A Dimensional Reduction Approach Based on the Application of Reduced Basis Methods in the Framework of Hierarchical Model Reduction

Many phenomena in nature have dominant spatial directions along which the essential dynamics occur. Examples are blood flow problems, fluid dynamics in pipes or river beds, and subsurface flow. Motivated by a project on adaptive hydrological modelling of coupled hydrological processes [2] we started to investigate a new dimensional reduction approach [5, 7] for problems with dominant direction which is based on the application of reduced basis (RB) techniques in the hierarchical model reduction (HMR) framework (cf. [8] and the refrences therein). In detail let Ω ⊂ R be a computational domain. We define the solution space V such that H 0 (Ω) ⊆ V ⊆ H(Ω) and consider the following general elliptic problem: Find p ∈ V : a(p, v) = f(v) ∀v ∈ V, where a(·, ·) is a coercive and continuous bilinear form and f a linear form. The idea of HMR, which goes back to the work of Vogelius and Babuska [10], is to perform a Galerkin projection onto a reduced space of rank m, i.e.