Dmitry Ammosov

Mohammed Al Kobaisi, Dmitry Ammosov, Yalchin Efendiev et al.

We develop a multicontinuum Generalized Multiscale Finite Element Method (MC-GMsFEM) for constructing coarse-scale models in heterogeneous media that simultaneously provide accurate numerical approximations and physically consistent macroscopic equations. Classical multiscale methods efficiently approximate fine-scale solutions on coarse grids using localized basis functions, but they do not offer a systematic pathway for deriving macroscopic governing equations. To overcome this limitation, we introduce a unified framework that integrates multiscale finite element constructions with multicontinuum representations. The proposed method builds on the structure of GMsFEM and exploits a representation of multiscale basis functions that separates coarse variables and their gradients. We construct continuum-dependent basis functions using auxiliary fields defined through local problems with integral constraints, ensuring that each basis function is associated with a specific continuum. This leads to a decomposition of the coarse-scale solution into continuum variables and their gradients, establishing a direct connection between multiscale discretizations and multicontinuum homogenization. Compared to existing multicontinuum approaches, the proposed framework provides greater flexibility in handling heterogeneous media with spatially varying numbers of continua and is naturally embedded within a standard finite element setting. This enables both systematic derivation of macroscopic equations and straightforward numerical implementation. We apply the proposed method to the upscaling of two-phase immiscible flow in heterogeneous porous media, where multiple interacting continua, including mobile and trapped phases, arise. With the proposed approaches, we derive new macroscopic models and show that if classical models are used, the errors can be large.

Djulustan Nikiforov, Leonardo A. Poveda, Dmitry Ammosov et al.

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.