Title: Balancing Neumann-Neumann Preconditioners for the Mixed Formulation of Almost-Incompressible Linear Elasticity 

(NYU-CS-TR847)

Author: Paulo Goldfeld


Abstract:


Balancing Neumann-Neumann methods are extended to the equations arising
from the mixed formulation of almost-incompressible linear elasticity problems
discretized with discontinuous-pressure finite elements. This family of domain
decomposition algorithms has previously been shown to be effective for large
finite element approximations of positive definite elliptic problems. Our
methods are proved to be scalable and to depend weakly on the size of the
local problems. Our work is an extension of previous work by Pavarino and
Widlund on BNN methods for Stokes equation.

Our iterative substructuring methods are based on the partition of the
unknowns into interior ones - including interior displacements and pressures
with zero average on every subdomain - and interface ones - displacements
on the geometric interface and constant-by-subdomain pressures. The
restriction of the problem to the interior degrees of freedom is then a
collection of decoupled local problems that are well-posed even in the
incompressible limit. The interior variables are eliminated and a hybrid
preconditioner of BNN type is designed for the Schur complement problem.
The iterates are restricted to a benign subspace, on which the preconditioned
operator is positive definite, allowing for the use of conjugate gradient
methods.

A complete convergence analysis of the method is presented for the constant
coefficient case. The algorithm is extended to handle discontinuous
coefficients, but a full analysis is not provided. Extensions of the algorithm and
of the analysis are also presented for problems combining pure-displacement 
and mixed finite elements in different subregions. An algorithm is also proposed
for problems with continuous discrete pressure spaces.

All the algorithms discussed have been implemented in parallel codes that
have been successfully tested on large sample problems on large parallel
computers; results are presented and discussed. Implementations issues are
also discussed, including a version of our main algorithm that does not require
the solution of any auxiliary saddle-point problem since all subproblems of the
preconditioner can be reduced to solving symmetric positive definite linear
systems.