Numerische Mathematik

, Volume 135, Issue 4, pp 1011–1043 | Cite as

Discontinuous Petrov–Galerkin boundary elements

Article

Abstract

Generalizing the framework of an ultra-weak formulation for a hypersingular integral equation on closed polygons in Heuer and Pinochet (SIAM J Numer Anal: 52(6), 2703–2721, 2014), we study the case of a hypersingular integral equation on open and closed polyhedral surfaces. We develop a general ultra-weak setting in fractional-order Sobolev spaces and prove its well-posedness and equivalence with the traditional formulation. Based on the ultra-weak formulation, we establish a discontinuous Petrov–Galerkin method with optimal test functions and prove its quasi-optimal convergence in related Sobolev norms. For closed surfaces, this general result implies quasi-optimal convergence in the \(L^2\)-norm. Some numerical experiments confirm expected convergence rates.

Mathematics Subject Classification

65N38 65N30 65N12 

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© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Facultad de MatemáticasPontificia Universidad Católica de ChileSantiagoChile
  2. 2.Departamento de MatemáticaUniversidad Técnica Federico Santa MaríaValparaísoChile

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