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

## Dr. Clemens Pechstein

**Dec. 10, 2013, 12:45 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.