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Optimal Operator preconditioning for hypersingular operator over 3D screens

by R. Hiptmair and C. Jerez-Hanckes and C. Urzúa-Torres

(Report number 2016-09)

Abstract
We propose a new Calderón-type preconditioner for the hypersingular integral operator for \(-\Delta\) on screens in \(\mathbb{R}^{3}\). We introduce a modified weakly singular operator, which is the exact inverse of the hypersingular operator on the unit disk. It forms the foundation for dual-mesh-based operator preconditioning. Applied to low-order boundary element Galerkin discretizations, it achieves \(h\)-uniformly bounded condition numbers. Heuristic extensions to general screens even with non-smooth boundaries are discussed. Their good performance is confirmed by numerical tests.

Keywords: open surface problems, boundary integral equations, Laplace equation, operator (Calderón) preconditioning, screen problems

@Techreport{HJU16_646,
  author = {R. Hiptmair and C. Jerez-Hanckes and C. Urzúa-Torres},
  title = {Optimal Operator preconditioning for hypersingular operator over 3D screens},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-09},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-09.pdf },
  year = {2016}
}

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