Preconditioned Minimum Residual Iteration for the h-p Version of
the Coupled FEM/BEM with Quasi-uniform Meshes
Norbert Heuer, Matthias Maischak, Ernst P. Stephan
Numer. Linear Algebra Appl., 6 (6), 435-456, 1999.
Abstract
We propose and analyze efficient preconditioners for the minimum residual
method to solve indefinite, symmetric systems of equations arising from the
h-p version for the finite element and boundary element coupling.
According to the structure of the Galerkin matrix we study 2- and 3-block
preconditioners corresponding to Neumann and Dirichlet problems
for the finite element discretization.
In the case of exact inversion of the blocks we obtain bounded
iteration numbers for the 2-block Jacobi solver and $O(h^{-3/4}p^{3/2})$
iteration numbers for the 3-block Jacobi solver.
Here, $h$ denotes the mesh size and $p$ the polynomial degree.
For the efficient 2-block method we analyze the influence of various
preconditioners which are based on further decomposing the trial functions
into nodal, edge, and interior functions.
By further splitting the ansatz space with respect to basis functions
associated with the edges we obtain a partially diagonal preconditioner.
The penultimate method requires $O(\log^2 p)$ iterations whereas
the latter method needs $O(p\log^2 p)$ iterations.
Numerical results are presented which support the theory.