Geodesic
Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1302-1313 Cet article a éte moissonné depuis la source Math-Net.Ru

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A nonlocal boundary value problem for Laplace’s equation on a rectangle is considered. Dirichlet boundary conditions are set on three sides of the rectangle, while the boundary values on the fourth side are sought using the condition that they are equal to the trace of the solution on the parallel midline of the rectangle. A simple proof of the existence and uniqueness of a solution to this problem is given. Assuming that the boundary values given on three sides have a second derivative satisfying a Hölder condition, a finite difference method is proposed that produces a uniform approximation (on a square mesh) of the solution to the problem with second order accuracy in space. The method can be used to find an approximate solution of a similar nonlocal boundary value problem for Poisson’s equation. 
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E. A. Volkov. Approximate grid solution of a nonlocal boundary value problem for Laplace’s equation on a rectangle. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 53 (2013) no. 8, pp. 1302-1313. http://geodesic.mathdoc.fr/item/ZVMMF_2013_53_8_a7/

[1] Bitsadze A. V., Samarskii A. A., “O nekotorykh prosteishikh obobscheniyakh lineinykh ellipticheskikh kraevykh zadach”, Dokl. AN SSSR, 185:4 (1969), 739–740 | Zbl

[2] Wasow W., “On the truncation error in the solution of Laplace's equation by finite differences”, J. Res. Nat. Bur. Stand., 48 (1952), 345–348 | DOI | MR

[3] Volkov E. A., “Asimptoticheski bystryi priblizhennyi metod nakhozhdeniya na setochnykh otrezkakh resheniya raznostnogo uravneniya Laplasa”, Tr. MIAN SSSR, 173, M., 1986, 69–89 | Zbl

[4] Ilin V. A., Moiseev E. I., “Dvumernaya nelokalnaya kraevaya zadacha dlya operatora Puassona v differentsialnoi i raznostnoi traktovkakh”, Mat. modelirovanie, 2:8 (1990), 139–156 | MR

[5] Skubachevskii A. L., “Neklassicheskie kraevye zadachi. I”, Sovremennaya matematika. Fundamentalnye napravleniya, 26, 2007, 3–132 | MR

[6] Skubachevskii A. L., “Neklassicheskie kraevye zadachi. II”, Sovremennaya matematika. Fundamentalnye napravleniya, 33 (2009), 3–179 | MR

[7] Keldysh M. V., “O razreshimosti i ustoichivosti zadachi Dirikhle”, Uspekhi matem. nauk, 1940, no. 8, 171–231

[8] Kellog O. D., “On the derivatives of harmonic functions on the boundary”, Trans. Amer. Math. Soc., 33:2 (1931), 486–510 | DOI | MR

[9] Volkov E. A., “O metode regulyarnykh sostavnykh setok dlya uravneniya Laplasa na mnogougolnikakh”, Tr. MIAN SSSR, 140, M., 1976, 68–102 | MR | Zbl

[10] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[11] Volkov E. A., “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravnenii Laplasa i Puassona na pryamougolnike”, Tr. MIAN SSSR, 77, M., 1965, 89–112 | Zbl

[12] Samarskii A. A., Andreev V. B., Raznostnye metody dlya ellipticheskikh uravnenii, Nauka, M., 1976 | MR | Zbl