Issue ESAIM: M2AN Volume 42, Number 5, September-October 2008 Page(s) 821 - 849 DOI 10.1051/m2an:2008025 Published online 04 July 2008

ESAIM: M2AN 42 (2008) 821-849
DOI: 10.1051/m2an:2008025

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

Alexei Bespalov and Norbert Heuer

Department of Mathematical Sciences, Brunel University, Uxbridge, West London UB8 3PH, UK. albespalov@yahoo.com; norbert.heuer@gmail.com

Received October 16, 2007. Published online July 4, 2008.

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(h^1/2p^-1), h being the mesh size and p denoting the polynomial degree. This result had been conjectured previously.

Mathematics Subject Classification. 41A10, 65N15, 65N38.

Key words: hp-version with quasi-uniform meshes, boundary element method, singularities.


© EDP Sciences, SMAI 2008

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