An Additive Schwarz Method for the h-p Version of the
Boundary Element Method
for Hypersingular Integral Equations in R³
Norbert Heuer, Ernst P. Stephan
IMA J. Numer. Anal. 21 (1), 265-283, 2001.
We study a preconditioner for the h-p version of the boundary element
method for hypersingular integral equations in three dimensions.
The preconditioner is based on a three-level decomposition of the underlying
ansatz space, the levels being piecewise bilinear functions on a coarse
grid, piecewise bilinear functions on a fine grid, and piecewise
polynomials of high degree on the fine grid.
We prove that the condition number of the preconditioned linear system
is bounded by $\max_j (1+\log\frac{H_j p_j}{h_j})^2$ where $H_j$ is
the diameter of an element $\G_j$ of the coarse grid, $h_j$ is the size
of the elements of the fine grid on $\Gamma_j$, and $p_j$ is the maximum
of the polynomial degrees used in $\Gamma_j$.
Numerical results supporting our theory are reported.