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Wavelet Galerkin Algorithms for Boundary Integral Equations

by C. Lage and Ch. Schwab

(Report number 1997-15)

Abstract
The implementation of a fast, wavelet-based Galerkin discretization of second kind integral equations on piecewise smooth surfaces $\Gamma\subset \IR^3$ is described. It allows meshes consisting of triangles as well as quadrilaterals. The algorithm generates a sparse, approximate stiffness matrix with $\cN=O(N(log N)^2)$ nonvanishing entries in $O(N(\log N)^4)$ operations where N is the number of degrees of freedom on the boundary while essentially retaining the asymptotic convergence rate of the full Galerkin scheme. A new proof of the matrix-compression estimates is given based on derivative-free kernel estimates. The condition number of the sparse stiffness matrices is bounded independently of the meshwidth. The data structure containing the compressed stiffness matrix is described in detail: it requires $O(\cN)$ memory and can be set up in $O(\cN)$ operations. Numerical experiments show that the asymptotic performance estimates apply for moderate N. Problems with $N=10^5$ degrees of freedom were computed in core on a workstation. The impact of various parameters in the compression scheme on the performance and the accuracy of the algorithm is studied.

Keywords:

@Techreport{LS97_220,
  author = {C. Lage and Ch. Schwab},
  title = {Wavelet Galerkin Algorithms for Boundary Integral Equations},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {1997-15},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports1997/1997-15.pdf },
  year = {1997}
}

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