov. On an iterative process for the grid conjugation problem with iterations on the boundary of the solution discontinuity. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 21 (2019) no. 3, pp. 329-342. http://geodesic.mathdoc.fr/item/SVMO_2019_21_3_a2/

[1] F. V. Lubyshev, “Finite difference approximations of optimal control problems for semilinear elliptic equations with discontinuous coefficients and solutions”, Computational mathematics and mathematical physics, 52:8 (2012), 1094–1114 | DOI | MR | Zbl

[2] A. A. Samarskij, P. N. Vabishchevich, Computational heat transfer, Editorial URSS, Moscow, 2003, 784 pp. (In Russ.)

[3] A. A. Samarskij, V. B. Andreev, Difference methods for elliptic equations, Nauka, Moscow, 1976, 352 pp. (In Russ.)

[4] V. S. Zarubin, Engineering methods for solving heat conduction problems, Energoatomizdat, Moscow, 1983, 328 pp. (In Russ.)

[5] E. M. Kartashov, Analytical methods in the theory of thermal conductivity of solids, Vysshaya shkola, Moscow, 1985, 480 pp. (In Russ.)

[6] V. A. Tsurko, “The accuracy of finite-difference schemes for parabolic equations with a discontinuous solution”, Differential Equations, 36:7 (2000), 1094–1101 | DOI | MR | Zbl

[7] V. A. Tsurko, “Finite-difference methods for convection-diffusion problems with discontinuous coefficients and solutions”, Differential Equations, 41:2 (2005), 290–296 | DOI | MR | Zbl

[8] A. A. Samarskij, R. D. Lazarov, V. L. Makarov, Difference schemes for differential equations with generalized solutions, Vysshaya shkola, Moscow, 1987, 296 pp. (In Russ.)

[9] H. Gajewski, K. Greger, K. Zacharias, Nichtlineare operatorgleichungen und operatordifferentialgleichungen, Akademie Verlag, Berlin, 1974, 281 pp. | MR | Zbl

[10] A. Kufner, S. Fuchik, Nonlinear differential equations, Elsevier, Prague, 1980, 359 pp. | MR | Zbl

[11] F. E.Brauder, “Nonlinear elliptic boundary value problems”, Nonlinear elliptic boundary value problems, Materialy k sovmestnomu sovetsko-amerikanskomu simpoziumu po uravneniyam s chastnymi proizvodnymi, Publishing House of the Siberian Branch of the Academy of Sciences of the USSR, Novosibirsk, 1963 (In Russ.)