Elsevier

Journal of Computational Physics

Volume 251, 15 October 2013, Pages 116-135
Journal of Computational Physics

Generalized multiscale finite element methods (GMsFEM)

https://doi.org/10.1016/j.jcp.2013.04.045Get rights and content

Abstract

In this paper, we propose a general approach called Generalized Multiscale Finite Element Method (GMsFEM) for performing multiscale simulations for problems without scale separation over a complex input space. As in multiscale finite element methods (MsFEMs), the main idea of the proposed approach is to construct a small dimensional local solution space that can be used to generate an efficient and accurate approximation to the multiscale solution with a potentially high dimensional input parameter space. In the proposed approach, we present a general procedure to construct the offline space that is used for a systematic enrichment of the coarse solution space in the online stage. The enrichment in the online stage is performed based on a spectral decomposition of the offline space. In the online stage, for any input parameter, a multiscale space is constructed to solve the global problem on a coarse grid. The online space is constructed via a spectral decomposition of the offline space and by choosing the eigenvectors corresponding to the largest eigenvalues. The computational saving is due to the fact that the construction of the online multiscale space for any input parameter is fast and this space can be re-used for solving the forward problem with any forcing and boundary condition. Compared with the other approaches where global snapshots are used, the local approach that we present in this paper allows us to eliminate unnecessary degrees of freedom on a coarse-grid level. We present various examples in the paper and some numerical results to demonstrate the effectiveness of our method.

Introduction

Many problems arising from various physical and engineering applications are multiscale in nature. Because of the presence of small scales and uncertainties in these problems, the direct simulations are prohibitively expensive. Moreover, these problems are typically solved for many source terms with input parameter coming from a high dimensional parameter space. For example, the flow in heterogeneous porous media described by Darcy’s equation is typically solved for multiple source terms. Moreover, the permeability usually has uncertainties which are parametrized in some sophisticated manner. In this case, one needs to solve many forward problems with different source terms and a wide range of permeabilities to make accurate predictions. These problems can be cast using an input–output relation (see Fig. 1) which is typically done in reduced-order modeling. For the example of flow problems, the input space consists of source terms and the permeability that takes a value from a large parameter space. The output space depends on the quantities of interest and may consist of coarse-grid solutions or some other integrated quantities with respect to the solution. In many applications, the output space is typically smaller than the input space. The design of a general multiscale finite element framework that takes advantage of the effective low dimensional solution space for multiscale problems with high dimensional input space is the main objective of this paper.

Due to a large number of forward simulations, the computational effort can be tremendous to learn and process the output space given the high dimensionality of the input parameter space. In many of these problems, the solution space can be approximated by a low dimensional manifold via some model reduction tools. The main objective of reduced-order models is to represent the solution space with a small dimensional space. However, many existing reduced-order methods fail to give a small dimensional output solution space when the physical solution has multiscale structures. Another major limitation of the current reduced-order methods is that the reduced solution space needs to be regenerated for different forcing or boundary conditions. The general multiscale finite element framework proposed in this paper is designed to overcome these two limitations by dividing the construction of reduced basis into the offline and online steps, and constructing our online multiscale bases from a reduced localized offline solution space.

Many local, global, and local–global model reduction techniques have been developed. The main idea of these methods is to find a small dimensional space that can represent the solution space given the input space.

Global model reduction techniques (see e.g., [37], [17], [18], [22], [6], [50]) construct a space of global fields that can approximate the solution space. One can, for example, consider a space of exhaustive global snapshots obtained by solving the global problem for many input parameters. This space can be further reduced using a spectral decomposition. In practice, the resulting space is constructed by solving global problems for some selected input parameters, right hand sides, and boundary conditions. These methods have been used with some success in practice. However, when the right hand sides or boundary conditions are changed, the resulting reduced space must be recomputed.

Local approaches (e.g., see [38], [41], [1], [2], [3], [4], [5], [7], [8], [19], [20], [32], [40], [33], [34], [44], [45], [51] for upscaling and multiscale methods) attempt to approximate the solution in local (coarse-grid) regions for all input parameters without computing global snapshots of solutions. Local approaches first compute an offline space (possibly small dimensional) which is used to compute multiscale basis functions at the online stage. The local approximation space at the online stage is computed by finding a subspace of offline space for a given input parameter (see Fig. 2).

Local approaches can be effective as they avoid the computation of global snapshots. Local approaches become more effective if the restriction of the solution space onto a local region has a small dimension. This is the case if the dimension of the space of solutions restricted to a coarse region is smaller than the dimension of the fine-grid space within this coarse region. For example, if the parameter is a coarse-grid scalar function, then at the coarse-grid level, this parameter is a scalar. While if we consider this problem from the point of view of a global model reduction, then the parameter belongs to a large dimensional space and this may not be amenable to computations.

One of the advantages of local approaches is that they eliminate the unnecessary degrees of freedom in the parameter space at the coarse-grid level. In global methods, one first needs to compute many expensive global snapshots and many snapshots may not contribute to the solution at the online stage. In local approaches, these values of the parameter are identified at the coarse level inexpensively. Moreover, local approaches can easily handle large-scale parameter space when the parameter is a coarse-grid function and local approximation spaces are usually independent of the source terms or boundary conditions. We will further elaborate these issues in the paper.

In this paper, we introduce a general multiscale framework, which we call the Generalized Multiscale Finite Element Method (GMsFEM). This method incorporates complex input space information and the input–output relation. It systematically enriches the coarse space through our local construction. Our approach, as in many multiscale and model reduction techniques, divides the computation into two stages: offline; and online. In the offline stage, we construct a small dimensional space that can be efficiently used in the online stage to construct multiscale basis functions. These multiscale basis functions can be re-used for any input parameter to solve the problem on a coarse-grid. Thus, this provides a substantial computational saving at the online stage. Below, we present an outline of the algorithm and a chart that depicts our algorithm in Fig. 2.

  • 1.Offline computation:

    • – 1.0. Coarse grid generation;

    • – 1.1. Construction of snapshot space that will be used to compute an offline space;

    • – 1.2. Construction of a small dimensional offline space by performing dimension reduction in the space of global snapshots.

  • 2. Online computations:

    • – 2.1. For each input parameter, compute multiscale basis functions;

    • – 2.2. Solution of a coarse-grid problem for any force term and boundary condition;

    • – 2.3. Iterative solvers, if needed.

In the offline computation, we first set up a coarse grid where each coarse-grid block consists of a connected union of fine-grid blocks. The construction of snapshot space in Step 1.1 involves solving local problems for various choices of input parameters. This space is used to construct the offline space in Step 1.2 via a spectral decomposition of the snapshot space. The snapshot space in a coarse region can be replaced by the fine-grid space associated with this coarse space; however, in many applications, one can judiciously choose the space of snapshots to avoid expensive offline space construction. The offline space in Step 1.2 is constructed by spectrally decomposing the space of snapshots. This spectral decomposition is typically based on the offline eigenvalue problem. The spectral decomposition enables us to select the high-energy elements from the offline space by choosing those eigenvectors corresponding to the largest eigenvalues. More precisely, we seek a subspace of the snapshot space such that it can approximate any element of the snapshot space in the appropriate sense defined via auxiliary bilinear forms.

In the online step 2.1 for a given input parameter, we compute the required online coarse space. In general, we want this to be a small dimensional subspace of the offline space. This space is computed by performing a spectral decomposition in the offline space via an eigenvalue problem. Furthermore, the eigenvectors corresponding to the largest eigenvalues are identified and used to form the online coarse space. The online coarse space is used within the finite element framework to solve the original global problem. Here, we propose several options such as the Galerkin coupling of multiscale basis functions, the Petrov–Galerkin coupling of multiscale basis functions, etc. In some of these coupling approaches, the choice of the initial partition of unity (that can be computed in the offline or online stage) is important and it will be discussed in the paper.

Our techniques differ from many previous approaches that are based on the homogenization theory. In the homogenization based methods, one usually constructs local approximation based on local solves and these approaches do not provide a systematic procedure to complement the local spaces. It is important to note that one needs to systematically complement the local spaces in order to converge to the fine-grid solution. How to develop an online systematic enrichment procedure and how to construct the initial partition of unity functions play a crucial role in obtaining a low dimensional offline space. These issues are central points of our proposed method.

We also discuss iterative solvers that use the coarse spaces and iterate on the residual to converge to the fine-scale solution. These iterative solvers serve as an online correction of the coarse-grid solution. We consider two-level domain decomposition preconditioners and some other methods where the importance of appropriately chosen multiscale coarse spaces has been demonstrated in the literature [49], [43]. We will discuss how the choice of coarse spaces yields optimal iterative solvers where the number of iterations is independent of the high contrast in the media properties.

Section snippets

A generalized multiscale finite element method

To describe the GMsFEM for linear problems, we considerLμ(u)=f,subject to some boundary conditions, where μ is the parameter. For example,Lμ(u)=-div(κ(x;μ)u).

Here, the operator L may depend on various spatial fields, e.g., heterogeneous conductivity fields, convection fields, reaction fields, and so on. The dependence of the solution from these fields is nonlinear, while the solution linearly depends on external source terms f and boundary conditions. We assume there is a bilinear form

Case studies and relation to existing methods. Discussions and applications

In this section, we illustrate basic concepts via some specific examples. We use existing methods in the literature for multiscale problems and show how these methods can be put under the general framework of the GMsFEM and show some numerical results.

Conclusions

In this paper, we propose a multiscale framework, the Generalized Multiscale Finite Element Method (GMsFEM), for solving PDEs with multiple scales. The main objective is to propose a framework that extends MsFEMs to more general problems with complex input space that includes parameters, high contrast, and right-hand-sides or boundary conditions. The GMsFEM starts with a family of snapshots for the local solutions. These snapshots can usually be generated based on the solutions of local

Acknowledgements

We would like to thank Ms. Guanglian Li for helping us with the computations and providing some computational results. Y. Efendiev’s work is partially supported by the DOE, US DoD Army ARO, and NSF (DMS 0934837 and DMS 0811180). J. Galvis would like to acknowledge partial support from DOE. This publication is based in part on work supported by Award No. KUS-C1-016-04, made by King Abdullah University of Science and Technology (KAUST).

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