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Discontinuous Petrov–Galerkin boundary elements

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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.

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Authors and Affiliations

  1. Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Avenida Vicuña Mackenna 4860, Macul, Santiago, Chile

    Norbert Heuer

  2. Departamento de Matemática, Universidad Técnica Federico Santa María, Avenida España 1680, Valparaíso, Chile

    Michael Karkulik

Authors
  1. Norbert Heuer
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  2. Michael Karkulik
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Correspondence to Norbert Heuer.

Additional information

Supported by CONICYT through FONDECYT projects 1150056, 3140614 and Anillo ACT1118 (ANANUM).

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Heuer, N., Karkulik, M. Discontinuous Petrov–Galerkin boundary elements. Numer. Math. 135, 1011–1043 (2017). https://doi.org/10.1007/s00211-016-0824-z

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  • DOI: https://doi.org/10.1007/s00211-016-0824-z

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