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

TL;DR

This paper introduces an adaptive hyper-reduction method for the MAC scheme to efficiently solve parametric Stokes and Navier-Stokes equations, enhancing computational speed while maintaining accuracy and stability.

Contribution

It develops a novel hyper-reduced MAC scheme with an adaptive enrichment strategy for parametric fluid flow problems, improving efficiency and robustness.

Findings

Demonstrates high efficiency in solving fluid flow problems.

Maintains stability and accuracy with the hyper-reduced scheme.

Effective handling of parametric variations in fluid dynamics.

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 {\em 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 online computation efficiency. To avoid this degradation, reduced residual reduced over-collocation (R2-ROC) was invented integrating empirical interpolation techniques on the solution snapshots and 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 Stokes and Navier-Stokes equations demonstrating its high efficiency, stability and accuracy.