The localized reduced basis multi-scale method with online enrichment

We are interested in the efficient and reliable numerical solution of parametric multi-scale problems, the multi-scale (parametric) character of which is indicated by ε (μ) if expressed in the general notation of (1). It is well known that solving parametric multi-scale problems accurately can be challenging and computationally costly for small scales ε and for a strong dependency of the solution on μ. Two traditional approaches exist to reduce this computational complexity: numerical multi-scale methods and model order reduction techniques. Numerical multi-scale methods reduce the complexity of multi-scale problems with respect to ε, while model order reduction techniques reduce the complexity of parametric problems with respect to μ (for both see [3] and references therein). The localized reduced basis multiscale (LRBMS) method is a combination of both to reduce the complexity of parametric multi-scale problems with respect to ε and μ simultaneously. It performs well, for instance in the context of two-phase flow problems (see [1]), but still requires solving (1) on the ε scale for several parameters μ, just like classical RB methods. Therefore, we propose an extension to the LRBMS method which requires a smaller number of full solutions of (1) by further incorporating localization ideas from numerical multi-scale methods. Following the notation of [3], we consider solutions ε μuh ∈ Uh of the parameterized variational multi-scale problem R μ[ ε μuh](vh) = 0 ∀vh ∈ Vh, (1)