Modern domain decomposition solvers: BDDC, deluxe scaling, and an algebraic approach

Dr. Clemens Pechstein

Dec. 10, 2013, 1:45 p.m. S2 059

This talk is on balancing domain decomposition by constraints (BDDC),
one of the leading non-overlapping domain decomposition (DD) solvers for
finite element systems. I will explain the main ingredients in a purely
algebraic way
(in particular the scaling and the constraints) and draw connections to
other DD methods.

For the standard PDEs like Poisson's equation and linear elasticity, a
rich theory is available on how to choose scaling and constraints in
order to obtain an overall quasi-optimal solver. However, already for
non-resolved jumping coefficients little is known, and even less for
more general systems of PDEs.

In a joint collaboration with Clark Dohrmann (which is still work in
progress), we pick up and modify an earlier heuristic approach by Mandel
and Sousedik, in order to filter out optimal constraints from local
generalized eigenproblems.
This may be paired with the so-called deluxe scaling, which is in a
certain sense an optimal choice of scaling.
At least for general SPD problems, we can provide theoretical estimates
for the condition number. Our theory uses an algebraic technique called
parallel sum of matrices which is interesting in itself.

The efficiency of the method has yet to be validated, but it seems
satisfactory for standard PDEs, based on a complexity analysis for Schur
complements provided by sparse direct solvers.