CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Recent Advances on Additive Schwarz Methods
 

SPEAKER: Marcus Sarkis, 
         University of Colorado at Boulder
         
DATE:    Monday, February 2, 1998  


PLACE:   Mathematics Conference Room 626 
         UCD Building, 1250 14th St., Denver


TIME:    noon (Refreshments served at 11:45 am) 



ABSTRACT

The original additive Schwarz method (AS) was introduced for solving 
symmetric positive definite elliptic finite element problems, and was
later extended to many other nonsymmetric and non-elliptic systems. In 
the first part of this presentation I discuss a modified overlapping 
additive Schwarz preconditioner for sparse linear systems.  We
proposed a very simple change on the (AS), and the resulting method  
is more effective in terms of both iteration numbers and CPU time on 
sequential and parallel computers. The method is referred to as the 
restricted additive Schwarz (RAS). I will discuss some theoretical 
results obtained for the laplacian operator and will address the
theoretical results that confirms that the RAS is superior to the AS. 
I will show numerical results for a wide range of problems including 
convection-diffusion equations, indefinite complex Helmholtz equations 
and the 3D compressible Euler's equation discretized on unstructured meshes. 
 
In the second  part of this talk, I will present discretizations and 
preconditioners for solving two-dimensional elliptic problems discretized 
on overlapping  non-matching grids. I first discuss ways to 
formulate the discrete problem. The formulations  are based either on 
Alternating Schwarz technique or on weighted bilinear forms. The
discrete problem also depends on how to  pass the values of a function 
from one grid to another; a simple pointwise interpolation or 
a mortar projection can be used. Theoretical results and numerical
experiments will be presented to understand how the accuracy and
the preconditioning depends on the mesh sizes, ratio of the mesh sizes,
and the size of the overlap of the  non-matching grids. Some of the 
presented results have been obtained in joint work with Prof Xiao-Chuan Cai, 
Maksymilian Dryja and Tarek Mathew.