Title: An Overlapping Schwarz Algorithm for Almost Incompressible
       Elasticity 

(NYU-CS-TR912)

Authors: Clark R. Dohrmann and Olof B. Widlund 

Abstract:
Overlapping Schwarz methods are extended to  mixed finite element
approximations of linear elasticity which use discontinuous pressure spaces.
The coarse component of the preconditioner is based on a low-dimensional space
previously developed for scalar elliptic problems and a domain decomposition 
method of iterative substructuring type, i.e., a method based on non-overlapping 
decompositions of the domain, while the local components of the preconditioner 
are based on solvers on a set of overlapping subdomains. 
A bound is established for the condition number of the algorithm which grows in 
proportion to the square of the logarithm of the number of degrees of freedom in
individual subdomains and the third power of the relative overlap between the 
overlapping subdomains, and which is independent of the Poisson ratio as well as
jumps in the Lam\'e parameters across the interface between the subdomains.
A positive definite reformulation of the discrete problem makes the use of the 
standard preconditioned conjugate gradient method straightforward. Numerical 
results, which include a comparison with problems of compressible elasticity, 
illustrate the findings.