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| We are interested in the efficient and reliable numerical approximation of elliptic parametric multi-scale problems which consist of finding $p_h(\mu) \in V_h$, such that $b_h( p_h(\mu) , q_h; \varepsilon, \mu) = l(q_h)$ for all $q_h \in V_h$ for an a-priori given multi-scale parameter $\varepsilon > 0 $ either in a multi-query context, where we want to solve for many parameters $\mu$, or in a real-time context, where we have to solve for some parameters $\mu$ as fast as possible.
Model reduction using \underline{r}educed \underline{b}asis (RB) methods is a well established and reliable technique to reduce the computational complexity of parametric problems with respect to $\mu$.
In the context of multi-scale problems, however, standard RB methods may become computationally too expensive.
The \underline{l}ocalized RB \underline{m}ulti-\underline{s}cale (LRBMS) method was introduced in [F. Albrecht and B. Haasdonk and S. Kaulmann and M. Ohlberger, ``The localized reduced basis multiscale method", \textit{Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry, Podbanske, September 9-14, 2012}, 393--403, (2012).] as a combination of model reduction and numerical multi-scale methods to overcome the shortcomings of classical RB methods.
We will present recent advances in the context of the LRBMS based on a recently derived efficient a-posteriori error estimator. |
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