Error Estimates of Reiterated Stokes Systems via Fourier Transform Methods

TL;DR

This paper develops error estimates for reiterated Stokes systems in bounded domains using Fourier transform methods to handle multi-scale issues, achieving specific error bounds for velocity and pressure terms.

Contribution

It introduces a Fourier transform approach to obtain error estimates in reiterated Stokes systems, addressing multi-scale homogenization challenges.

Findings

O(ε) error estimate for velocity

O(ε^{1/2}) error estimate for pressure

Fourier transform method effectively separates different scales

Abstract

In this paper, we are interested in the error estimates of the reiterated Stokes systems in a bounded $C^{1,1}$ domain with Dirichlet boundary conditions. And we have obtained the $O(\varepsilon)$ error estimates for the velocity term and $O(\varepsilon^{1/2})$ error estimates for the pressure term. Compared to the general homogenization of Stokes systems problems, the difficulty in the reiterated homogenization is that we need to handle the different scales of $x$. To overcome this difficulty, we use the Fourier transform methods which was firstly introduced by the author in [10] to separate these different scales. We also note that this method may be adapted to a more general multi-scale homogenization problem.