Multiscale Methods for Boundary Integral Equations and their Application to Boundary Value Problems in Scattering Theory and Geodesy
- Kleemann, Bernd H.
- Rathsfeld, Andreas
- Schneider, Reinhold
2010 Mathematics Subject Classification
- 45L10 65R20 65N38
- Multiscale methods, wavelet algorithm, pseudodifferential equations, Galerkin method, collocation
In the present paper we give an overview on multiscale algorithms for the solution of boundary integral equations which are based on the use of wavelets. These methods have been introduced first by Beylkin, Coifman, and Rokhlin . They have been developed and thoroughly investigated in the work of Alpert , Dahmen, Proessdorf, Schneider [16-19], Harten, Yad-Shalom , v.Petersdorff, Schwab [33-35], and Rathsfeld [39-40]. We describe the wavelet algorithm and the theoretical results on its stability, convergence, and complexity. Moreover, we discuss the application of the method to the solution of a two-dimensional scattering problem of acoustic or electromagnetic waves and to the solution of a fixed geodetic boundary value problem for the gravity field of the earth. The computational tests confirm the high compression rates and the saving of computation time predicted by the theory.
- Notes on Numerical Fluid Mechanics Vol. 54, Proceedings of the 12th GAMM-Seminar Kiel on Boundary Elements: Implementation and Analysis of Advanced Algorithms, eds.: W.Hackbusch, G. Wittum, Vieweg-Verlag, Braunschweig, Wiesbaden, 1996