Wavelet based fast solution of boundary integral equations

This paper presents a wavelet Galerkin scheme for the fast solution of boundary integral equations. Wavelet Galerkin schemes employ appropriate wavelet bases for the discretization of boundary integral operators which yields quasi-sparse system matrices. These matrices can be compressed such that the complexity for solving a boundary integral equation scales linearly with the number of unknowns without compromising the accuracy of the underlying Galerkin scheme. Based on the wavelet Galerkin scheme we present also an adaptive algorithm. By numerical experiments we provide results which demonstrate the performance of our algorithm. AMS Subject Classification: 47A20, 65F10, 65F50, 65N38, 65R20.

[1]  W. Dahmen,et al.  Multilevel preconditioning , 1992 .

[2]  Wolfgang Dahmen,et al.  A multiscale method for the double layer potential equation on a polyhedron. , 2005 .

[3]  Wolfgang Dahmen,et al.  Wavelets on Manifolds I: Construction and Domain Decomposition , 1999, SIAM J. Math. Anal..

[4]  Ernst P. Stephan,et al.  Adaptive multilevel BEM for acoustic scattering , 1997 .

[5]  W. Dahmen.  Wavelet and multiscale methods for operator equations , 1997, Acta Numerica.

[6]  Birgit Faermann,et al.  Local a-posteriori error indicators for the Galerkin discretization of boundary integral equations , 1998 .

[7]  Wolfgang Dahmen,et al.  Compression Techniques for Boundary Integral Equations - Optimal Complexity Estimates , 2006 .

[8]  Christoph Schwab.  Variable order composite quadrature of singular and nearly singular integrals , 2005, Computing.

[9]  Wolfgang Dahmen,et al.  Composite wavelet bases for operator equations , 1999, Math. Comput..

[10]  Christoph Schwab,et al.  Fully Discrete Multiscale Galerkin BEM , 1997 .

[11]  Wolfgang Dahmen,et al.  Adaptive wavelet methods for elliptic operator equations: Convergence rates , 2001, Math. Comput..

[12]  Claudio Canuto,et al.  The wavelet element method. Part I: Construction and analysis. , 1997 .

[13]  W. Dahmen,et al.  Biorthogonal Spline Wavelets on the Interval—Stability and Moment Conditions , 1999 .

[14]  Reinhold Schneider,et al.  Biorthogonal wavelet approximation for the coupling of FEM-BEM , 2002, Numerische Mathematik.

[15]  Ernst P. Stephan,et al.  An adaptive two-level method for hypersingular integral equations in \(R^3\) , 2000 .

[16]  Reinhold Schneider,et al.  Multiwavelets for Second-Kind Integral Equations , 1997 .

[17]  Olaf Steinbach,et al.  A new a posteriori error estimator in adaptive direct boundary element method , 2000 .

[18]  Leslie Greengard,et al.  A fast algorithm for particle simulations , 1987 .

[19]  Wolfgang Dahmen,et al.  Adaptive Wavelet Methods II—Beyond the Elliptic Case , 2002, Found. Comput. Math..

[20]  L. Villemoes.  Wavelet analysis of refinement equations , 1994 .

[21]  R. Coifman,et al.  Fast wavelet transforms and numerical algorithms I , 1991 .

[22]  J. Nédélec.  Acoustic and Electromagnetic Equations : Integral Representations for Harmonic Problems , 2001 .

[23]  Wolfgang Dahmen,et al.  Nonlinear functionals of wavelet expansions – adaptive reconstruction and fast evaluation , 2000, Numerische Mathematik.

[24]  Carsten Carstensen,et al.  Adaptive Boundary Element Methods for Some First Kind Integral Equations , 1996 .

[25]  I. Daubechies,et al.  Biorthogonal bases of compactly supported wavelets , 1992 .

[26]  Albert Cohen,et al.  Wavelet adaptive method for second order elliptic problems: boundary conditions and domain decomposition , 2000, Numerische Mathematik.

[27]  H. Harbrecht,et al.  Wavelet Galerkin Schemes for 2D-BEM , 2001 .

[28]  Olaf Steinbach,et al.  A new a posteriori error estimator in adaptive direct boundary element methods: the Dirichlet problem , 2000 .

[29]  M. G. Duffy,et al.  Quadrature Over a Pyramid or Cube of Integrands with a Singularity at a Vertex , 1982 .

[30]  H. Harbrecht,et al.  Numerische Simulation Auf Massiv Parallelen Rechnern Least Squares Methods for the Coupling of Fem and Bem Preprint-reihe Des Chemnitzer Sfb 393 , 2022 .

[31]  W. Hackbusch,et al.  On the fast matrix multiplication in the boundary element method by panel clustering , 1989 .