By Alexander Mielke

This booklet studies contemporary mathematical advancements within the DFG precedence Programme ''Analysis, Modeling and Simulation of Multiscale Problems'', which all started as a German examine initiative in 2006. the sector of multiscale difficulties happens in lots of fields of technological know-how, resembling microstructures in fabrics, sharp-interface versions, many-particle platforms and motions on diverse spatial and temporal scales in quantum mechanics or in molecular dynamics. lately constructed instruments are defined in a entire demeanour. This publication presents the cutting-edge at the mathematical foundations of the modeling and the effective numerical therapy of such difficulties.

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Garcke, B. Stinner Numerical simulations of appropriate test problems reveal an enormous gain in eﬃciency thanks to a better approximation. In [Al99] the analysis was extended to the case of diﬀerent diﬀusivities in the phases and both classical and thin interface asymptotics were discussed. 20)) an approximation of second order could still be achieved but the gradient structure of the model and thermodynamic consistency were lost. Based on those ideas it was shown in [An02] that even an approximation of third order is possible by using high order polynomials for the interpolation.

7. 6). 8. Let g ∈ L∞ (U η ) with g ∈ C 1 (Ω + ∩ U η ) and g ∈ C 1 (Ω − ∩ U η ), and let z ∈ R be given. 13). By g + the limit of g in x ∈ Γ when approximated from the side Ω + is denoted. Analogously g − is deﬁned when approximating x ∈ Γ from Ω − , and [g]+ − = g + − g − is the diﬀerence. Proof. The ﬁrst identity follows from the divergence theorem applied to the two parts U η ∩ Ω + and U η ∩ Ω − of U η using that ξ η vanishes on the external boundary ∂U η . For the limiting behavior consider the functions ξ˜η := z 0 2 on U η−η , 2 on U η \U η−η .

Uecker. Exponential averaging and traveling waves in rapidly varying periodic media Mathematische Nachrichten, to appear 2006. I. Neishtadt. The separation of motions in systems with rapidly rotating phase. J. Appl. Math. , 48:133–139, 1984. [Nek79] N. N. Nekhoroˇsev, An exponential estimate of the time of stability of nearly integrable Hamiltonian systems, (Russian) Uspehi Mat. Nauk, 32:5–66, 1977. [Neu00] A. Neuss-Radu. A result on the decay of the boundary layers in the homogenization theory.