# Johannes Kepler Symposium on Mathematics

As part of the Johannes Kepler symposium on mathematics
**Prof. Olof B.
Widlund**, Courant Institute of Mathematical Sciences, New York University,
will give a public talk (followed by a discussion) on **Wed, May 28, 2008**
at **15:15 o'clock** at **HS 9**
on the topic of
"Coarse Space Components of Domain Decomposition Algorithms"
. The organziers of the symposium,

Univ.-Prof. Dr. Gerhard Larcher

A.Univ.-Prof. Dr. Jürgen Maaß, and

die ÖMG (Österreichische Mathematische Gesellschaft),

hereby cordially invite you.

Series B - Mathematical Colloquium:

The intention is to present new mathematical results for an audience interested in general mathematics.

### Coarse Space Components of Domain Decomposition Algorithms

Domain decomposition algorithms now provide powerful preconditioners for the often very large systems of linear and nonlinear algebraic equations which must be solved in large scale finite element simulations. These preconditioners are combined with a conjugate gradient or other Krylov space method. Successful domain decomposition algorithms almost always require the use a coarse space component as well as many local components. The latter are often simply the restrictions of the given problem to many subdomains into which the domain of the original problem has been subdivided. On large scale computing systems, as many as twenty and forty thousand degrees of freedom per subdomain are used while the dimension of the coarse problem is a small multiple of the number of subdomains; for elasticity in three dimensions, six degrees of freedom per subdomain is the minimum that is required.

The development of the coarse components of the preconditioners has always been at the core of domain decomposition work. Early work by Bramble, Pasciak, and Schatz has been quite influential and has led to the development of some quite exotic coarse spaces and also to important technical tools that have been used in many studies. The theory has much in common with work on multigrid but the emphasis has also differed in several respects. We will examine two families of domain decomposition methods and compare the findings with those for multigrid methods.

The two families are the iterative substructuring methods, based on subdomains which do not overlap, and the two- or multi-level overlapping Schwarz methods. We will also discuss methods which combine components from both families. The coarse component of a domain decomposition method can also serve purposes other than providing some global interchange of information, in each step of the iteration, across the entire domain. Recent results on mixed finite element approximation of almost incompressible elasticity will serve as an example.