Title:        Domain Decomposition Algorithms for PDE problems with
              Large Scale Variations
Authors:      Luc Giraud and Ray Tuminaro
Organization: CERFACS
              42 Ave. Gustave Coriolis
              31057 Toulouse Cedex
              France
Phone:        (33) 61 19 30 21 or (33) 61 19 30 25
fax:          (33) 61 19 30 00
email:        tuminaro@cerfacs.fr  or giraud@cerfacs.fr


We consider a Schur complement BPS-like domain decomposition algorithm 
for PDE problems whose coefficients vary greatly. In particular,
we focus on two problems: anisotropic phenomena and large changes
in the PDE coefficients as one moves spatial within the domain.
We consider both Fourier and probe based interface preconditioners. However, 
our focus is primarily on probing as the large scale variations can make 
the Fourier techniques somewhat less efficient. To cope with anisotropic 
phenomena, we introduce additional band matrices (in the context of the 
probe preconditioner) to approximate the coupling between neighboring 
interfaces. To address coefficient variations over the domain, we make use
of the close connection between domain decomposition and multigrid and
introduce specialized interpolation, projection, and averaging techniques 
(as in multigrid methods) to develop an accurate coarse grid approximation. 
We demonstrate the benefits of the new approach using both analysis on 
model problems as well as computational experiments on difficult 
application problems such as the Drift-Diffusion equations arising from 
semiconductor modeling.