# FETI/BETI domain decomposition solvers for nonlinear problems in (un)bounded domains

## Dr. Clemens Pechstein

**Oct. 17, 2006, 1:30 p.m. T 1010**

Domain decomposition (DD) methods, such as the wideley used finite elementtearing and interconnecting (FETI) methods, dual-primal FETI (FETI-DP) methods and balanced domain decomposition by constraints (BDDC), offer par-allel solvers for large-scale problems, stemming from standard partial differential equations (PDEs) like the Poisson problem or linear elasticity. The BETI method, a boundary element method (BEM) counterpart of the FETI method, can be coupled into this framework, resulting in fast solvers which benefit fromthe advantages of both discretization techniques, FEM and BEM. The condition number of the preconditioned system can be bounded by $C(1 + log(H/h))^2$, where the constant $C$ is independent of the mesh size $h$, the average subdomaindiameter $H$ and possible jumps in the coefficients of the PDE across subdomaininterfaces.

For the application of FETI/BETI methods to magnetostatic problems, twoextensions are of major interest: First, the modelling of nonlinearities is of upmost importance. Applying the FETI/BETI solver to the linearized Newton-type problems, the coefficient field on a subdomain may involve high variation. Standard preconditioners run into problems due to this variation. We propose anew preconditioner that addresses such variation much better than straightforward techniques would do, which can be well-observed in numerical examples.

Secondly, one wishes to consider problems in unbounded domains together with suitable radiation conditions. We give some analysis for linear problemsin case that one of the subdomains is the unbounded exterior of its boundary. Here, we need to prove elementary results on the exterior Dirichlet-to-Neumannmap using Sobolev interpolation techniques.

This is joint work with Prof. Ulrich Langer (JKU Linz), in collaborationwith Prof. Olaf Steinbach and Dr. Günther Of (TU Graz)