Analysis of FETI methods for multiscale PDEs

Abstract

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)) 2 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| Ω i varies only mildly in a layer Ω i,η of width η near the boundary of each of the subdomains Ω i , then C(α) = O((H/η) 2 ), independent of the variation of α in the remainder Ω i \Ω i,η of each subdomain and independent of any jumps of α across subdomain interfaces. The quadratic dependence of C(α) on H/η can be relaxed to a linear dependence under stronger assumptions on the behaviour of α in the interior of the subdomains. Our theoretical findings are confirmed in numerical tests. Mathematics Subject Classification (2000)65F10 · 65N22 · 65N30 · 65N55

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