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IMA Journal of Numerical Analysis Advance Access published online on November 24, 2008
IMA Journal of Numerical Analysis, doi:10.1093/imanum/drn052
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© The author 2008. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.

The hp-version of the boundary element method with quasi-uniform meshes for weakly singular operators on surfaces

Alexei Bespalov{dagger}

Department of Mathematical Sciences, Brunel University, Uxbridge, West London UB8 3PH, UK

Norbert Heuer{ddagger}

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

1 Email: albespalov{at}yahoo.com

2 Corresponding author. Email: nheuer{at}mat.puc.cl

Received on 18 May 2007. Accepted for publication 20 November 2007.


  Abstract

We prove an a priori error estimate for the hp-version of the boundary element method with weakly singular operators in three dimensions. The underlying meshes are quasi-uniform. Our model problem is that of the Laplacian exterior to an open surface, where the solution has strong singularities that are not L2-regular. Our results confirm previously conjectured convergence rates in h (the mesh size) and p (the polynomial degree) and these rates are given explicitly in terms of the exponents of the singular functions. In particular, for sufficiently smooth given data we prove a convergence in the energy norm like O(h1/2p–1).

Key Words: hp-version with quasi-uniform meshes; boundary element method; weakly singular operators; singularities

Dedicated to Prof. Gabriel N. Gatica on the occasion of his 50th birthday


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