By Bibhutibhushan Datta

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Theorem 3. Under the above assumptions, the decomposition V = VI + ... + Vrn with the local FE spaces (22) fulfils the assumptions (A1), (A2) with the constants (23) 26 R. Blaheta Remark 3. The proof of Theorem 3 can be found in the works of M. Dryja and O. Widlund, see the books [13, 20] and the references given there. We sketch the proof here, because it contains ideas important for understanding the further development of the method in the next sections. Sketch of the proof. - For the overlapping domain decomposition, there is a partition of the unity 8 1, ...

N}, n E N. (4) For i E 1, ... , N -1, one takes F21-! = g(wi, wi+1)' where 9 is the numerical flux. It has to satisfy, in particular, the classical consistency condition, namely g(a, a) = F(a), and needs to be chosen in order to obtain some stability properties for the numerical scheme under a so called CFL condition on the time step (see Sect. 3 for the study of a scalar model). In the case of two phase flow in a pipeline, the classical numerical fluxes such as the Godunov flux (see [9]) or the Roe flux (see [11]) may not be implemented, because of computational difficulties.

M . 30 R. Blaheta m L m Ilvkll~ ~ 1\:2 k=1 L m IVkl~l(DO k=1 ~ 1\:2 fL me [5- ~ 1\:2 fL me [()~ 5- ~ c [1 + h- I H + 5- 2 H2] 2 1I v - vaIIL(D) + Iv - Val~l(D)] 2 H2 + 2 + 2i7r ~] IvIJ-J1(D) Ilvll~. - The above estimates give (30). The estimate of KI is the same as in Theorem 4. g. [3] and the references therein. The properties of the basis functions {'ljJd can be improved by smoothing and the smoothed aggregations can be again used in two-level Schwarz preconditioners, see [11]. Note that the paper [15] describes the use of massive aggregation, which means aggregation of all degrees of freedom in each subdomain.