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Analysis of FETI methods for multiscale elliptic PDEs

In this paper we study a variant of the finite element tearing and interconnecting (FETI) method which is suitable for elliptic PDEs with highly heterogeneous (multiscale) coefficients α(x); in particular, coefficients with strong variation within subdomains and/or jumps that are not aligned with the subdomain interfaces. Using energy minimisation and cut-off arguments we can show rigorously that for an arbitrary (positive) coefficient function α ∈ L∞(Ω) the condition number of the preconditioned FETI system can be bounded by C(α) (1 + log(H/h)) where H is the subdomain diameter and h is the mesh size, and where the function C(α) depends only on the coefficient variation in the vicinity of subdomain interfaces. In particular, if α|Ωi varies only mildly in a layer Ωi,η of width η near the boundary of each of the subdomains Ωi, then C(α) = O((H/η)), independent of the variation of α in the remainder Ωi\Ωi,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependency of C(α) on H/η can be relaxed to a linear dependency under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests.