Convergence analysis for GMsFEM approximation of elliptic eigenvalue problems

In this paper, we construct a class of Mixed Generalized Multiscale Finite Element Methods for the approximation on a coarse grid for an elliptic problem in thin two-dimensional domains. We consider the elliptic equation with homogeneous boundary conditions on the domain walls. For reference solution of the problem, we use a Mixed Finite Element Method on a fine grid that resolves complex geometry on the grid level. To construct a lower dimensional model, we use the Mixed Generalized Multiscale Finite Element Method, which is based on some multiscale basis functions for velocity fields. The construction of the basis functions is based on the local snapshot space that takes all possible flows on the interface between coarse cells into account. In order to reduce the size of the snapshot space and obtain the multiscale approximation, we solve a local spectral problem to identify dominant modes in the snapshot space. We present a convergence analysis of the presented multiscale method. Numerical results are presented for two-dimensional problems in three testing geometries along with the errors associated to different numbers of the multiscale basis functions used for the velocity field. Numerical investigations are conducted for problems with homogeneous and heterogeneous properties respectively.

A two-stage ensemble Kalman filter based on multiscale model reduction for inverse problems in time fractional diffusion-wave equations

In this paper we present and analyze a constraint energy minimizing generalized multiscale finite element method for convection diffusion equations. To define the multiscale basis functions, we first build an auxiliary multiscale space by solving local spectral problems motivated by analysis. Then a constraint energy minimization performed in the oversampling domains is exploited to construct the multiscale space. The resulting multiscale basis functions have a good decay property even for high contrast diffusion and convection coefficients. Furthermore, if the number of oversampling layers is chosen properly, we can prove that the convergence rate is proportional to the coarse meshsize. Our analysis also indicates that the size of the oversampling domain weakly depends on the contrast of the heterogeneous coefficients. Several numerical experiments are presented illustrating the performance of our method. © 2023 Society for Industrial and Applied Mathematics Publications. All rights reserved.

An exponential integration generalized multiscale finite element method for parabolic problems

In this paper we consider a stochastic inverse flow problem with a high-contrast and stochastic permeability coefficient. This problem is solved within the context of the Markov chain Monte Carlo method in which a number of forward simulations are needed to construct the posterior distribution. Due to the large number of forward solves, a constrained version of the generalized multiscale finite element method (GMsFEM) is used to reduce the global dimension of the pressure solutions. Furthermore, since the permeability coefficient exhibits a large variation in scales, GMsFEM allows us to construct coarse solutions that are able to accurately capture the underlying fine effects of the system. To ensure a local mass conservation property, Lagrange multipliers are used in the problem formulation and discretized solution technique. Since the construction of GMsFEM spaces involves solving a number of localized eigenvalue problems that can become burdensome in the context of random sampling, we employ the use of a sparse-grid collocation technique for local stochastic reduction. In particular, rather than recomputing the eigenfunctions and basis functions for each realization, a fixed set of collocation points is used for eigenfunction pre-computations. Then, for each realization a sparse interpolant is used in the place of the direct computations for the coarse space constructions. In doing so, we are able to incorporate a reduced stochastic model (that combines GMsFEM-FV and sparse-grid collocation) in order to present a suitable alternative for determining input permeability distributions conditioned to available tracer cut data.

We applied the generalized multiscale finite-element method (GMsFEM) to simulate seismic wave propagation in fractured media. Fractures are represented explicitly on a fine-scale triangular mesh, and they are incorporated using the linear-slip model. The motivation for applying GMsFEM is that it can reduce computational costs by using basis functions computed from the fine-scale fracture model to simulate propagation on a coarse grid. First, we apply the method to a simple model that has a uniform distribution of parallel fractures. At low frequencies, the results could be predicted using a homogeneous, effective medium, but at higher frequencies, GMsFEM allows simulation of more complex, scattered wavefields generated by the fractures without assuming a specific form of anisotropy. A second, more complex model has two fracture corridors in addition to a few sparsely distributed fractures. Simulations compare scattered wavefields for different acquisition geometries. The third test case represents a vertical section of subsurface structures and is designed to test the influence of fractures on the surface seismic. We compared the fine-scale solution with multiscale solution to demonstrate the accuracy and efficiency of computations. Given the simulation results of three different test cases, GMsFEM allows a reduction of computation time of approximately 80% compared with a conventional finite-element result computed directly from the fine-scale grid, and it can predict seismic signal variations useful for the interpretation of fracture distributions.

In the reservoir simulation, mathematical modeling of the fluid flow and geo - mechanics in the fractured porous media plays an important role. Fracture networks have complex geometries, exist in the multiple scales and typically have very small thickness compared to typical reservoir sizes. Due to high permeability, fractures have a significant impact on the flow processes. In this work, we consider a discrete fracture model for coupled flow and mechanics problems. We construct coarse grid approximation using Generalized Multiscale Finite Element method (GMsFEM). In this method, we solve local spectral problems for construction to the multiscale basis functions for pressure and displacements. We present numerical results for two - dimensional model problem.

This paper presents the Meshfree Generalized Multiscale Finite Element Method (MFGMsFEM) for solving parabolic problems in multicontinuum media. Multicontinuum models are widely used in various fields, such as petroleum engineering, hydrogeology, and biomechanics, to describe fluid flow in fractured porous materials. The proposed approach leverages the advantages of both mesh free methods and the Generalized Multiscale Finite Element Method (GMsFEM), allowing for efficient and accurate numerical modeling. The novelty of this work lies in the development of multiscale basis functions, constructed using both split and coupled strategies. Numerical experiments demonstrate the effectiveness of the method in capturing complex flow dynamics while significantly reducing computational costs compared to traditional fine-grid simulations. The results show that the coupled multiscale basis functions provide higher accuracy in modeling flow interactions within multicontinuum media.

In this work, we consider the elasticity problem in fractured media. For the efficient numerical solution, we present a Generalized Multiscale Finite Element Method (GMsFEM). GMsFEM is used for the construction of a coarse grid approximation of the problem by solution of the local spectral problems. We consider two types of the multiscale basis functions: (1) CG-GMsFEM with continuous multiscale basis functions and (2) DG-GMsFEM with discontinuous multiscale basis functions. The result of the numerical solution for the two-dimensional model problem is presented to show the performance of the presented multiscale method for fractured media. We compute error between the multiscale solution with the fine-scale solutions by choosing different numbers of multiscale basis functions.

The paper explores the dual-continuum unsaturated filtration model in heterogeneous porous media with rough surface topography. The rough surface topography plays a crucial role in determining the flow process and incorporates multiscale features. The mathematical model is based on the Richard’s equation with a Robin boundary condition for the rough surface boundary. The Online Generalized Multiscale Finite Element Method (Online GMsFEM) is used for coarse-grid discretization. Multiscale basis functions that encompass small-scale heterogeneities are created. Additional basis functions are generated to accommodate the impact of boundary conditions on rough surfaces. We present a consrtuction of online mulsticale basis functions that based on local residuals. All multiscale basis functions are coupled and take into account transfer term between continuums. Numerical results for a two-dimensional problem are provided, and relative errors between multiscale and reference solutions are calculated for varying numbers of multiscale basis functions to verify the results. The influence of online bases on the algorithm’s accuracy is also discussed.

A hybridizable discontinuous Galerkin Generalized Multiscale Finite element method for highly heterogeneous linear elasticity problems

In this paper, we consider a coupled system of equations that describes simplified magnetohydrodynamics (MHD) problem in perforated domains. We construct a fine grid that resolves the perforations on the grid level in order to use a traditional approximation. For the solution on the fine grid, we construct approximation using the mixed finite element method. To reduce the size of the fine grid system, we will develop a Mixed Generalized Multiscale Finite Element Method (Mixed GMsFEM). The method differs from existing approaches and requires some modifications to represent the flow and magnetic fields. Numerical results are presented for a two-dimensional model problem in perforated domains. This model problem is a special case for the general 3D problem. We study the influence of the number of multiscale basis functions on the accuracy of the method and show that the proposed method provides a good accuracy with few basis functions.

Multiscale simulations for multi-continuum Richards equations

A deep learning based reduced order modeling for stochastic underground flow problems

Multiscale model reduction technique for fluid flows with heterogeneous porous inclusions