H-Matrix Preconditioning for Domain Decomposition with Brick-and-Mortar Coefficients
Boris Khoromskij (MPI Leipzig)
A class of hierarchical matrices
(-matrices) allows the data-sparse
approximation to integral and more general nonlocal operators
(say, the Poincaré-Steklov operators)
with almost linear cost. We consider
the -matrix-based approximation to the Schur
complement on the interface  corresponding to the
of an elliptic operator with jumping coefficients
As with the standard Schur complement domain decomposition methods,
we split the elliptic inverse as a sum of local inverses
associated with subdomains (this can be implemented in parallel), and
the corresponding Poincaré-Steklov operator on the interface.
Using the hierarchical
formats based on either standard or weakened admissibility
criteria (cf. ) we elaborate the
approximate Schur complement inverse in an explicit form
that is proved to have a linear-logarithmic cost
is the number of degrees of freedom on the interface.
The -matrix-based preconditioner can be also applied.
Numerical tests confirm the
almost linear cost of our parallel direct Schur complement method.
In particular, we consider examples with the brick-and-mortar structure
of coefficients arising in the skin modeling problem.
 W. Hackbusch, B.N. Khoromskij and R. Kriemann. Hierarchical Matrices
Based on Weak Admissibility Criterion. Computing 73 (2004), 207-243.
 W. Hackbusch, B.N. Khoromskij and R. Kriemann.
Direct Schur Complement Method by Domain Decomposition
based on Hierarchical Matrix Approximation.
Preprint MPI MIS No. 25, Leipzig, 2004 (submitted).