An Iterative Substructuring Method for the p-Version of
the Boundary Element Method for Hypersingular Integral Operators
in Three Dimensions
Norbert Heuer
Numer. Math., 79 (3), 371-396, 1998.
Erratum published in 87 (4), 793-794, 2001.
Abstract
We study preconditioners for the p-version of the boundary element method
for hypersingular integral equations in three dimensions. The preconditioners
are based on iterative substructuring of the underlying ansatz spaces
which are constructed by using discretely harmonic basis functions.
We consider a so-called wire basket preconditioner and a non-overlapping
additive Schwarz method based on the complete natural splitting,
i.e. with respect to the nodal, edge and interior functions, as well as
an almost diagonal preconditioner.
In any case we add the space of piecewise bilinear functions which eliminate
the dependence of the condition numbers on the mesh size.
For all these methods we prove that the resulting condition numbers are
bounded by $C(1+\log p)^2$. Here, $p$ is the polynomial degree of the
ansatz functions and
$C$ is a constant which is independent of $p$ and the mesh size of the
underlying boundary element mesh.
Numerical experiments supporting these results are reported.