Mixed multiscale finite element methods using approximate global information based on partial upscaling

Abstract

The use of limited global information in multiscale simulations is needed when there is no scale separation. Previous approaches entail fine-scale simulations in the computation of the global information.The computation of the global information is expensive. In this paper, we propose the use of approximate global information based on partial upscaling. A requirement for partial homogenization is to capture long-range (non-local) effects present in the finescale solution, while homogenizing some of the smallest scales. The local information at these smallest scales is captured in the computation of basis functions. Thus, the proposed approach allows us to avoid the computations at the scales that can be homogenized. This results in coarser problems for the computation of global fields. We analyze the convergence of the proposed method. Mathematical formalism is introduced, which allows estimating the errors due to small scales that are homogenized. The proposed method is applied to simulate two-phase flows in heterogeneous porous media. Numerical results are presented for various permeability fields, including those generated using two-point correlation functions and channelized permeability fields from the SPE Comparative Project (Christie and Blunt, SPE Reserv Evalu Eng 4:308-317, 2001). We consider simple cases where one can identify the scales that can be homogenized. For more general cases, we suggest the use of upscaling on the coarse grid with the size smaller than the target coarse grid where multiscale basis functions are constructed. This intermediate coarse grid renders a partially upscaled solution that contains essential non-local information. Numerical examples demonstrate that the use of approximate global information provides better accuracy than purely local multiscale methods.

Stochastic mixed multiscale finite element methods and their applications in random porous media

“…This idea extends the framework of mixed MsFEM proposed in [24]. The second numerical example in Section 5 confirms the effectiveness of the idea.…”

“…The mathematical definition of G-convergence and its applications can be found in [25]. By the proof of Theorem 3.9 in [24], we obtain the following proposition. …”

“…(3.11) and we use this choice in the proof. If we take q(x, y r )j K = f(x, y r )j K in (3.11), then div(P h (I m u)) = f. Since interpolation operator I m is linear and bounded, we have kdivðI m u À P h ðI m uÞÞk L 2 ðDÞL 2 q ðCÞ 6 kf À hf i K k L 2 ðDÞ 6 Chkrf k L 2 ðDÞ : By following the similar procedure in the Proofs of Theorems 3.5 and 4.2 in [24], we obtain the following theorem. …”

Local–global model reduction method for stochastic optimal control problems constrained by partial differential equations

“…So it is desirable to develop an inexpensive low-fidelity model to obtain the snapshot functions. As a local model reduction method, Multiscale finite element method (MsFEM) [25,27] is an efficient method to achieve a good trade-off between accuracy and efficiency. The main idea is to divide the fine scale problem into many local problems and use the solutions of the local problems to form a coarse scale model [27].…”

“…As a local model reduction method, Multiscale finite element method (MsFEM) [25,27] is an efficient method to achieve a good trade-off between accuracy and efficiency. The main idea is to divide the fine scale problem into many local problems and use the solutions of the local problems to form a coarse scale model [27]. MsFEM incorporates the small-scale information to multiscale basis functions and capture the impact of small-scale features on the coarse-scale through a variational formulation.…”