Reduced Basis Method for the Stokes Equations in Decomposable Parametrized Domains Using Greedy Optimization

Flow simulations in pipelined channels and several kinds of parametrized configurations have a growing interest in many life sciences and industrial applications. Applications may be found in the analysis of the blood flow in specific compartments of the circulatory system that can be represented as a combination of few deformed vessels from reference ones, e.g. pipes. We propose a solution approach that is particularly suitable for the study of internal flows in hierarchical parametrized geometries. The main motivation is for applications requiring rapid and reliable numerical simulations of problems in domains involving parametrized complex geometries. The classical reduced basis (RB) method is very effective to address viscous flows equations in parametrized geometries (see, e.g., [10]). An interesting alternative foresees a combination of RB with a domain decomposition approach. In this respect, preliminary efforts to reduce the global parametrized problem to local ones have led to the introduction of the so-called reduced basis element method to solve the Stokes problem [6], and more recently to the reduced basis hybrid method [3] and to the static condensation method [7]. In general, we are interested in defining a method able to maintain the flexibility of dealing with arbitrary combinations of subdomains and several geometrical deformations of the latter. A further new contribution to this field is the computation of the reduced basis functions through an optimization greedy algorithm

Variational Multiscale Stabilization and the Exponential Decay of Fine-scale Correctors

This paper addresses the variational multiscale stabilization of standard finite element methods for linear partial differential equations that exhibit multiscale features. The stabilization is of Petrov-Galerkin type with a standard finite element trial space and a problem-dependent test space based on pre-computed fine-scale correctors. The exponential decay of these correctors and their localisation to local cell problems is rigorously justified. The stabilization eliminates scale-dependent pre-asymptotic effects as they appear for standard finite element discretizations of highly oscillatory problems, e.g., the poor $L^2$ approximation in homogenization problems or the pollution effect in high-frequency acoustic scattering