Meshfree GMsFEM-based exponential integration for multiscale 3D advection-diffusion problems

TL;DR

This paper extends a meshfree multiscale exponential integration method to 3D advection-diffusion problems in complex media, improving stability and efficiency for large-scale simulations.

Contribution

It introduces new multiscale basis functions incorporating advection and addresses 3D computational challenges, enhancing simulation accuracy and stability.

Findings

Method preserves accuracy in 3D multiscale simulations.

Enables larger time steps compared to standard methods.

Demonstrates robustness in heterogeneous media.

Abstract

In this work, we extend the meshfree generalized multiscale exponential integration framework introduced in Nikiforov et al. (2025) to the simulation of three-dimensional advection--diffusion problems in heterogeneous and high-contrast media. The proposed approach combines meshfree generalized multiscale finite element methods (GMsFEM) for spatial discretization with exponential integration techniques for time advancement, enabling stable and efficient computations in the presence of stiffness induced by multiscale coefficients and transport effects. We introduce new constructions of multiscale basis functions that incorporate advection either at the snapshot level or within the local spectral problems, improving the approximation properties of the coarse space in advection-dominated regimes. The extension to three-dimensional settings poses additional computational and methodological challenges, including increased complexity in basis construction, higher-dimensional coarse representations, and stronger stiffness effects, which we address within the proposed framework. A series of numerical experiments in three-dimensional domains demonstrates the viability of the method, showing that it preserves accuracy while allowing for significantly larger time steps compared to standard time discretizations. The results highlight the robustness and efficiency of the proposed approach for large-scale multiscale simulations in complex heterogeneous media.