Preconditioners for interface problems in Eulerian formulations<br/><br/><br/><br/>

Wednesday, December 1, 2010 - 11:00am - 11:45am
Keller 3-180
David Keyes (King Abdullah University of Science & Technology)
Eulerian formulations of problems with interfaces avoid the subtleties
of tracking and remeshing, but do they complicate solution of the
discrete equations, relative to domain decomposition methods that
respect the interface? We consider two different interface problems –
one involving cracking and one involving phase separation. Crack
problems can be formulated by extended finite element methods (XFEM),
in which discontinuous fields are represented via special degrees of
freedom. These DOFs are not properly handled in a typical AMG
coarsening process, which leads to slow convergence. We propose a
Schwarz approach that retains AMG advantages on the standard DOFs and
avoids coarsening the enriched DOFs. This strategy allows reasonably
mesh-independent convergence rates, though the convergence degradation
of the (lower dimensional) set of crack DOFs remains to be addressed.
Phase separation problems can be formulated by the Cahn-Hilliard
approach, in which the level set of a continuous Eulerian field
demarcates the phases. Here, scalable preconditioners follow
naturally, once the subtlety of the temporal discretization is sorted
out. The first project is joint with R. Tuminaro and H. Waisman and
the second with X.-C. Cai and C. Yang.

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