A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

Abstract

In this paper, we develop a reduced generalized multiscale basis method for efficiently solving the parametrized groundwater flow problems in heterogeneous porous media. Recently proposed generalized multiscale finite element (GMsFE) method is one of the accurate and efficient methods to solve the multiscale problems on a coarse grid. However, the GMsFE basis functions usually depend on the random parameters in the parametrized model, which substantially impacts on the computation efficiency when the parametrized equation needs to be solved many times for many instances of parameters. To enhance computation efficiency, we construct reduced generalized multiscale basis functions independent of parameters from the high‐dimensional GMsFE space and generate a reduced‐order multiscale model by projecting the parameterized flow governing equation onto the low‐dimensional reduced GMsFE space. Then, in order to improve the online computation of reduced‐order model, we apply a matrix discrete empirical interpolation method to affinely decompose the nonaffine parametrized systems arising from the discretization of governing equation. Finally, a few numerical experiments are carried out for the parametrized transient flow problems in heterogeneous porous media to illustrate the efficiency and accuracy of the proposed model reduction method. The results show that the proposed reduced generalized multiscale basis method can significantly improve computation efficiency while maintaining comparative accuracy for the groundwater flow problems by selecting a suitable coarse mesh size and an optimal number of reduced GMsFE basis functions at each of coarse nodes.

A local-global generalized multiscale finite element method for highly heterogeneous stochastic groundwater flow problems

“…, N in } be the index set over some non-overlapping coarse neighborhoods. For each i ∈ I, we obtain a residual-driven basis function β i ∈ V i by solving (17) and define…”

“…Solve the residual-driven basis and add them to the space. Solve (17) in the above k local domains with above residuals r k1 , • • • , r km respectively to obtain…”

“…To obtain the KLE of Y , we need to solve an eigenvalue problem using the covariance function C Y (x, z). In this paper, we choose a two-point exponential covariance function C Y used in [17], that is,…”

“…However, for stochastic permeability field case, special treatment should be considered when using the GMsFEM. Previous efforts can be seen [21,17,27,22,10,12]. All these methods show high accuracy and efficiency by designing parameter-independent multiscale basis functions.…”

AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems

“…However, the formed dictionaries may be too large and only a sparse selection of the bases in the dictionaries are needed for the solution approximation. Some model reduction techniques such as reduced basis method or greedy algorithm [5,29,23,20,19,25,38] has been applied to solve parameterized elliptic PDEs. We aim to design an adaptive sparse learning algorithm with the help of precomputed basis functions as the prior dictionary, and apply it to the coupled two-phase flow systems.…”