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On the evaluation of layer potentials close to their sources. (English) Zbl 1135.65404
Summary: When solving elliptic boundary value problems using integral equation methods one may need to evaluate potentials represented by a convolution of discretized layer density sources against a kernel. Standard quadrature accelerated with a fast hierarchical method for potential field evaluation gives accurate results far away from the sources. Close to the sources this is not so. Cancellation and nearly singular kernels may cause serious degradation.
This paper presents a new scheme based on a mix of composite polynomial quadrature, layer density interpolation, kernel approximation, rational quadrature, high polynomial order corrected interpolation and differentiation, temporary panel mergers and splits, and a particular implementation of the GMRES solver. Criteria for which mix is fastest and most accurate in various situations are also supplied. The paper focuses on the solution of the Dirichlet problem for Laplace’s equation in the plane. In a series of examples we demonstrate the efficiency of the new scheme for interior domains and domains exterior to up to 2000 close-to-touching contours. Densities are computed and potentials are evaluated, rapidly and accurate to almost machine precision, at points that lie arbitrarily close to the boundaries.

MSC:
65N38 Boundary element methods for boundary value problems involving PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Software:
Algorithm 788
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[1] Atkinson, K.; Jeon, Y., Algorithm 788: automatic boundary integral equation program for the planar Laplace equation, ACM trans. math. software, 24, 4, 395-417, (1998) · Zbl 0934.65131
[2] Barrett, R.; Berry, M.; Chan, T.F.; Demmel, J.; Donato, J.; Dongarra, J.; Eijkhout, V.; Pozo, R.; Romine, C.; Van der Vorst, H., Templates for the solution of linear systems: building blocks for iterative methods, (1994), SIAM Philadelphia, PA
[3] Beale, J.T.; Lai, M.-C., A method for computing nearly singular integrals, SIAM J. numer. anal., 38, 6, 1902-1925, (2001) · Zbl 0988.65025
[4] Biros, G.; Ying, L.; Zorin, D., A fast solver for the Stokes equations with distributed forces in complex geometries, J. comput. phys., 193, 1, 317-348, (2004) · Zbl 1047.76065
[5] Cheng, H.; Greengard, L., On the numerical evaluation of electrostatic fields in dense random dispersions of cylinders, J. comput. phys., 136, 2, 629-639, (1997) · Zbl 0903.65099
[6] Cristini, V.; Lowengrub, J.; Nie, Q., Nonlinear simulation of tumor growth, J. math. biol., 46, 3, 191-224, (2003) · Zbl 1023.92013
[7] Dejoie, A.; Mogilevskaya, S.G.; Crouch, S.L., A boundary integral method for multiple circular holes in an elastic half-plane, Eng. anal. bound. elem., 30, 6, 450-464, (2006) · Zbl 1195.74223
[8] Dutt, A.; Gu, M.; Rokhlin, V., Fast algorithms for polynomial interpolation, integration, and differentiation, SIAM J. numer. anal., 33, 5, 1689-1711, (1996) · Zbl 0862.65005
[9] Englund, J., A Nyström scheme with rational quadrature applied to edge crack problems, Commun. numer. methods eng., 23, 10, 945-960, (2007) · Zbl 1130.74056
[10] Englund, J., A higher order scheme for two-dimensional quasi-static crack growth simulations, Comput. methods appl. mech. eng., 196, 21-24, 2527-2538, (2007) · Zbl 1173.74470
[11] Farina, L., Evaluation of single layer potentials over curved surfaces, SIAM J. sci. comp., 23, 1, 81-91, (2001) · Zbl 0990.65131
[12] Gautschi, W., The use of rational functions in numerical quadrature, J. comput. appl. math., 133, 1-2, 111-126, (2001) · Zbl 0985.65017
[13] Golub, G.H.; Van Loan, C.F., Matrix computations, (1989), The John Hopkins University Press Baltimore · Zbl 0733.65016
[14] Greengard, L.; Kropinski, M.C., Integral equation methods for Stokes flow in doubly-periodic domains, J. eng. math., 48, 2, 157-170, (2004) · Zbl 1046.76011
[15] Greengard, L.; Lee, J.-Y., Electrostatics and heat conduction in high contrast composite materials, J. comput. phys., 211, 1, 64-76, (2006) · Zbl 1129.78005
[16] Greengard, L.; Rokhlin, V., A fast algorithm for particle simulations, J. comput. phys., 73, 2, 325-348, (1987) · Zbl 0629.65005
[17] 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
[18] Heath, M.T., Scientific computing: an introductory survey, (2002), McGraw-Hill New York · Zbl 0903.68072
[19] Helsing, J., Thin bridges in isotropic electrostatics, J. comput. phys., 127, 1, 142-151, (1996) · Zbl 0862.65079
[20] Helsing, J.; Wadbro, E., Laplace’s equation and the dirichlet – neumann map: a new mode for mikhlin’s method, J. comput. phys., 202, 2, 391-410, (2005) · Zbl 1063.65130
[21] Ioakimidis, N.I.; Papadakis, K.E.; Perdios, E.A., Numerical evaluation of analytic functions by cauchy’s theorem, BIT numer. math., 31, 2, 276-285, (1991) · Zbl 0737.65011
[22] Jonsson, A., Discrete dislocation dynamics by an O(N) algorithm, Comput. mater. sci., 27, 3, 271-288, (2003)
[23] Jou, H.-J.; Leo, P.H.; Lowengrub, J.S., Microstructural evolution in inhomogeneous elastic media, J. comput. phys., 131, 1, 109-148, (1997) · Zbl 0880.73050
[24] Khayat, M.A.; Wilton, D.R., Numerical evaluation of singular and near-singular potential integrals, IEEE trans. antennas propagat., 53, 10, 3180-3190, (2005)
[25] Martinsson, P.G., Fast evaluation of electro-static interactions in multi-phase dielectric media, J. comput. phys., 211, 1, 289-299, (2006) · Zbl 1079.78007
[26] Mayo, A., Fast high-order accurate solution of laplace’s equation on irregular regions, SIAM J. sci. stat. comput., 6, 1, 144-157, (1985) · Zbl 0559.65082
[27] Mayo, A.; Greenbaum, A., Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions, J. comput. phys., 220, 2, 900-914, (2007) · Zbl 1109.65028
[28] McKenney, A., An adaptation of the fast multipole method for evaluating layer potentials in two dimensions, Comput. math. appl., 31, 1, 33-57, (1996) · Zbl 0853.65124
[29] Monegato, G.; Palamara Orsi, A., Product formulas for Fredholm integral equations, (), 140-156
[30] Saad, Y.; Schultz, M.H., GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. sci. stat. comput., 7, 3, 856-869, (1986) · Zbl 0599.65018
[31] Sethian, J.A.; Wilkening, J., A numerical model of stress driven grain boundary diffusion, J. comput. phys., 193, 1, 275-305, (2004) · Zbl 1117.74302
[32] Thornton, K.; Akaiwa, N.; Voorhees, P.W., Large-scale simulations of ostwald ripening in elastically stressed solids: I. development of microstructure, Acta mater., 52, 5, 1353-1364, (2004)
[33] Tsamasphyros, G.; Theotokoglou, E.E., A quadrature formula for integrals with nearby singularities, Int. J. numer. methods eng., 67, 8, 1082-1093, (2006) · Zbl 1113.74086
[34] Walker, H.F., Implementation of the GMRES method using Householder transformations, SIAM J. sci. stat. comput., 9, 1, 152-163, (1988) · Zbl 0698.65021
[35] Weideman, J.A.C.; Laurie, D.P., Quadrature rules based on partial fraction expansions, Numer. alg., 24, 1-2, 159-178, (2000) · Zbl 0962.65020
[36] Yarvin, N.; Rokhlin, V., Generalized Gaussian quadratures and singular value decompositions of integral operators, SIAM J. sci. comput., 20, 2, 699-718, (1998) · Zbl 0932.65020
[37] Ying, L.; Biros, G.; Zorin, D., A high-order 3D boundary integral equation solver for elliptic PDEs in smooth domains, J. comput. phys., 219, 1, 247-275, (2006) · Zbl 1105.65115
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