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Analysis of FETI methods for multiscale PDEs

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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 \({\alpha \in L^\infty(\Omega)}\) 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 \({\alpha|_{\Omega_{i}}}\) varies only mildly in a layer Ω i,η of width η near the boundary of each of the subdomains Ω i , then \({C(\alpha) = \mathcal{O}((H/\eta)^2)}\) , 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.

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Authors and Affiliations

  1. Institute of Computational Mathematics, Johannes Kepler University, Altenberger Str. 69, 4040, Linz, Austria

    Clemens Pechstein

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Robert Scheichl

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  1. Clemens Pechstein
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  2. Robert Scheichl
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Correspondence to Clemens Pechstein.

Additional information

C. Pechstein was supported by the Austrian Science Funds (FWF) under grant F1306.

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Pechstein, C., Scheichl, R. Analysis of FETI methods for multiscale PDEs. Numer. Math. 111, 293–333 (2008). https://doi.org/10.1007/s00211-008-0186-2

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  • DOI: https://doi.org/10.1007/s00211-008-0186-2

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