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Corner singularities for elliptic problems: Integral equations, graded meshes, quadrature, and compressed inverse preconditioning. (English) Zbl 1152.65114

Summary: We take a fairly comprehensive approach to the problem of solving elliptic partial differential equations numerically using integral equation methods on domains where the boundary has a large number of corners and branching points. Use of non-standard integral equations, graded meshes, interpolatory quadrature, and compressed inverse preconditioning are techniques that are explored, developed, mixed, and tested on some familiar problems in materials science.

The recursive compressed inverse preconditioning, the major novelty of the paper, turns out to be particularly powerful and, when it applies, eliminates the need for mesh grading completely. In an electrostatic example for a multiphase granular material with about two thousand corners and triple junctions and a conductivity ratio between phases up to a million we compute a common functional of the solution with an estimated relative error of 10 -12 . In another example, five times as large but with a conductivity ratio of only a hundred, we achieve an estimated relative error of 10 -14 .

65N38Boundary element methods (BVP of PDE)
35Q60PDEs in connection with optics and electromagnetic theory
78A30Electro- and magnetostatics
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
78M15Boundary element methods (optics)
[1]Atkinson, K. E.: The numerical solution of integral equations of the second kind, (1997)
[2]Cheng, H.; Greengard, L.: On the numerical evaluation of electrostatic fields in dense random dispersions of cylinders, J. comput. Phys. 136, No. 2, 629-639 (1997) · Zbl 0903.65099 · doi:10.1006/jcph.1997.5787
[3]Cheng, H.; Rokhlin, V.; Yarvin, N.: Nonlinear optimization, quadrature, and interpolation, SIAM J. Optim. 9, No. 4, 901-923 (1999) · Zbl 1032.90528 · doi:10.1137/S1052623498349796
[4]Chen, Y. Z.: Stress intensity factors for curved and kinked cracks in plane extension, Theor. appl. Fract. mech. 31, No. 3, 223-232 (1999)
[5]Craster, R. V.; Obnosov, Y. V.: Checkerboard composites with separated phases, J. math. Phys. 42, No. 11, 5379-5388 (2001) · Zbl 1063.82016 · doi:10.1063/1.1398336
[6]Edwards, R. G.; Goodman, J.; Sokal, A. D.: Multigrid method for the random-resistor problem, Phys. rev. Lett. 61, No. 12, 1333-1335 (1988)
[7]J. Englund, Large scale computations for cracks with corners, in: R. Gallego, M.H. Aliabadi (Eds.), Advances in Boundary Element Techniques, BeTeQ IV, Queen Mary University of London, 2003, pp. 71 – 76.
[8]Englund, J.: Fast accurate and stable algorithm for the stress field around a zig-zag-shaped crack, Eng. fract. Mech. 70, No. 2, 355-364 (2003)
[9]Englund, J.: A higher order scheme for two-dimensional quasi-static crack growth simulations, Comput. methods appl. Mech. eng. 196, No. 21-24, 2527-2538 (2007) · Zbl 1173.74470 · doi:10.1016/j.cma.2007.01.007
[10]Fel, L. G.; Kaganov, I. V.: Relation between effective conductivity and susceptibility of two-component rhombic checkerboard, J. phys. A 36, No. 19, 5349-5358 (2003) · Zbl 1032.78013 · doi:10.1088/0305-4470/36/19/311
[11]Greengard, L.; Lee, J. -Y.: Electrostatics and heat conduction in high contrast composite materials, J. comput. Phys. 211, No. 1, 64-76 (2006) · Zbl 1129.78005 · doi:10.1016/j.jcp.2005.05.004
[12]Greengard, L.; Moura, M.: On the numerical evaluation of electrostatic fields in composite materials, acta numerica 1994, (1994) · Zbl 0815.73047
[13]Greengard, L.; Rokhlin, V.: A fast algorithm for particle simulations, J. comput. Phys. 73, No. 2, 325-348 (1987) · Zbl 0629.65005 · doi:10.1016/0021-9991(87)90140-9
[14]Helsing, J.; Jonsson, A.: On the computation of stress fields on polygonal domains with V-notches, Int. J. Num. meth. Eng. 53, No. 2, 433-453 (2002) · Zbl 1112.74385 · doi:10.1002/nme.291
[15]Helsing, J.; Jonsson, A.: A seventh order accurate and stable algorithm for the computation of stress inside cracked rectangular domains, Int. J. Multiscale comput. Eng. 2, No. 1, 47-68 (2004)
[16]Helsing, J.; Ojala, R.: On the evaluation of layer potentials close to their sources, J. comput. Phys. 227, No. 5, 2899-2921 (2008) · Zbl 1135.65404 · doi:10.1016/j.jcp.2007.11.024
[17]Helsing, J.; Peters, G.: Integral equation methods and numerical solutions of crack and inclusion problems in planar elastostatics, SIAM J. Appl. math. 59, No. 3, 965-982 (1999) · Zbl 0934.74006 · doi:10.1137/S0036139998332938
[18]Henderson, H. V.; Searle, S. R.: On deriving the inverse of a sum of matrices, SIAM rev. 23, No. 1, 53-60 (1981) · Zbl 0451.15005 · doi:10.1137/1023004
[19]Kolm, P.; Jiang, S. D.; Rokhlin, V.: Quadruple and octuple layer potentials in two dimensions I: Analytical apparatus, Appl. comput. Harmon. anal. 14, No. 1, 47-74 (2003) · Zbl 1139.35397 · doi:10.1016/S1063-5203(03)00004-6
[20]Kumar, P.; Nukala, V. V.; Šimunović, S.; Guddati, M. N.: An efficient algorithm for modelling progressive damage accumulation in disordered materials, Int. J. Num. meth. Eng. 62, No. 14, 1982-2008 (2005) · Zbl 1080.74565 · doi:10.1002/nme.1257
[21]Kress, R.; Sloan, I. H.; Stenger, F.: A sinc quadrature method for the double-layer integral equation in planar domains with corners, J. integr. Eqn. appl. 10, No. 3, 291-317 (1998) · Zbl 0916.65135 · doi:10.1216/jiea/1181074232 · doi:http://math.la.asu.edu/~rmmc/jie/JIE10-3/CONT10-3/CONT10-3.html
[22]Martinsson, P. G.; Rokhlin, V.: A fast direct solver for boundary integral equations in two dimensions, J. comput. Phys. 205, No. 1, 1-23 (2005) · Zbl 1078.65112 · doi:10.1016/j.jcp.2004.10.033
[23]Mayo, A.; Greenbaum, A.: Fourth order accurate evaluation of integrals in potential theory on exterior 3D regions, J. comput. Phys 220, No. 2, 900-914 (2007) · Zbl 1109.65028 · doi:10.1016/j.jcp.2006.05.042
[24]Muskhelishvili, N. I.: Some basic problems of the mathematical theory of elasticity, (1953) · Zbl 0052.41402
[25]Saad, Y.; Schultz, M. H.: GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. stat. Comp. 7, No. 3, 856-869 (1986) · Zbl 0599.65018 · doi:10.1137/0907058
[26]Sethian, J. A.; Wilkening, J.: A numerical model of stress driven grain boundary diffusion, J. comput. Phys. 193, No. 1, 275-305 (2004) · Zbl 1117.74302 · doi:10.1016/j.jcp.2003.08.015
[27]Tuncer, E.; Serdyuk, Y. V.; Gubanski, S. M.: Dielectric mixtures: electrical properties and modeling, IEEE trans. Diel. electr. Insul. 9, No. 5, 809-828 (2002)
[28]Yavuz, A. K.; Phoenix, S. L.; Termaath, S. C.: An accurate and fast analysis for strongly interacting multiple crack configurations including kinked (V) and branched (Y) cracks, Int. J. Solids struct. 43, No. 22-23, 6727-6750 (2006) · Zbl 1120.74763 · doi:10.1016/j.ijsolstr.2006.02.005
[29]Ying, L.; Biros, G.; Zorin, D.: A high-order 3D boundary integral equation solver for elliptic pdes in smooth domains, J. comput. Phys. 219, No. 1, 247-275 (2006) · Zbl 1105.65115 · doi:10.1016/j.jcp.2006.03.021