zbMATH — the first resource for mathematics

Boundary integral equations with the generalized Neumann kernel for Laplace’s equation in multiply connected regions. (English) Zbl 1211.65159
The paper is concerned with study of a boundary integral method for solving Laplace’s equation with a Dirichlet boundary condition or a Neumann condition on both bounded and unbounded multiply connected regions. The integral equations are solved numerically by the Nyström method with the trapezoidal rule. Numerical results illustrate the efficiency of the proposed method.

65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Algorithm 788
Full Text: DOI
[1] Atkinson, K.E., The solution of non-uniquely linear integral equations, Numer. math., 10, 117-124, (1967) · Zbl 0183.18305
[2] Atkinson, K.E., The numerical solution of the eigenvalue problem for compact integral operators, Trans. am. math. soc., 129, 458-465, (1967) · Zbl 0177.18803
[3] Atkinson, K.E., The numerical solution of integral equations of the second kind, (1997), Cambridge University Press Cambridge · Zbl 0155.47404
[4] Atkinson, K.E.; Jeon, Y., Algorithm 788: automatic boundary integral equation programs for the planar Laplace equation, ACM trans. math. software, 24, 4, 395-417, (1998) · Zbl 0934.65131
[5] Crowdy, D., The Schwarz problem in multiply connected domains and the schottky – klein prime function, Complex var. elliptic equ., 53, 221-236, (2008) · Zbl 1133.30302
[6] Gakhov, F.D., Boundary value problems, English translation of Russian edition 1963, (1966), Pergamon Press Oxford · Zbl 0141.08001
[7] Greenbaum, A.; Greengard, L.; McFadden, G.B., Laplace’s equation and the dirichlet – neumann map in multiply connected domains, J. comput. phys., 105, 2, 267-278, (1993) · Zbl 0769.65085
[8] Helsing, J.; Ojala, R., On the evaluation of layer potentials close to their sources, J. comput. phys., 227, 2899-2921, (2008) · Zbl 1135.65404
[9] Helsing, J.; Wadbro, E., Laplace’s equation and the dirichlet – neumann map: a new mode for mikhlin’s method, J. comput. phys., 202, 391-410, (2005) · Zbl 1063.65130
[10] Henrici, P., Applied and computational complex analysis, Vol. 3, (1986), John Wiley New York
[11] Kellogg, O., Foundations of potential theory, (1967), Springer-Verlag Berlin · Zbl 0152.31301
[12] Krantz, S.G., Geometric function theory: explorations in complex analysis, (2006), Birkhäuser Boston · Zbl 1086.30001
[13] Kress, R., Linear integral equations, (1989), Springer-Verlag Berlin
[14] Mayo, A., Rapid methods for the conformal mapping of multiply connected regions, J. comput. appl. math., 14, 143-153, (1986) · Zbl 0586.30009
[15] Mikhlin, S.G., Integral equations and their applications to certain problems in mechanics, mathematical physics and technology, English translation of Russian edition 1948, (1957), Pergamon Press Armstrong · Zbl 0077.09903
[16] Muskhelishvili, N.I., Singular integral equations, English translation of Russian edition 1953, (1977), Noordhoff Leyden
[17] Nasser, M.M.S., A boundary integral equation for conformal mapping of bounded multiply connected regions, Comput. methods func. theor., 9, 127-143, (2009) · Zbl 1159.30007
[18] Nasser, M.M.S., Numerical conformal mapping via a boundary integral equation with the generalized Neumann kernel, SIAM J. sci. comput., 31, 1695-1715, (2009) · Zbl 1198.30009
[19] Nasser, M.M.S., The Riemann-Hilbert problem and the generalized Neumann kernel on unbounded multiply connected regions, The university researcher (IBB university journal), 20, 47-60, (2009)
[20] Nasser, M.M.S.; Murid, A.H.M.; Zamzamir, Z., A boundary integral method for the riemann – hilbert problem in domains with corners, Complex var. elliptic equ., 53, 11, 989-1008, (2008) · Zbl 1159.30023
[21] Pogorzelski, W., Integral equation and their applications, Vol. 1, (1966), Pergamon Press Oxford · Zbl 0137.30502
[22] Saad, Y.; Schultz, M.H., GMRES: a generalized minimum residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[23] Vekua, I.N., Generalized analytic functions, (1992), Pergamon London · Zbl 0698.47036
[24] Wegmann, R.; Murid, A.H.M.; Nasser, M.M.S., The riemann – hilbert problem and the generalized Neumann kernel, J. comput. appl. math., 182, 388-415, (2005) · Zbl 1070.30017
[25] Wegmann, R.; Nasser, M.M.S., The riemann – hilbert problem and the generalized Neumann kernel on multiply connected regions, J. comput. appl. math., 214, 36-57, (2008) · Zbl 1157.45303
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.