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

TL;DR

This paper addresses the challenge of solving heterogeneous Stokes problems on coarse meshes by extending the Localized Orthogonal Decomposition (LOD) to a high-order multiscale framework, ensuring accurate velocity and pressure approximations without resolving all fine-scale variations.A carefully designed set of quantities of interest (QOIs) and a complementary space decomposition enable high-order convergence, while an exponential decay of basis functions justifies localizable computations and a stabilized localization strategy prevents deterioration under refinement.The authors provide rigorous a priori error analysis showing velocity convergence of order m+2 in the H^1-norm and m+3 in the L^2-norm, exponential convergence for the pressure under localization, and a post-processing step that yields high-order pressure accuracy.Numerical experiments corroborate the theory, demonstrate substantial improvements over previous lowest-order approaches, and illustrate practical benefits of the proposed pressure post-processing and localization design for heterogeneous Stokes problems.

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.

Figures (5)

• Figure 7.1: Initial mesh $\mathcal{T}_{2^{-0}}$ for the mesh generation (left), barycentric refinement of mesh $\mathcal{T}_{2^{-3}}$, and multiscale coefficient used in all numerical experiments (right).
• Figure 7.2: Error plots of the velocity approximation for polynomial degrees $m \in \{0,1,2\}$ (from left to right). For fixed localization parameters $\ell$, the $H^1$-norm (top row) and $L^2$-norm (bottom row) errors are plotted as functions of the coarse mesh size $H$.
• Figure 7.3: Error plots of the pressure approximation for polynomial degrees $m \in \{0,1,2\}$ (from left to right). For fixed coarse mesh sizes $H$, the $L^2$-norm error relative to $\Pi_H p_h$ is shown as a function of the localization parameter $\ell$.
• Figure 7.4: Error plots of the post-processed pressure approximation for polynomial degrees $m \in \{0,1,2\}$ (from left to right). For fixed localization parameters $\ell$, the $L^2$-norm errors are plotted as functions of the coarse mesh size $H$.
• Figure 7.5: Error plots for the lowest-order multiscale method from Hauck2025: velocity $H^1$-error, velocity $L^2$-error, pressure $L^2$-error with respect to $\Pi_H p_h$, and post-processed pressure $L^2$-error (arranged from left to right, top to bottom).

Theorems & Definitions (36)

• Example 3.2: Construction of the complement space
• Lemma 3.3: Local Ladyzhenskaya-type bubble
• proof
• Lemma 3.4: Bubble functions
• proof
• Remark 3.5: Elimination of gradients in QOIs
• Lemma 3.6: Prototypical basis
• proof
• Lemma 3.7: Local Poincare-type inequality
• proof