A Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems

TL;DR

This paper introduces a multiscale Localized Orthogonal Decomposition method for heterogeneous Stokes problems, achieving approximation accuracy independent of coefficient regularity and demonstrating optimal convergence through theoretical analysis and numerical validation.

Contribution

The paper develops a novel LOD-based multiscale approach for Stokes problems with heterogeneous coefficients, featuring exponentially decaying basis functions and proven optimal convergence rates.

Findings

Exponential decay of basis functions enables localization.

Optimal convergence rates are achieved for velocity and pressure approximations.

Numerical experiments confirm theoretical error estimates.

Abstract

In this paper, we propose a multiscale method for heterogeneous Stokes problems. The method is based on the Localized Orthogonal Decomposition (LOD) methodology and has approximation properties independent of the regularity of the coefficients. We apply the LOD to an appropriate reformulation of the Stokes problem, which allows us to construct exponentially decaying basis functions for the velocity approximation while using a piecewise constant pressure approximation. The exponential decay motivates a localization of the basis computation, which is essential for the practical realization of the method. We perform a rigorous a priori error analysis and prove optimal convergence rates for the velocity approximation and a post-processed pressure approximation, provided that the supports of the basis functions are logarithmically increased with the desired accuracy. Numerical experiments support the theoretical results of this paper.