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 reused 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.

Abstract. In this paper we propose a generalization of multiscale finite element methods (Ms-FEM) to nonlinear problems. We study the convergence of the proposed method for nonlinear elliptic equations and propose an oversampling technique. Numerical examples demonstrate that the oversampling technique greatly reduces the error. The application of MsFEM to porous media flows is considered. Finally, we describe further generalizations of MsFEM to nonlinear time-dependent equations and discuss the convergence of the method for various kinds of heterogeneities.

Abstract. The multiscale finite element method (MsFEM) [T. Y. Hou, X. H. Wu, and Z. Cai, Math. Comp., 1998, to appear; T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coefficients. This is accomplished by constructing the multiscale base functions from the local solutions of the elliptic operator. Our previous study reveals that the leading order error in this approach is caused by the "resonant sampling," which leads to large error when the mesh size is close to the small scale of the continuous problem. Similar difficulty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this difficulty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169-189]. A consequence of the oversampling method is that the resulting finite element method is no longer conforming. Here we give a detailed analysis of the nonconforming error. Our analysis also reveals a new cell resonance error which is caused by the mismatch between the mesh size and the wavelength of the small scale. We show that the cell resonance error is of lower order. Our numerical experiments demonstrate that the cell resonance error is generically small and is difficult to observe in practice.Key words. multiscale, finite element, homogenization, resonant sampling AMS subject classifications. 65F10, 65F30PII. S0036142997330329 1. Introduction. Multiscale problems in science and engineering are often described by partial differential equations (PDEs) with highly oscillatory coefficients. Typical examples include flows in porous media and turbulent transport problems. Solving these problems numerically is difficult because an accurate solution usually requires a very fine resolution and hence tremendous amount of computer memory and CPU time. Parallel computing relieves the difficulty to some degree, but the size of computation is not reduced in the traditional approaches which directly solve the equations on fine meshes.Recently, a multiscale finite element method (MsFEM) has been developed [10, 8] for capturing the large scale solutions of multiscale problems on a coarse mesh (with mesh size larger than a certain cut-off scale of the problem). The main idea of the method is to build the local small scale information of the leading order differential operator into the finite element base functions. It is through these multiscale bases and the finite element formulation that the effect of small scales on the large scales are correctly captured. A key feature of MsFEM is that the construction of the base functions is a local operation within the elements. Thus, the construction in one element is decoupled from that in another element. In other words, a large scale computation is broken into many small and independent pieces. This results in many computational advantages [8], such as saving in computer memory and good parallel

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In this paper, we study domain decomposition preconditioners for multiscale flows in high contrast media. Our problems are motivated by porous media applications where low conductivity regions play an important role in determining flow patterns. We consider flow equations governed by elliptic equations in heterogeneous media with large contrast between high and low conductivity regions. This contrast brings an additional small scale (in addition to small spatial scales) into the problem expressed as the ratio between low and high conductivity values. Using weighted coarse projections, we show that the condition number of the preconditioned systems using domain decomposition methods is independent of the contrast. For this purpose, Poincaré inequalities for weighted norms are proved in the paper. The results are further generalized by employing extension theorems from homogenization theory. Our numerical observations confirm the theoretical results.

Abstract.We study the preconditioning of Markov Chain Monte Carlo (MCMC) methods using coarse-scale models with applications to subsurface characterization. The purpose of preconditioning is to reduce the fine-scale computational cost and increase the acceptance rate in the MCMC sampling. This goal is achieved by generating Markov chains based on two-stage computations. In the first stage, a new proposal is first tested by the coarse-scale model based on multiscale finite-volume method. The full fine-scale computation will be conducted only if the proposal passes the coarse-scale screening. For more efficient simulations, an approximation of the full fine-scale computation using precomputed multiscale basis functions can also be used. Comparing with the regular MCMC method, the preconditioned MCMC method generates a modified Markov chain by incorporating the coarse-scale information of the problem. The conditions under which the modified Markov chain will converge to the correct posterior distribution are stated in the paper. The validity of these assumptions for our application, and the conditions which would guarantee a high acceptance rate are also discussed. We would like to note that coarse-scale models used in the simulations need to be inexpensive, but not necessarily very accurate, as our analysis and numerical simulations demonstrate. We present numerical examples for sampling permeability fields using two-point geostatistics. The Karhunen-Loeve expansion is used to represent the realizations of the permeability field conditioned to the dynamic data, such as production data, as well as some static data. Our numerical examples show that the acceptance rate can be increased by more than ten times if MCMC simulations are preconditioned using coarse-scale models.

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