Optimal Coarse Space Components for Domain Decomposition Methods Based on Multiscale Elements

Martin Gander
Date and time
Workshop - Multiscale numerical methods for differential equations

Domain decomposition methods for elliptic problems need a coarse space
to be scalable, and there are well established convergence results for
these so called two level domain decomposition methods, for both
overlapping and non-overlapping subdomains. These results however
always contain constants which remain unspecified. I explain in this
talk how specific choices of coarse space components can influence
these constants. I first show for a simple, one dimensional model
problem a coarse space correction which leads together with a Schwarz
method to convergence after one coarse correction step; a truly
optimal coarse correction. I will then show that such an optimal
coarse correction can also be defined for higher dimensional problems,
where it however becomes too expensive to be used in practice. I will
thus propose approximations of the optimal coarse space, based on
multiscale finite element techniques, and show numerical experiments,
both for model problems and more realistic high contrast examples.