A hyper-reduced MAC scheme for the parametric Stokes and Navier-Stokes equations

Abstract

The need for accelerating the repeated solving of certain parametrized systems motivates the development of more efficient reduced order methods. The classical reduced basis method is popular due to an offline-online decomposition and a mathematically rigorous a posterior error estimator which guides a greedy algorithm offline. For nonlinear and nonaffine problems, hyper reduction techniques have been introduced to make this decomposition efficient. However, they may be tricky to implement and often degrade the offline and online computational efficiency.To avoid this degradation, reduced residual reduced over-collocation (R2-ROC) was invented. It integrates the empirical interpolation techniques on the solution snapshots and the well-chosen residuals, the collocation philosophy, and the simplicity of evaluating the hyper-reduced well-chosen residuals. In this paper, we introduce an adaptive enrichment strategy for R2-ROC rendering it capable of handling parametric fluid flow problems. Built on top of an underlying Marker and Cell (MAC) scheme, a novel hyper-reduced MAC scheme is therefore presented and tested on lid-driven cavity and flow past a backward-facing step problems demonstrating its high efficiency, stability and accuracy.