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Highlights► A new IsogEometric Tearing and Interconnecting (IETI) method is proposed. ► Exact geometry representation of IGA and solver design of FETI methods are combined. ► Coupling conditions for interfaces, including hanging knots, are discussed. ► Efficient preconditioning techniques for the interface problem are presented. ► Some local refinement options for IGA are discussed.
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
Poincaré type inequalities are a key tool in the analysis of partial differential equations. They play a particularly central role in the analysis of domain decomposition and multilevel iterative methods for second-order elliptic problems. When the diffusion coefficient varies within a subdomain or within a coarse grid element, then condition number bounds for these methods based on standard Poincaré inequalities may be overly pessimistic. In this paper we present new results on weighted Poincaré type inequalities for very general classes of coefficients that lead to sharper bounds independent of any possible large variation in the coefficients. The main requirement on the coefficients is some form of quasi-monotonicity which we will carefully describe and analyse. The Poincaré constants depend on the topology and the geometry of regions of relatively high and/or low coefficient values, and we will study these dependencies in detail. Applications of the inequalities in the analysis of the geometric multigrid, the two-level overlapping Schwarz and the FETI methods can be found in [25, 30].
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