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Dual Mesh Operator Preconditioning On 3D Screens: Low-Order Boundary Element Discretization.

by R. Hiptmair and C. Urzúa-Torres

(Report number 2016-14)

Abstract
We provide key estimates that provide the theoretical foundation for dual mesh operator preconditioning [R. Hiptmair: Operator preconditioning. Computers and Mathematics with Applications, 52 (2006)] of the weakly singular boundary integral operator and the hypersingular boundary integral operator arising from \(-\Delta\) on screens in \(\mathbb{R}^{3}\). For each related boundary integral equation (BIE), this entails the construction of a dual discrete boundary element space (BE space), such that it has the same dimension as its corresponding primal discrete BE space. We discuss this for triangular elements following Buffa and Christiansen [A. Buffa and S. Christiansen: A dual finite element complex on the barycentric refinement. Mathematics of Computation, 76 (2007)], and extend their approach also to quadrilateral elements. Furthermore, we adapt Steinbach's work [O. Steinbach: Stability estimates for hybrid coupled domain decomposition methods. Vol. 1809 of Lecture Notes in Mathematics, Springer-Verlag, Berlin, 2003] in order to establish mesh assumptions under which our operator preconditioning policy remains applicable to locally refined meshes.

Keywords: open surface problems, Laplace equation, operator (Calder\'on) preconditioning, screen problems

@Techreport{HU16_651,
  author = {R. Hiptmair and C. Urzúa-Torres},
  title = {Dual Mesh Operator Preconditioning On 3D Screens: Low-Order Boundary Element Discretization.},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2016-14},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2016/2016-14.pdf },
  year = {2016}
}

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