Finite and Boundary Element Tearing and Interconnecting Methods for Multiscale Elliptic Partial Diﬀerential Equations

Dr. Clemens Pechstein

Jan. 20, 2009, 3:30 p.m. MZ 005A

This talk is a 30 minutes overview over my doctoral dissertation. There, I have studied a special class of domain decomposition solvers for ﬁnite and boundary element discretizations of elliptic PDEs, namely the ﬁnite and boundary element tearing and interconnecting (FETI/BETI) methods. I was able to generalize the theory of FETI/BETI methods in two directions, unbounded domains and highly heterogeneous coeﬃcients.

The basic idea of FETI/BETI methods is to subdivide the computational domain into smaller subdomains, where the corresponding local problems can still be handled eﬃciently by direct solvers, if feasible, in parallel. The global solution is then constructed iteratively from the repeated solution of local problems.

Here, suitable preconditioners are needed in order to ensure that the number of iterations depends only weakly on the size of the local problems. Furthermore, the incorporation of a coarse solve ensures the scalability of the method, which means that the number of iterations is independent of the number of subdomains.

For scalar second-order elliptic equations given in a bounded domain where the diﬀusion coeﬃcient is constant on each subdomain, FETI/BETI methods are proved to be quasi-optimal. In particular, the condition number of the corresponding preconditioned system is bounded in terms of a logarithmic expression in the local problem size. Furthermore, the bound is independent of jumps in the diﬀusion coeﬃcient across subdomain interfaces.

First, we consider the case of unbounded domains, where one subdomain corresponds to an exterior problem, while the other subdomains are bounded. The exterior problem is approximated using the boundary element method. The fact that this exterior domain can touch arbitrarily many interior subdomains and that the diameter of its boundary is larger than those of the interior subdomains leads to special diﬃculties in the analysis. We provide explicit condition number bounds that depend on a few geometric parameters, and which are quasi-optimal in special cases. Our results are conﬁrmed in numerical experiments.

Second, we consider elliptic equations with highly heterogeneous coeﬃcient distributions. We prove rigorous bounds for the condition number of the preconditioned FETI system that depend only on the coeﬃcient variation in the vicinity of the subdomain interfaces. To be more precise, if the coeﬃcient varies only moderately in a layer near the boundary of each subdomain, the method is proved to be robust with respect to arbitrary variation in the interior of each subdomain and with respect to coeﬃcient jumps across subdomain interfaces. In our analysis we develop and use new technical tools such as generalized Poincaré and discrete Sobolev inequalities. Our results are again conﬁrmed in numerical experiments. We also demonstrate that FETI preconditioners can lead to robust behavior even for certain coeﬃcient distributions that are highly varying in the vicinity of the subdomain interfaces.

Finally, we consider nonlinear stationary magnetic ﬁeld problems in two dimensions, as an important application of our preceding analysis. There, the Newton linearization leads to problems with highly heterogeneous coeﬃcients, which can be eﬃciently solved using the proposed FETI/BETI methods.