Meshfree Generalized Multiscale Exponential Integration Method for Parabolic Problems

TL;DR

This paper introduces a novel meshfree multiscale exponential integration method combining MFGMsFEM and exponential time schemes to efficiently and stably solve complex multiscale flow problems in heterogeneous porous media.

Contribution

It develops a new combined approach that enhances spatial and temporal approximation for multiscale problems without coarse grid construction, with rigorous convergence analysis.

Findings

Significant reduction in computational cost.

Enhanced stability allowing larger time steps.

Validated effectiveness through numerical experiments.

Abstract

This paper considers flow problems in multiscale heterogeneous porous media. The multiscale nature of the modeled process significantly complicates numerical simulations due to the need to compute huge and ill-conditioned sparse matrices, which negatively affect both the computational cost and the stability of the numerical solution. We propose a novel combined approach of the meshfree Generalized Multiscale Finite Element Method (MFGMsFEM) and exponential time integration for solving such problems. MFGMsFEM provides a robust and efficient spatial approximation, allowing us to consider complex heterogeneities without constructing a coarse computational grid. At the same time, exponential integration, using the cost-effective MFGMsFEM matrix, provides a robust temporal approximation for stiff multiscale problems, allowing larger time steps. For the proposed multiscale approach, we provide a rigorous convergence analysis, including the new analysis of the MFGMsFEM spatial approximation. We conduct numerical experiments to computationally verify the proposed approach by solving linear and semi-linear flow problems in multiscale media. Numerical results demonstrate that the proposed multiscale method achieves significant reductions in computational cost and improved stability, even with larger time steps, confirming the theoretical analysis.