Mode Decomposition Methods for Flows in High-Contrast Porous Media. Part II. Local-Global Approach

In this paper, we combine concepts of the generalized multiscale finite element method and mode decomposition methods to construct a robust local-global approach for model reduction of flows in high-contrast porous media. This is achieved by implementing proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) techniques on a coarse grid. The resulting reduced-order approach enables a significant reduction in the flow problem size while accurately capturing the behavior of fully resolved solutions. We consider a variety of high-contrast coefficients and present the corresponding numerical results to illustrate the effectiveness of the proposed technique. This paper is a continuation of the first part where we examine the applicability of POD and DMD to derive simplified and reliable representations of flows in high-contrast porous media. In the current paper, we discuss how these global model reduction approaches can be combined with local techniques to speed-up the simulations. The speed-up is due to inexpensive, while sufficiently accurate, computations of global snapshots.

💡 Research Summary

This paper presents a novel local‑global model reduction framework for simulating flows in highly heterogeneous, high‑contrast porous media. The authors combine the Generalized Multiscale Finite Element Method (GMsFEM) with two popular global dimensionality‑reduction techniques—Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD)—to achieve substantial computational savings while preserving the essential dynamics of the full‑order solution.

Problem Setting
The governing equation is a time‑dependent single‑phase pressure diffusion model: ∂u/∂t − ∇·(κ(x)∇u) = f, where κ(x) denotes a permeability‑to‑viscosity ratio that can vary by several orders of magnitude, creating channels of high conductivity and inclusions of low conductivity. Direct fine‑grid finite element discretization leads to a very large system (N_f degrees of freedom) that is prohibitive for repeated forward solves required in sensitivity analysis or uncertainty quantification.

Local Reduction via GMsFEM
A coarse mesh T_H is introduced, and an initial multiscale space V_initial^0 is built from standard partition‑of‑unity functions χ_i that satisfy local homogeneous equations. To enrich this space, the authors compute an energy weight e_κ based on the coarse mesh size H and solve localized Neumann eigenvalue problems on each coarse neighborhood ω_i: −∇·(κ∇ψ) = λ e_κ ψ. Eigenvectors associated with the smallest eigenvalues (which correspond to the most conductive channels or inclusions) are multiplied by χ_i to form the enriched basis functions Φ_{i,l}=χ_i ψ_{i,l}. The collection of all Φ_{i,l} spans the coarse space V_0 of dimension N_c ≪ N_f. The fine‑scale mass and stiffness matrices are projected onto V_0 using the restriction operator R_0, yielding reduced matrices M_0 and A_0. Time integration on the coarse level uses a backward‑Euler scheme: U^{n+1}_0 = (M_0+Δt A_0)^{-1}