A-posteriori error estimates for the localized reduced basis multi-scale method

Abstract: We present a localized a-posteriori error estimate for the localized reduced basis multi-scale (LRBMS) method [Albrecht, Haasdonk, Kaulmann, Ohlberger (2012): The localized reduced basis multiscale method]. The LRBMS is a combination of numerical multi-scale methods and model reduction using reduced basis methods to efficiently reduce the computational complexity of parametric multi-scale problems with respect to the multi-scale parameter $\varepsilon$ and the online parameter $\mu$ simultaneously. We formulate the LRBMS based on a generalization of the SWIPDG discretization presented in [Ern, Stephansen, Vohralik (2010): Guaranteed and robust discontinuous Galerkin a posteriori error estimates for convection-diffusion-reaction problems] on a coarse partition of the domain that allows for any suitable discretization on the fine triangulation inside each coarse grid element. The estimator is based on the idea of a conforming reconstruction of the discrete diffusive flux, that can be computed using local information only. It is offline/online decomposable and can thus be efficiently used in the context of model reduction.