Yating Wang

Online adaptive local multiscale model reduction for heterogeneous problems in perforated domains

In this paper, we develop and analyze an adaptive multiscale approach for heterogeneous problems in perforated domains. We consider commonly used model problems including the Laplace equation, the elasticity equation, and the Stokes system in perforated regions. In many applications, these problems have a multiscale nature arising because of the perforations, their geometries, the sizes of the perforations, and configurations. Typical modeling approaches extract average properties in each coarse region, that encapsulate many perforations, and formulate a coarse-grid problem. In some applications, the coarse-grid problem can have a different form from the fine-scale problem, e.g. the coarse-grid system corresponding to a Stokes system in perforated domains leads to Darcy equations on a coarse grid. In this paper, we present a general offline/online procedure, which can adequately and adaptively represent the local degrees of freedom and derive appropriate coarse-grid equations. Our approaches start with the offline procedure, which constructs multiscale basis functions in each coarse region and formulates coarse-grid equations. We presented the offline simulations without the analysis and adaptive procedures, which are needed for accurate and efficient simulations. The main contributions of this paper are (1) the rigorous analysis of the offline approach, (2) the development of the online procedures and their analysis, and (3) the development of adaptive strategies. We present an online procedure, which allows adaptively incorporating global information and is important for a fast convergence when combined with the adaptivity. We present online adaptive enrichment algorithms for the three model problems mentioned above. Our methodology allows adding and guides constructing new online multiscale basis functions adaptively in appropriate regions. We present the convergence analysis of the online adaptive enrichment algorithm for the Stokes system. In particular, we show that the online procedure has a rapid convergence with a rate related to the number of offline basis functions, and one can obtain fast convergence by a sufficient number of offline basis functions, which are computed in the offline stage. The convergence theory can also be applied to the Laplace equation and the elasticity equation. To illustrate the performance of our method, we present numerical results with both small and large perforations. We see that only a few (1 or 2) online iterations can significantly improve the offline solution.

Multiscale model reduction for transport and flow problems in perforated domains

Abstract Convection-dominated transport phenomenon is important for many applications. In these applications, the transport velocity is often a solution of heterogeneous flow problems, which results to a coupled flow and transport phenomena. In this paper, we consider a coupled flow (Stokes problem) and transport (unsteady convection–diffusion problem) in perforated domains. Perforated domains (see Fig. 1) represent void space outside hard inclusions as in porous media, filters, and so on. We construct a coarse-scale solver based on Generalized Multiscale Finite Element Method (GMsFEM) for a coupled flow and transport. The main idea of the GMsFEM is to develop a systematic approach for computing multiscale basis functions. We use a mixed formulation and appropriate multiscale basis functions for both flow and transport to guarantee a mass conservation. For the transport problem, we use Petrov–Galerkin mixed formulation, which provides a stability. As a first approach, we use the multiscale flow solution in constructing the basis functions for the transport equation. In the second approach, we construct multiscale basis functions for coupled flow and transport without solving global flow problem. The novelty of this approach is to construct a coupled multiscale basis function. Numerical results are presented for computations using offline basis. We also present an algorithm for adaptively adding online multiscale basis functions, which are computed using the residual information. Numerical examples using online GMsFEM show the speed up of convergence.

A conservative local multiscale model reduction technique for Stokes flows in heterogeneous perforated domains

In this paper, we present a new multiscale model reduction technique for the Stokes flows in heterogeneous perforated domains. The challenge in the numerical simulations of this problem lies in the fact that the solution contains many multiscale features and requires a very fine mesh to resolve all details. In order to efficiently compute the solutions, some model reductions are necessary. To obtain a reduced model, we apply the generalized multiscale finite element approach, which is a framework allowing systematic construction of reduced models. Based on this general framework, we will first construct a local snapshot space, which contains many possible multiscale features of the solution. Using the snapshot space and a local spectral problem, we identify dominant modes in the snapshot space and use them as the multiscale basis functions. Our basis functions are constructed locally with non-overlapping supports, which enhances the sparsity of the resulting linear system. In order to enforce the mass conservation, we propose a hybridized technique, and uses a Lagrange multiplier to achieve mass conservation. We will mathematically analyze the stability and the convergence of the proposed method. In addition, we will present some numerical examples to show the performance of the scheme. We show that, with a few basis functions per coarse region, one can obtain a solution with excellent accuracy.

Non-local multi-continua upscaling for flows in heterogeneous fractured media

Abstract In this paper, we propose a rigorous and accurate non-local (in the oversampled region) upscaling framework based on some recently developed multiscale methods [10] . Our proposed method consists of identifying multi-continua parameters via local basis functions and constructing non-local (in the oversampled region) transfer and effective properties. To achieve this, we significantly modify our recent work proposed within Generalized Multiscale Finite Element Method (GMsFEM) in [10] and derive appropriate local problems in oversampled regions once we identify important modes representing each continuum. We use piecewise constant functions in each fracture network and in the matrix to write an upscaled equation. Thus, the resulting upscaled equation is of minimal size and the unknowns are average pressures in the fractures and the matrix. Note that the use of non-local upscaled model for porous media flows is not new, e.g., in [14] , the authors derive non-local approach. Our main contribution is identifying appropriate local problems together with local spectral modes to represent each continuum. The model problem for fractures assumes that one can identify fracture networks. The resulting non-local equation (restricted to the oversampling region, which is several times larger compared to the target coarse block) has the same form as [14] with much smaller local regions. We present numerical results, which show that the proposed approach can provide good accuracy.

A locally conservative Multiscale Finite Element Method for multiphase flow simulation through heterogeneous and fractured porous media

Abstract A Multiscale Locally Conservative Galerkin (MsLCG) method is proposed to accurately simulate multiphase flow in heterogeneous and fractured porous media. MsLCG employs a coarse partition of the fine grids and multiscale basis function for mapping the fine-scale information to the coarse-scale unknowns. Different from standard Multiscale Finite Element Method (MsFEM), the main improvement of our MsLCG is to use the property of local conservation at steady state conditions to define a numerical flux at element boundaries. MsLCG provides a way to extend standard MsFEM to handle challenging multiphase flow problems in heterogeneous and fractured porous media. MsLCG preserves all the advantages of the standard MsFEM while it provides explicitly conservative fluxes through each element. We present a number of representative numerical examples to demonstrate that our method is efficient and accurate for simulating multiphase flow through heterogeneous and fractured porous media.