Analysis of FETI methods for multiscale PDEs |
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Authors: | Clemens Pechstein Robert Scheichl |
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Institution: | (1) Institute of Computational Mathematics, Johannes Kepler University, Altenberger Str. 69, 4040 Linz, Austria;(2) Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK |
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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 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 varies only mildly in a layer Ω
i,η
of width η near the boundary of each of the subdomains Ω
i
, then , 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.
C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306. |
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Keywords: | Mathematics Subject Classification (2000)" target="_blank">Mathematics Subject Classification (2000) 65F10 65N22 65N30 65N55 |
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