Multipole method for the Dirichlet problem on doubly connected domains of complex geometry: A general description of the method
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 40 (2000) no. 11, pp. 1633-1647 Cet article a éte moissonné depuis la source Math-Net.Ru

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     title = {Multipole method for the {Dirichlet} problem on doubly connected domains of complex geometry: {A} general description of the method},
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V. I. Vlasov; S. L. Skorokhodov. Multipole method for the Dirichlet problem on doubly connected domains of complex geometry: A general description of the method. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 40 (2000) no. 11, pp. 1633-1647. http://geodesic.mathdoc.fr/item/ZVMMF_2000_40_11_a6/

[1] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[2] Kondratev V. A., Oleinik O. A., “Kraevye zadachi dlya uravnenii s chastnymi proizvodnymi v negladkikh oblastyakh”, Uspekhi matem. nauk, 38:2 (1983), 3–76 | MR

[3] Grisvard P., Elliptic problems in nonsmooth domains, Pitman Publ., England, 1985 | MR | Zbl

[4] Kufner A., Sändig A.-M., Some applications of wieghted Sobolev spaces, Europaische Hochschulschriften, 100, Teubner-Texte zur Math., Leipzig, 1987 | MR | Zbl

[5] Dauge M., Elliptic boundary value problems in corner domains – smoothness and asymptotics of solutions, Lect. Notes Math., 1341, Springer, Heidelberg, 1988 | MR | Zbl

[6] Nazarov C. A., Plamenevskii B. A., Ellipticheskie zadachi v oblastyakh s kusochno-gladkoi granitsei, Nauka, M., 1991

[7] Maz'ya V., Solov'ev A., “Boundary integral equations of the logarithmic potential theory for domains with peaks”, Atti Accad. Nazi. Lincei. Matem. e Applic. Ser. IX, 6:4 (1995), 211–236 | MR

[8] Khuskivadze G., Kokilashvili V., Paatashvili V., Boundary value problems for analytic and harmonic functions in domains with non-smooth boundaries. Application to conformal mappings, Memo. Different. Equations and Math. Phys., 14, 1998, 195 pp. | MR | Zbl

[9] Streng G., Fiks Dzh., Teoriya metoda konechnykh elementov, Mir, M., 1973 | Zbl

[10] Godunov S. K., Ryabenkii B. C., Raznostnye skhemy, Nauka, M., 1973 | Zbl

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

[12] Samarskii A. A., Teoriya raznostnykh skhem, Nauka, M., 1977 | MR | Zbl

[13] T. Kruz, F. Ritstso (red.), Metody granichnykh integralnykh uravnenii, Vychislitelnye aspekty i prilozheniya v mekhanike. Mekhanika, 15, Mir, M., 1978

[14] Oganesyan L. A., Rukhovets L. A., Variatsionno-raznostnye metody resheniya ellipticheskikh uravnenii, Izd-vo AN ArmSSR, Erevan, 1979

[15] Marchuk G. I., Agoshkov V. I., Vvedenie v proektsionno-setochnye metody, Nauka, M., 1981 | MR

[16] Brebbia K., Telles Zh., Vroubel L., Metody granichnykh elementov, Mir, M., 1987 | MR

[17] Szabó B. A., Babuška I., Finite element analysis, John Wiley and Sons, New York, 1991 | MR | Zbl

[18] Golovin G. T., Makarov M. H., Sablin M. N. i dr., “Sravnenie razlichnykh metodov resheniya zadachi Dirikhle dlya uravneniya Latslasa v slozhnykh.oblastyakh”, Zh. vychisl. matem. i matem. fiz., 27:11 (1987), 1662–1679 | MR

[19] Blum H., “Numerical treatment of corner and crack singularities”, Finite Element and Boundary Element Techn. from Math. and Engng Point of View, CISM Courses and Lect., 301, Springer, Vienna, 1988, 171–212 | MR

[20] Krauthammer Th., “Accuracy of the finite element method near acurved boundary”, Comput. and Structures, 10 (1979), 921–929 | DOI | Zbl

[21] MSC/PROBE, Techn. Overview: Release 4.1. The MacNeal-Schwengler Corp., January, 1990

[22] Vorovich I. I., “Nekotorye problemy kontsentratsii napryazhenii”, Kontsentratsiya napryazhenii, Vyp. 2, Hayk. dumka, Kiev, 1968, 45–53

[23] Sherman D. I., “O stanovlenii metoda integralnykh uravnenii v teorii uprugosti”, Predislovie k kn.: Parton V. Z., Perlin P. I., Integralnye uravneniya v teorii uprugosti, Nauka, M., 1977, 5–14

[24] Zielinski A. P., “On trial functions applied in the generalized Trefftz method”, Advances Engng Software, 24 (1995) | Zbl

[25] Melenk J. M., Babuška I., “Approximation with harmonic and generalized harmonic polynomials in the partition of unity method”, Comput. Assisted Mech. and Engng Sci., 4 (1997), 607–632 | Zbl

[26] Mikhlin S. G., Variatsionnye metody v matematicheskoi fizike, Nauka, M. | MR

[27] Vekua I. N., Novye metody resheniya ellipticheskikh uravnenii, Gostekhteorizdat, M., 1948 | MR

[28] Ryabenkii B. C., Metod raznostnykh potentsialov dlya nekotorykh zadach mekhaniki sploshnykh sred, Nauka, M., 1987 | MR

[29] Lebedev V. I., Agoshkov V. I., Operatory Puankare–Steklova i ikh primenenie v analize, Izd-vo AN SSSR, M., 1983 | Zbl

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

[31] Papamichael N., Kokkinos C. A., “The use of singular functions for the approximate conformal mapping of doublyconriecteddomains”, SIAM J. Sci. Statist. Comput., 5 (1984), 684–700 | DOI | MR | Zbl

[32] Costabel M., Dauge M., “Computation of corner singularities tin linear elasticity”, Boundary Value Problems and Integral Equations in Nonsmooth Domains, Marcel Dekker, New York etc., 1987, 59–68

[33] Leguillon D., Sanchez-Palencia E., Computation of singular splutions in elliptic problems andrelasticity, John Wiley and Sons, New York, 1987 | MR | Zbl

[34] Yosibash Z., Szabó B. A., “Numerical analysis of singularities in two-dimensions. Part 1: Computation of eigenpairs”, Internal. J. Numer. Meth. Engng., 38:12 (1995), 2655–2082 | MR

[35] Yosibash Z., “Numerical analysis on singular solutions of the Poisson equation in twordimensions”, Comput. Mech., 20:4 (1997), 320–330 | DOI | MR | Zbl

[36] Papamichael N., Pritsker I. E., Saff E. B., Stylianopoulos N. S., “Approximation of conformal mapping of annular regions”, Numer. Math., 76 (1997), 489–513 | DOI | MR | Zbl

[37] Babuška I., Kellog R. B., Pitkaranta J., “Direct and inverse error estimates for finite elements with mesh refinements”, Numer. Math., 33 (1979), 447–471 | DOI | MR | Zbl

[38] Apel Th., Sändig A.-M., Whiteman J. R., “Graded mesh refinement and error estimates for finite element solutions of elliptic boundary value problems in non-smooth domains”, Math. Meth. Appl. Sci., 19 (1996), 65–85 | 3.0.CO;2-S class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR

[39] Vlasov V. I., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, VTs AN SSSR, M., 1987 | MR

[40] Vlasov V. M., Kraevye zadachi v oblastyakh s krivolineinoi granitsei, Dis. $\dots$ dokt. fiz.-matem. nauk, VTs AN SSSR, M., 1990 | MR

[41] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1973 | MR

[42] Brelo M., Osnovy klassicheskoi teorii potentsiala, Mir, M., 1964 | MR | Zbl

[43] Henrici P., Applied and computational complex analysis, v. 1–3, John Wiley and Sons, New York, 1991 | MR

[44] Mazya V. G., Plamenevskii B. A., “O koeffitsientakh v asimptotike reshenii ellipticheskikh kraevykh zadach v oblastyakh s konicheskimi tochkami”, Math. Nachricht., 76 (1997), 29–60

[45] Szabó B. A., Yosibash Z., “Numerical analysis of singularities in two-dimensions. Part 2: Computation of the generalized flux/stress intensity factors”, Internat. J. Numer. Meth. Engng., 39:3 (1996), 409–434 | 3.0.CO;2-P class='badge bg-secondary rounded-pill ref-badge extid-badge'>DOI | MR | Zbl

[46] Trefethen L. N., Driscoll T. A., Proc. Internat. Congr. Math. (Berlin, 1998) | MR

[47] Blum H., Dobrovolski M., “On finite element methods for elliptic equations on domains with corners”, Computing, 28 (1982), 53–63 | DOI | MR | Zbl

[48] Babuška I., Miller A., “The post-processing approach in the finite element method – Part 2: The calculation of stress intensity factors”, Internat. J. Numer. Meth. Engng., 20 (1984), 1111–1129 | DOI | Zbl

[49] Brenner C. S., “Multigrid methods for the computation of singular solutions and stress intensity factors I: Corner singularities”, Math. Comput., 32 (1996), 135–148 | MR

[50] Gaier D., “Conformal modulus and their computation”, Comput. Meth. and Functional Theory (CMFT'94). World Scient., 1995, 159–171 | MR | Zbl

[51] Betsakos D., Samuelsson K., Vuorinen M., Computation of capacity of planar condensers, Preprint 222, May 1999, Univ. Helsinki, 1999 | MR

[52] Gaier D., Konstructive Methoden der konformen Abbildung, Berlin, 1964 | MR | Zbl

[53] Opfer G., Untere, beliebig verbesserbare Schranken für den Modul eines zweifach zusammenhängendeh Gebietes mit Hilfe von Differenzenverfahren, Diss., Hamburg, 1967

[54] Mizumoto H., “An application of Green's formula of a discrete function: Determination of periodicity moduli. I”, Kodai Math. Sem. Rept., 22 (1970), 231–243 ; II 244–249 | DOI | MR | Zbl | Zbl

[55] Weisel J., “Losung singularen Variationsprobleme durch die Verfahfen von Ritz und Galerkin mit finiten Elementen. Anwendungen in der konformen Abbildung”, Mitt. Math. Sem. Giessen, 138 (1979), 1–150 | MR

[56] Daeppen H., Die Schwarz–Christoffel–Abbildung fur zweifach zusammenhangende Gebiete mit Anwendungen, Ph.D. thesis, E.T.H., Zurich, 1988 | Zbl

[57] Volkov E. A., Block method for solving the Laplace equation and for constructing conformal mapping, RCR Press, London, 1994 | MR

[58] Betsakos D., Vuorinen M., Estimates of conformal capacity, Preprint 205, Univ. Helsinki. Rept. Dept. Math., 1998 | MR

[59] Hu C., “A software package for computing Schwarz–Christoffel conformal transformation for doubly connected polygonal regions”, ACM Trans. Math. Software, 24 (1998)

[60] Samuelsson K., A finite element method for computing conformal capacities, Preprint 252, Univ. Helsinki, 2000

[61] Laugesen R. I., Extremal problems involving logarithmic and Green capacity, Diss. St., Washington Univ., Louis, 1993 | MR

[62] Markushevich A. M., Teoriya analiticheskikh funktsii, v. I, II, Nauka, M., 1968 | Zbl

[63] Gilbarg D., Trudinger M., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[64] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973 | MR

[65] Guschin A. K., Mikhailov V. P., “O granichnykh znacheniyakh resheniya ellipticheskikh uravnenii”, Tr. Mezhdunar. konf. “Obobschennye funktsii i ikh primeneniya v matem. fiz.” (Moskva, 24–28 noyabrya 1980), Izd-vo AN SSSR, M., 1981, 189–206

[66] Goluzin G. M., Geometricheskaya teoriya funktsii kompleksnogo peremennogo, Nauka, M., 1966 | MR

[67] Duren P. L., Theory of $H^p$ spaces, Acad. Press, New York-London, 1970 | MR

[68] Gofman K., Banakhovy prostranstva analiticheskie funktsii, Izd-vo inostr. lit., M., 1963

[69] Tumarkin G. Ts., Khavinson S. Ya., “K opredeleniyu analiticheskikh funktsii klassa $E_p$ v mnogosvyaznykh oblastyakh”, Uspekhi matem. nauk, 13:1 (1958), 201–206 | MR | Zbl

[70] Vlasov V. I., “O prostranstvakh tipa Khardi garmonicheskikh funktsii”, Dokl. AN SSSR, 299:2 (1988), 272–276 | MR | Zbl

[71] Vlasov V. I., Rachkov A. V., “O vesovykh prostranstvakh tipa Khardi”, Dokl. RAN, 328:2 (1993), 281–284 | MR | Zbl

[72] Keldisch M., Lawrentieff M., “Sur la representation conforme des domaines limités par des courbes rectifiables”, Ann. Ecole Normale Super., 59 (1937), 1–38

[73] Akhiezer N. I., Elementy teorii ellipticheskikh funktsii, Nauka, M., 1970 | MR | Zbl