The hp-version of the boundary element method with quasi-uniform meshes in three dimensions

Alexei Bespalov; Norbert Heuer

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

  • Volume: 42, Issue: 5, page 821-849
  • ISSN: 0764-583X

Abstract

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We prove an a priori error estimate for the hp-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H1-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in H1. In this paper we cover both cases and, in particular, prove an a priori error estimate for the h-version with quasi-uniform meshes. For open surfaces we prove a convergence like O(h1/2p-1), h being the mesh size and p denoting the polynomial degree. This result had been conjectured previously.

How to cite

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Bespalov, Alexei, and Heuer, Norbert. "The hp-version of the boundary element method with quasi-uniform meshes in three dimensions." ESAIM: Mathematical Modelling and Numerical Analysis 42.5 (2008): 821-849. <http://eudml.org/doc/250357>.

@article{Bespalov2008,
abstract = { We prove an a priori error estimate for the hp-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H1-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in H1. In this paper we cover both cases and, in particular, prove an a priori error estimate for the h-version with quasi-uniform meshes. For open surfaces we prove a convergence like O(h1/2p-1), h being the mesh size and p denoting the polynomial degree. This result had been conjectured previously. },
author = {Bespalov, Alexei, Heuer, Norbert},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis},
keywords = {hp-version with quasi-uniform meshes; boundary element method; singularities.; -version with quasi-uniform meshes; singularities},
language = {eng},
month = {7},
number = {5},
pages = {821-849},
publisher = {EDP Sciences},
title = {The hp-version of the boundary element method with quasi-uniform meshes in three dimensions},
url = {http://eudml.org/doc/250357},
volume = {42},
year = {2008},
}

TY - JOUR
AU - Bespalov, Alexei
AU - Heuer, Norbert
TI - The hp-version of the boundary element method with quasi-uniform meshes in three dimensions
JO - ESAIM: Mathematical Modelling and Numerical Analysis
DA - 2008/7//
PB - EDP Sciences
VL - 42
IS - 5
SP - 821
EP - 849
AB - We prove an a priori error estimate for the hp-version of the boundary element method with hypersingular operators on piecewise plane open or closed surfaces. The underlying meshes are supposed to be quasi-uniform. The solutions of problems on polyhedral or piecewise plane open surfaces exhibit typical singularities which limit the convergence rate of the boundary element method. On closed surfaces, and for sufficiently smooth given data, the solution is H1-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in H1. In this paper we cover both cases and, in particular, prove an a priori error estimate for the h-version with quasi-uniform meshes. For open surfaces we prove a convergence like O(h1/2p-1), h being the mesh size and p denoting the polynomial degree. This result had been conjectured previously.
LA - eng
KW - hp-version with quasi-uniform meshes; boundary element method; singularities.; -version with quasi-uniform meshes; singularities
UR - http://eudml.org/doc/250357
ER -

References

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