Decoupling scales via localized subspace iteration and temporal splitting for multiscale parabolic equations

TL;DR

This paper introduces an advanced multiscale method combining localized subspace iteration and temporal splitting to efficiently simulate diffusion in complex heterogeneous media, ensuring stability and accuracy over long times.

Contribution

The authors extend the Localized Subspace Iteration framework with new eigenspace approximation techniques and a contrast-independent temporal splitting scheme for improved multiscale parabolic equation simulation.

Findings

The method accurately captures slow-decaying modes in multiscale media.

The approach guarantees stability without restrictive time-step constraints.

Numerical experiments demonstrate superior efficiency and accuracy in high-contrast media.

Abstract

Simulating diffusion in heterogeneous media presents a significant computational challenge, as resolving microscopic physical scales traditionally demands excessively fine computational grids. To overcome this barrier, we extend the Localized Subspace Iteration (LSI) framework to multiscale parabolic equations. The proposed method constructs optimal, low-dimensional trial spaces by iteratively approximating the dominant eigenspaces of local inverse operators via Localized Standard Subspace Iteration (LSSI) or Localized Krylov Subspace Iteration (LKSI). Because these LSI basis functions are inherently tailored to capture the slow-decaying, low-frequency modes of the parabolic solution, they naturally suppress error accumulation over long-term integration. To further improve computational efficiency, we decouple the basis construction into an offline phase and implement a contrast-independent, partially explicit temporal splitting scheme for online time-stepping. By explicitly advancing the dominant macroscopic modes while implicitly treating high-frequency microscopic corrections, this scheme guarantees stability without imposing restrictive time-step constraints. We establish rigorous a priori error estimates in both the energy and $L^2$ norms. Numerical experiments illustrate the accuracy and efficiency of the LSI framework, particularly highlighting the LKSI method's advantages in handling high-contrast, complex multiscale media.