A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains

TL;DR

This paper introduces a new multiscale model reduction method for simulating Stokes flows in complex perforated domains, enabling efficient and accurate solutions with fewer basis functions.

Contribution

It develops a localized multiscale finite element approach with a hybridized technique for mass conservation, improving computational efficiency and accuracy in heterogeneous perforated domains.

Findings

Achieves high accuracy with few basis functions per coarse region

Ensures mass conservation through a hybridized Lagrange multiplier method

Demonstrates stability and convergence of the proposed scheme

Abstract

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy.