  In this contribution we consider localized, robust and efficient a-posteriori
error estimation of the localized reduced basis multi-scale (LRBMS) method for
parametric elliptic problems with possibly heterogeneous diffusion coefficient.
The numerical treatment of such parametric multi-scale problems are
characterized by a high computational complexity, arising from the multi-scale
character of the underlying differential equation and the additional parameter
dependence. The LRBMS method can be seen as a combination of numerical
multi-scale methods and model reduction using reduced basis (RB) methods to
efficiently reduce the computational complexity with respect to the multi-scale
as well as the parametric aspect of the problem, simultaneously. In contrast to
the classical residual based error estimators currently used in RB methods, we
are considering error estimators that are based on conservative flux
reconstruction and provide an efficient and rigorous bound on the full error
with respect to the weak solution. In addition, the resulting error estimator
is localized and can thus be used in the on-line phase to adaptively enrich the
solution space locally where needed. The resulting certified LRBMS method with
adaptive on-line enrichment thus guarantees the quality of the reduced solution
during the on-line phase, given any (possibly insufficient) reduced basis that
was generated during the offline phase. Numerical experiments are given to
demonstrate the applicability of the resulting algorithm with online enrichment
to single phase flow in heterogeneous media.
