Wavelet collocation methods for a first kind boundary integral equation in acoustic scattering

Advances in Computational Mathematics

Volumn 17, Issue 4, 2002, Pages 281-308

  Hsiao, George C ^a   Rathsfeld, Andreas ^b  

^a UNIVERSITY OF DELAWARE   (United States)

^b Weierstra Inst Angew Anal/S   (Germany)

Author keywords

First kind integral operators; Piecewise constant collocation; Wavelet algorithm

Indexed keywords

EID: 0141514730     PISSN: 10197168     EISSN: None     Source Type: Journal    
DOI: 10.1023/A:1016249415373     Document Type: Article

References (30)

1. G. Beylkin, R. Coifman and V. Rokhlin, Fast wavelet transforms and numerical algorithms I, Comm. Pure Appl. Math. 44 (1991) 141-183.
2. C.K. Chui, An Introduction to Wavelets (Academic Press, Boston, 1992).
3. A. Cohen and R. Masson, Wavelet methods for second order problems, preconditioning, and adaptivity, SIAM J. Sci. Comput. 21 (1999) 1006-1026.
4. D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Applied Mathematical Sciences, Vol. 93 (Springer, Berlin/Heidelberg, 1992).
5. W. Dahmen, Stability of multiscale transformations, J. Fourier Anal. Appl. 2 (1996) 341-361.
6. W. Dahmen and A. Kunoth, Multilevel preconditioning, Numer. Math. 63 (1992) 315-344.
7. W. Dahmen, S. Prößdorf and R. Schneider, Wavelet approximation methods for pseudo-differential equations II: Matrix compression and fast solution, Adv. Comput. Math. 1 (1993) 259-335.
8. W. Dahmen, S. Prößdorf and R. Schneider, Wavelet approximation methods for pseudo-differential equations I: Stability and convergence, Math. Zeitschr. 215 (1994) 583-620.
9. T. Daubechies, Ten Lectures on Wavelets, CBMS Lecture Notes, Vol. 61 (SIAM, Philadelphia, PA, 1992).
10. S. Ehrich and A. Rathsfeld, Piecewise linear wavelet collocation on triangular grids, approximation of the boundary manifold and quadrature, WIAS preprint, No. 433 (1998).
11. L. Greengard and V. Rokhlin, A new version of the fast multipole method for the Laplace equation in three dimensions, Acta Numerica (1997) 229-269.
12. A. Haar, Zur Theorie der orthogonalen Funktionensysteme, Math. Ann. 69 (1910) 331-371.
13. W. Hackbusch and Z.P. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math. 54 (1989) 463-491.
14. A. Harten and I. Yad-Shalom, Fast multiresolution algorithms for matrix-vector multiplication, SIAM J. Numer. Anal. 31 (1994) 1191-1218.
15. G.C. Hsiao, The finite element method for a class of improperly posed integral equations, in: Improperly Posed Problems and Their Numerical Treatment, eds. G. Hammerlin and K.H. Hoffmann (Birkhäuser, Basel, 1983) pp. 117-131.
16. G.C. Hsiao and W.L. Wendland, Boundary integral equation methods in low frequency acoustics, J. Chinese Inst. Engrg. 23 (2000) 369-375.
17. C. Lage and C. Schwab, Wavelet Galerkin algorithm for boundary integral equations, SIAM J. Sci. Comput. 20 (1999) 2195-2222.
18. P.L. Levin, R. Schneider and M. Spasojević, Creation of sparse boundary element matrices and axi-symmetric electrostatics problems using biorthogonal Haar wavelet, IEEE Trans. Dielectrics Electrical Insulation 5 (1998) 469-484.
19. P.L. Levin, R. Schneider and M. Spasojević, Multiscale compression of BEM equations for electrostatic systems, IEEE Trans. Dielectrics Electric Insulation 3 (1996) 482-493.
20. P.L. Levin, R. Schneider and M. Spasojević, On the creation of sparse boundary element matrices for two-dimensional electrostatics problems using the orthogonal Haar wavelet, IEEE Trans. Dielectrics Electric Insulation 4 (1997) 249-258.
21. C.A. Micchelli, Y. Xu and Y. Zhao, Wavelet Galerkin methods for second-kind integral equations, J. Comput. Appl. Math. 86 (1997) 251-270.
22. T.V. Petersdorff, R. Schneider and C. Schwab, Multiwavelets for second-kind integral equations, SIAM J. Numer. Anal. 34 (1997) 2212-2227.
23. T.V. Petersdorff and C. Schwab, Fully discrete multiscale Galerkin BEM, in: Multiresolution Analysis and PDE, eds. A.J. Kurdial and P. Oswald, Series: Wavelet Analysis and its Applications (Academic Press, San Diego, CA, 1997) pp. 287-346.
24. T.V. Petersdorff and C. Schwab, Wavelet approximations for first kind boundary integral equations on polygons, Numer. Math. 74 (1996) 479-516.
25. A. Rathsfeld, A wavelet algorithm for the boundary element solution of a geodetic boundary value problem, Comput. Methods Appl. Mech. Engrg. 157 (1998) 267-287.
26. V. Rokhlin, End-point corrected trapezoidal quadrature rules for singular functions, Comput. Math. Appl. 20 (1990) 51-62.
27. V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys. 60 (1983) 187-207.
28. Y. Saad and M.H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Statist. Comput. 7 (1986) 856-869.
29. R. Schneider, Multiskalen- und Waveletkompression: Analysisbasierte Methoden zur effizienten Lösung großer vollbesetzter Gleichungssysteme, Advances in Numerical Analysis (Teubner, Stuttgart/ Leipzig, 1998).
30. T. Tran, E.P. Stephan and S. Zaprianov, Wavelet based preconditioners for boundary integral operators, Adv. Comput. Math. 9 (1998) 233-249.