Quadrature methods for 2D and 3D problems

Journal of Computational and Applied Mathematics

Volumn 125, Issue 1-2, 2000, Pages 439-460

  Rathsfeld, A ^a  

^a WEIERSTRASS INSTITUTE FOR APPLIED ANALYSIS AND STOCHASTICS   (Germany)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTATIONAL COMPLEXITY; CONVERGENCE OF NUMERICAL METHODS; NUMERICAL ANALYSIS;

PSEUDODIFFERENTIAL EQUATIONS; QUADRATURE METHODS;

INTEGRAL EQUATIONS;

EID: 0034501943     PISSN: 03770427     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0377-0427(00)00484-2     Document Type: Article

References (60)

1. B.K. Alpert, Hybrid Gauss-trapezoidal quadrature rules, SIAM J. Sci. Statist. Comput. 20 (1999) 1551-1584.
2. B.A. Amosov, On the approximate solution of elliptic pseudodifferential equations on smooth curves, Z. Anal. Angew. 9 (1990) 545-563.
3. K.E. Atkinson, The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997.
4. C.T.H. Baker, The Numerical Treatment of Integral Equations, Oxford University Press, London, 1977.
5. S.M. Belotserkovskij, I.K. Lifanov, Numerical Methods in Singular Integral Equations and their Application to Aerodynamics, Elasticity Theory and Electrodynamics, Nauka, Moscow, 1985 (in Russian).
6. B. Bialecki, Y. Yan, A rectangular quadrature method for logarithmically singular integral equations of the first kind, J. Integral Equations Appl. 4 (1992) 337-369.
7. K. Bühring, A quadrature method for a hypersingular integral equation on an interval, J. Integral Equations Appl. 7 (1995) 263-301.
8. G.A. Chandler, C.B. Schneider, B. Weilbächer, A quadrature method for a Cauchy singular integral equation with weakly singular solution, Z. Angew. Math. Mech. 75 (Suppl. 2) (1995) 619-620.
9. G. Chen, J. Zhou, Boundary Element Methods, Academic Press, London, 1992.
10. D. Chien, Numerical evaluation of surface integrals in three dimensions, Math. Comp. 64 (1995) 727-743.
11. D. Chien, K. Atkinson, A discrete Galerkin method for a hypersingular boundary integral equation, IMA J. Numer. Anal. 17 (1997) 463-478.
12. M. Costabel, W. McLean, Spline collocation for strongly elliptic equations on the torus, Numer. Math. 62 (1992) 511-538.
13. W. Dahmen, S. Prößdorf, R. Schneider, Wavelet approximation for pseudodifferential equations I: Stability and convergence, Math. Z. 215 (1994) 583-620.
14. S. Ehrich, A. Rathsfeld, Piecewise linear wavelet collocation on triangular grids, Approximation of the boundary manifold and quadrature, WIAS preprint, 434, Berlin, 1998.
15. J. Elschner, On spline approximation for a class of non-compact integral equations, Math. Nachr. 146 (1990) 271-321.
16. J. Elschner, I.G. Graham, Numerical methods for integral equations of Mellin type, this issue, J. Comp. Appl. Meth. 250 (2000) 423-437.
17. A. Gerasoulis, Nyström's iterative variant methods for the solution of Cauchy singular integral equations, SIAM J. Numer. Anal. 26 (1989) 430-441.
18. M.A. Golberg, C.S. Chen, Discrete Projection Methods, Computational Mechanics Publications, Southampton, 1997.
19. I. Graham, W. Hackbusch, S. Sauter, Discrete boundary element methods on general meshes in 3D, Bath mathematics preprints 97/05, 1997.
20. I. Graham, W. Hackbusch, S. Sauter, Hybrid Galerkin boundary elements on degenerate meshes, Proceedings of Minisymposium on Mathematical Aspects of Boundary Elements Methods in honour of V.G. Maz'ya, Paris, 1998.
21. I. Graham, W. Hackbusch, S. Sauter, Hybrid Galerkin boundary elements: Theory and Implementation, Ber. Math. Sem. Kiel 98-6, 1998.
22. 3, SIAM J. Numer. Anal. 29 (1992) 1347-1369.
23. W. Hackbusch, Integral Equations. Theory and Numerical Treatment, Birkhäuser, Basel, 1995.
24. W. Hoppe, Stabilität von Splineapproximationsverfahren für singuläre Integralgleichungen auf kompakten, glatten Mannigfaltigkeiten ohne Rand, Dissertationsschrift, TU Chemnitz, 1994.
25. L. Hörmander, The Analysis of Linear Partial Differential Equations. III Pseudo-Differential Operators, Springer, Berlin, 1985.
26. C.G.L. Johnson, L.R. Scott, An analysis of quadrature errors in second-kind boundary integral methods, SIAM J. Numer. Anal. 26 (1989) 1356-1382.
27. R. Kieser, B. Kleemann, A. Rathsfeld, On a full discretization scheme for a hypersingular boundary integral equation over smooth curves, Z. Anal. Angew. 4 (1992) 385-396.
28. R. Kieser, C. Schwab, W.L. Wendland, Numerical evaluation of singular and finite-part integrals on curved surfaces using symbolic manipulation, Computing 49 (1992) 279-301.
29. R. Kress, On the numerical solution of a hypersingular integral equation in scattering theory, J. Comput. Appl. Math. 61 (1995) 345-360.
30. R. Kress, I.H. Sloan, On the numerical solution of a logarithmic integral equation of the first kind, Numer. Math. 66 (1993) 199-214.
31. R. Kussmaul, Ein numerisches Verfahren zur Lösung des Neumannschen Außenraumproblems für die Helmholtzsche Schwingungsgleichung, Computing 4 (1969) 246-273.
32. I.K. Lifanov, Singular Integral Equations and Discrete Vortices, Utrecht, VSP ix, 1996.
33. G. Mastroianni, G. Monegato, Nyström interpolants based on the zeros of Legendre polynomials for a non-compact integral operator equation, IMA J. Numer. Anal. 14 (1993) 81-95.
34. V.G. Mazya, Boundary Integral Equations, Encyclopaedia of Math. Sciences, Vol. 27, Analysis IV, Springer, Berlin, 1991.
35. W. McLean, Fully-discrete collocation methods for an integral equation of the first kind, J. Integral Equations Appl. 6 (1994) 537-571.
36. W. McLean, I.H. Sloan, A fully discrete and symmetric boundary element method, IMA J. Numer. Anal. 14 (1994) 311-345.
37. G. Monegato, L. Scuderi, High order methods for weakly singular integral equation with non-smooth input functions, Math. Comp. 67 (1998) 1493-1515.
38. H.N. Mülthei, C.B. Schneider, Discrete collocation for a first kind Cauchy singular integral equation with weakly singular solution, J. Integral Equations Appl. 9 (1997) 341-371.
39. E.J. Nyström, Über die praktische Auflösung von linearen Integralgleichungen mit Anwendungen auf Randwertaufgaben der Potentialtheorie, Comment. Phys.-Math. 4 (1928) 1-25.
40. F. Penzel, Error estimates for discretized Galerkin method for a boundary integral equation in two dimensions, Numer. Methods Partial Differential Equations 8 (1992) 405-421.
41. 3, Numerical Analysis and Mathematical modelling, Banach Center Publications 29 (1994) 115-134.
42. S. Prößdorf, R. Schneider, A spline collocation method for multidimensional strongly elliptic pseudodifferential operators of order zero, Integral Equations Operator Theory 14 (1991) 399-435.
43. S. Prößdorf, R. Schneider, Spline approximation methods for multidimensional periodic pseudodifferential equations, Integral Equations Operator Theory 15 (1992) 287-626
44. S. Prößdorf, B. Silbermann, Numerical Analysis of Integral and Related Operator Equations, Akademie Verlag, Berlin, 1991.
45. A. Rathsfeld, On quadrature methods for the double layer potential equation over the boundary of a polyhedron, Numer. Math. 66 (1993) 67-95.
46. A. Rathsfeld, Piecewise polynomial collocation for the double layer potential equation over polyhedral boundaries, in: M. Dauge, M. Costabel, S. Nicaise (Eds.), Boundary Value Problems and Integral Equations in Non-Smooth Domains, Lecture Notes in Applied Mathematics, Marcel Dekker, Basel, 1994, pp. 219-253.
47. A. Rathsfeld, Nyström's method and iterative solvers for the solution of the double layer potential equation over polyhedral boundaries, SIAM J. Numer. Anal. 32 (1995) 924-951.
48. I. Saad Abdel-Fattah, Stability analysis of quadrature methods for two-dimensional singular equations, Ph.D. Thesis, University of El-Mansoura, 1996.
49. J. Saranen, On the effect of numerical quadratures in solving boundary integral equations, Notes on Numerical Fluid Mechanics 21 (1988) 196-209.
50. J. Saranen, The modified quadrature method for logarithmic-kernel integral equations on closed curves, J. Integral Equations Appl. 3 (1991) 575-600.
51. J. Saranen, L. Schroderus, Quadrature methods for strongly elliptic equations of negative order on smooth closed curves, SIAM J. Numer. Anal. 30 (1993) 1769-1795.
52. J. Saranen, I.H. Sloan, Quadrature methods for logarithmic-kernel integral equations on closed curves, IMA J. Numer. Anal. 12 (1992) 167-187.
53. S. Sauter, Cubature techniques for 3-D Galerkin BEM, in: W. Hackbusch, G. Wittum (Eds.), Boundary Elements: Implementation and Analysis of Advanced Algorithms, Notes on Numerical Fluid Mechanics, Vol. 54, Vieweg Verlag, Braunschweig, Wiesbaden, 1996.
54. C.B. Schneider, Inversion formulas for the discrete Hubert transform on the unit circle, SIAM J. Numer. Anal. 35 (1998) 71-77.
55. C. Schwab, Variable order composite quadrature of singular and nearly singular integrals, Computing 53 (1994) 173-194.
56. C. Schwab, W.L. Wendland, On numerical cubatures of singular surface integrals in boundary element methods, Numer. Math. 62 (1992) 343-369.
57. I.H. Sloan, B.J. Burn, An unconventional quadrature method for logarithmic kernel integral equations on closed curves, J. Integral Equations Appl. 4 (1992) 117-151.
58. F. Stenger, Numerical Methods Based on Sine and Analytic Functions, Springer Series in Computational Mathematics, Springer, New York, 1993.
59. G. Vainikko, Periodic integral and pseudodifferential equations, Helsinki Univ. of Technology, Inst. of Math., Research Report C13, Helsinki, 1996.
60. 3 mit Hilfe von Einfach- und Doppelschichtpotentialen, Numer. Math. 11 (1968) 308-404.