# BDDC Deluxe Domain Decomposition Algorithms

**Olof B. Widlund
**

Courant Institute of Mathematical Sciences

251 Mercer Street

New York, NY 10012, USA

### Abstract:

When designing domain decomposition algorithms, approximate inverses,
also known as preconditioners, are constructed for very large matrices
by using solvers for many smaller linear systems often obtained from much
smaller instances of the given problem. The problems considered often
arise in continuum mechanics, e.g., in linear elasticity or electro-magnetics.
The preconditioners are used in Krylov space iterations. In addition,
for fast convergence with a rate of convergence independent of the number
of local problems, a coarse component of the preconditioner will be needed.

The BDDC algorithms, first developed by Clark Dohrmann,
have proven to be very successful domain decomposition
algorithms for a variety of elliptic problems. For any particular application,
the success of such an algorithm depends on the choice of a set of primal
constraints and the choice of an averaging operator, which is used to
restore the continuity of certain intermediate vectors in each iteration.

In the deluxe version, a new averaging procedure is used; it was first developed
in joint work with Dohrmann on H(curl) problems. A theory will be outlined
and a variety of successful applications will be discussed in particular
to problems formulated in H(div) and H(curl). This work has been developed
jointly with Clark Dohrmann, Duk-Soon Oh, and Stefano Zampini. Lately, there
has been an important development of adaptive algorithms for the choice of
the primal constraints.

In recent joint work with Stefano Zampini, PETSc programs have been developed
for massively parallel computing systems. This work involves the adaptive
selection of the primal constraints and exploring three-level BDDC algorithms
for problems with more than a billion degrees of freedom.