Ulrich Langer : Algebraic Multigrid Preconditioners for Boundary Element Matrices
Efficient preconditioners for boundary element (BE) matrices are
needed in many application. In the primal as well in the dual
boundary element domain decomposition (DD) methods preconditioners
for the discrete single layer potential integral operator,
arising from the BE approximation
of interior or exterior Dirichlet boundary value
problems, are required. In the primal DD method,
preconditioners for the assembled discrete
hypersingular integral operator are needed.
Geometric and algebraic multigrid techniques
based on sparse approximations of the
corresponding boundary element matrices
are a powerful technique for the construction
of robust and at least almost optimal
preconditioners for these BE matrices.
In this talk we present new algebraic multigrid
for sparse boundary element matrices arising from the
Adaptive-Cross-Approximation (ACA) of dense boundary element matrices.
As model problem we consider the single layer potential integral equation
resulting from the interior Dirichlet boundary value
problem for the Laplace equation.
The standard collocation, or Galerkin boundary element discretization
leads to fully populated system matrices which require O(N2)
complexity for the memory and the matrix-by-vector multiplication,
where N denotes the number of boundary unknowns. Sparse approximations
such as ACA reduce this complexity to O(N) up to some polylogarithmical factor.
Since the single layer potential operator is a
pseudodifferential operator of the order -1,
the resulting boundary element matrices are ill-conditioned.
Iterative solvers dramatically suffers from this property for growing N.
Our AMG preconditioners avoid this dramatical grow in the number
of iterations and lead to almost optimal solvers
with respect to the total complexity.
This behaviour is confirmed by our numerical experiments.
We mention that our AMG preconditioners use only single grid information
provided by the usual mesh data and by the ACA anyway.
Analogous results have been obtained for the