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Rapid solution of first kind boundary integral equations in R^3

by G. Schmidlin and C. Lage and Ch. Schwab

(Report number 2002-07)

Abstract
Weakly singular boundary integral equations (BIEs) of the first kind on polyhedral surfaces \Gamma in IR3 are discretized by Galerkin BEM on shape-regular, but otherwise unstructured meshes of meshwidth h. Strong ellipticity of the integral operator is shown to give nonsingular stiffness matrices and, for piecewise constant approximations, up to O(h3) convergence of the farfield. The condition number of the stiffness matrix behaves like O(h-1) in the standard basis. An O(N) agglomeration algorithm for the construction of a multilevel wavelet basis on \Gamma is introduced resulting in a preconditioner which reduces the condition number to O(| log h|). A class of kernel-independent clustering algorithms (containing the fast multipole method as special case) is introduced for approximate matrix-vector multiplication in O(N(log N)3) memory and operations. Iterative approximate solution of the linear system by CG or GMRES with wavelet preconditioning and clustering-acceleration of matrix-vector multiplication is shown to yield an approximate solution in log-linear complexity which preserves the O(h3) convergence of the potentials. Numerical experiments are given which confirm the theory.

Keywords:

@Techreport{SLS02_293,
  author = {G. Schmidlin and C. Lage and Ch. Schwab},
  title = {Rapid solution of first kind boundary integral equations in R^3},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2002-07},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2002/2002-07.pdf },
  year = {2002}
}

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