Quasi-Monte Carlo methods for elliptic PDEs with random coefficients and applications
Journal of Computational Physics
A Coarse Space Construction Based on Local Dirichlet-to-Neumann Maps
SIAM Journal on Scientific Computing
Original Article: Adaptive BDDC in three dimensions
Mathematics and Computers in Simulation
Ensemble level multiscale finite element and preconditioner for channelized systems and applications
Journal of Computational and Applied Mathematics
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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.