An exponentially convergent method for the solution of Laplace's equation on polygons

E. A. Volkov

Sbornik. Mathematics, Tome 37 (1980) no. 3, pp. 295-325

Résumé

A new approximate method of solving a mixed boundary value problem for Laplace's equation on an arbitrary polygon is presented and substantiated for the case when the right sides in the boundary conditions of the first and second kind on the sides of the polygon are given by algebraic polynomials in the arc length of the boundary of the polygon. By means of this method, an approximate solution of a boundary value problem on a closed polygon can be found with uniform accuracy $\varepsilon>0$ at the expense of $O(|\ln^3\varepsilon|)$ arithmetic operations. Bibliography: 15 titles.

Bibliographie

[1] E. A. Volkov, “Bystro skhodyaschiisya metod kvadratur resheniya uravneniya Laplasa na mnogougolnikakh”, DAN SSSR, 238:5 (1978), 1036–1039 | MR | Zbl

[2] E. A. Volkov, “O neizbezhnoi pogreshnosti metoda setok”, Matem. zametki, 4:6 (1968), 621–627 | MR | Zbl

[3] E. A. Volkov, “Priblizhennoe reshenie uravnenii Laplasa i Puassona v vesovykh prostranstvakh Gëldera”, Trudy Matem. in-ta im. V. A. Steklova, CXXVIII (1972), 76–112

[4] E. A. Volkov, “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravneniya Laplasa na mnogougolnikakh”, Trudy Matem. in-ta im. V. A. Steklova, LXXVII (1965), 113–142 | MR

[5] E. A. Volkov, “Metod setok dlya konechnykh i beskonechnykh oblastei s kusochno-gladkoi granitsei”, DAN SSSR, 168:5 (1966), 978–981 | Zbl

[6] E. A. Volkov, “Metod setok dlya konechnykh i beskonechnykh mnogougolnikov i otsenka pogreshnosti cherez izvestnye velichiny”, Trudy Matem. in-ta im. V. A. Steklova, XCVI (1968), 149–187

[7] E. A. Volkov, “O metode regulyarnykh sostavnykh setok dlya uravneniya Laplasa na mnogougolnikakh”, Trudy Matem. in-ta im. V. A. Steklova, CXL (1976), 68–102 | MR

[8] E. A. Volkov, “Raznostno-analiticheskii metod rascheta potentsialnogo polya na mnogougolnikakh”, DAN SSSR, 237:6 (1977), 1265–1268 | MR | Zbl

[9] M. A. Lavrentev, B. V. Shabat, Metody teorii funktsii kompleksnogo peremennogo, izd-vo “Nauka”, Moskva, 1973 | MR

[10] S. L. Sobolev, Nekotorye primeneniya funktsionalnogo analiza v matematicheskoi fizike, izd-vo Sibirskogo otdeleniya AN SSSR, Novosibirsk, 1962

[11] V. P. Mikhailov, Differentsialnye uravneniya v chastnykh proizvodnykh, izd-vo “Nauka”, Moskva, 1976 | MR

[12] M. V. Keldysh, “O razreshimosti i ustoichivosti zadachi Dirikhle”, Uspekhi matem. nauk, 1941, no. 8, 171–231 | MR | Zbl

[13] V. N. Malozemov, “Otsenka tochnosti odnoi kvadraturnoi formuly dlya periodicheskikh funktsii”, Vestnik LGU, matem., mekhanika, astronomiya, 1967, no. 1, 52–59 | MR | Zbl

[14] E. A. Volkov, “K voprosu o reshenii metodom setok vnutrennei zadachi Dirikhle dlya uravneniya Laplasa”, Vychislitelnaya matematika, no. 1, izd-vo AN SSSR, Moskva, 1957, 34–61

[15] E. A. Volkov, “O differentsialnykh svoistvakh reshenii kraevykh zadach dlya uravnenii Laplasa i Puassona na pryamougolnike”, Trudy Matem. in-ta im. V. A. Steklova, LXXVII (1965), 89–112