# 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.