A High-Order Localized Orthogonal Decomposition Method for Heterogeneous Stokes Problems

TL;DR

This paper introduces a high-order localized orthogonal decomposition method for heterogeneous Stokes problems, improving accuracy and efficiency through better localization and pressure reconstruction, with proven exponential decay and high-order convergence.

Contribution

It extends the multiscale method to high-order accuracy for Stokes problems, incorporating improved localization, pressure reconstruction, and rigorous error analysis.

Findings

Achieves optimal convergence orders for velocity and pressure.

Proves exponential decay of basis functions enabling localization.

Numerical experiments confirm theoretical high-order convergence.

Abstract

In this paper, we propose a high-order extension of the multiscale method introduced by the authors in [SIAM J. Numer. Anal., 63(4) (2025), pp. 1617--1641] for heterogeneous Stokes problems, while also providing several other improvements, including a better localization strategy and a more precise pressure reconstruction. The proposed method is based on the Localized Orthogonal Decomposition methodology and achieves optimal convergence orders under minimal structural assumptions on the coefficients. A key feature of our approach is the careful design of so-called quantities of interest, defining functionals of the solution whose values the multiscale approximation aims to reproduce exactly. Their selection is particularly delicate in the context of Stokes problems due to potential conflicts arising from the divergence-free constraint. We prove the exponential decay of the problem-adapted basis functions, justifying their localized computation in practical implementations. A rigorous a priori error analysis proves high-order convergence for both velocity and pressure, if the basis supports grow logarithmically with the desired accuracy. Numerical experiments confirm the theoretical findings.