The hp-version of the boundary element method with quasi-uniform meshes
in three dimensions
Alexei Bespalov and Norbert Heuer
ESAIM Math. Model. Num. Anal. 42 (5), 821-849, 2008.
We prove an optimal 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 optimal a priori
error estimate for the h-version with quasi-uniform meshes.
For open surfaces we prove a convergence like O(h½p-1),
h being the mesh size and p denoting the polynomial degree.