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

Bespalov, Alexei; Heuer, Norbert

ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 821-849.

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 $H^1$-regular whereas, on open surfaces, edge singularities are strong enough to prevent the solution from being in $H^1$. 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 $O\left(h^{1/2}{p}^{-1}\right)$, $h$ being the mesh size and $p$ denoting the polynomial degree. This result had been conjectured previously.

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