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Optimal Operator Preconditioning for Boundary Elements on Open Curves

by R. Hiptmair and C. Jerez-Hanckes and C. Urzua

(Report number 2013-48)

Abstract
Boundary value problems for the Poisson equation in the exterior of an open bounded Lipschitz curve \(\mathcal{C}\) can be recast as first-kind boundary integral equations featuring weakly singular or hypersingular boundary integral operators (BIEs). Based on the recent discovery in \([\)C.~Jerez-Hanckes and J.~Nedelec, Explicit variational forms for the inverses of integral logarithmic operators over an interval, SIAM Journal on Mathematical Analysis, 44 (2012), pp. 2666-2694.\(]\) of inverses of these BIEs for \(\mathcal{C}=[-1,1]\), we pursue operator preconditioning of the linear systems of equations arising from Galerkin-Petrov discretization by means of zeroth and first order boundary elements. The preconditioners rely on boundary element spaces defined on dual meshes and they can be shown to perform uniformly well independently of the number of degrees of freedom even for families of locally refined meshes.

Keywords: Operator (Calderon) preconditioning, screen problems, fracture problems, boundary integral operators

@Techreport{HJU13_548,
  author = {R. Hiptmair and C. Jerez-Hanckes and C. Urzua},
  title = {Optimal Operator Preconditioning for Boundary Elements on Open Curves},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2013-48},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2013/2013-48.pdf },
  year = {2013}
}

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