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33
A mixed multiscale finite element method for elliptic problems with oscillating coefficients
 MATH. COMP
, 2002
"... The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly h ..."
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Cited by 81 (11 self)
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The recently introduced multiscale finite element method for solving elliptic equations with oscillating coefficients is designed to capture the largescale structure of the solutions without resolving all the finescale structures. Motivated by the numerical simulation of flow transport in highly heterogeneous porous media, we propose a mixed multiscale finite element method with an oversampling technique for solving second order elliptic equations with rapidly oscillating coefficients. The multiscale finite element bases are constructed by locally solving Neumann boundary value problems. We provide a detailed convergence analysis of the method under the assumption that the oscillating coefficients are locally periodic. While such a simplifying assumption is not required by our method, it allows us to use homogenization theory to obtain the asymptotic structure of the solutions. Numerical experiments are carried out for flow transport in a porous medium with a random lognormal relative permeability to demonstrate the efficiency and accuracy of the proposed method.
A Multiscale Finite Element Method For Numerical Homogenization
, 2004
"... This paper is concerned with a multiscale finite element method for numerically solving second order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and of a fine local ..."
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Cited by 43 (3 self)
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This paper is concerned with a multiscale finite element method for numerically solving second order scalar elliptic boundary value problems with highly oscillating coefficients. In the spirit of previous other works, our method is based on the coupling of a coarse global mesh and of a fine local mesh, the latter one being used for computing independently an adapted finite element basis for the coarse mesh. The main new idea is the introduction of a composition rule, or change of variables, for the construction of this finite element basis. In particular, this allows for a simple treatment of high order finite element methods. We provide
Analysis of a twoscale, locally conservative subgrid upscaling for elliptic problems
 SIAM J. Numer. Anal
"... Abstract. We present a twoscale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid, but limits on computational power require that computations be performed on ..."
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Cited by 40 (2 self)
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Abstract. We present a twoscale theoretical framework for approximating the solution of a second order elliptic problem. The elliptic coefficient is assumed to vary on a scale that can be resolved on a fine numerical grid, but limits on computational power require that computations be performed on a coarse grid. We consider the elliptic problem in mixed variational form over W × V ⊂ L2 × H(div). We base our scale expansion on local mass conservation over the coarse grid. It is used to define a direct sum decomposition of W ×V into coarse and “subgrid ” subspaces Wc×Vc and δW × δV such that (1) ∇·Vc =Wc and ∇ · δV = δW, and (2) the space δV is locally supported over the coarse mesh. We then explicitly decompose the variational problem into coarse and subgrid scale problems. The subgrid problem gives a welldefined operator taking Wc ×Vc to δW × δV, which is localized in space, and it is used to upscale, that is, to remove the subgrid from the coarsescale problem. Using standard mixed finite element spaces, twoscale mixed spaces are defined. A mixed approximation is defined, which can be viewed as a type of variational multiscale method or a residualfree bubble technique. A numerical Green’s function approach is used to make the approximation to the subgrid operator efficient to compute. A mixed method πoperator is defined for the twoscale approximation spaces and used to show optimal order error estimates.
Subgrid upscaling and mixed multiscale finite elements
 SIAM J. Numer. Anal
"... Abstract. Second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale can be approximated on coarse meshes of spacing H only if one uses special techniques. The mixed variational multiscale method, also called subgrid upscaling, can be used, and ..."
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Cited by 37 (1 self)
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Abstract. Second order elliptic problems in divergence form with a highly varying leading order coefficient on the scale can be approximated on coarse meshes of spacing H only if one uses special techniques. The mixed variational multiscale method, also called subgrid upscaling, can be used, and this method is extended to allow oversampling of the local subgrid problems. The method is shown to be equivalent to the multiscale finite element method when one uses the lowest order Raviart–Thomas spaces and provided that there are no fine scale components in the source function f. In the periodic setting, a multiscale error analysis based on homogenization theory of the more general subgrid upscaling method shows that the error is O(+Hm+ /H), wherem = 1. Moreover, m = 2 if one uses the second order Brezzi–Douglas–Marini or Brezzi–Douglas–Durán–Fortin spaces and no oversampling. The error bounding constant depends only on the Hm−1norm of f and so is independent of small scales when m = 1. When oversampling is not used, a superconvergence result for the pressure approximation is shown.
A framework for adaptive multiscale methods for elliptic problems. Multiscale Modeling
 Simulation
"... Abstract. We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorit ..."
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Cited by 25 (0 self)
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Abstract. We describe a projection framework for developing adaptive multiscale methods for computing approximate solutions to elliptic boundary value problems. The framework is consistent with homogenization when there is scale separation. We introduce an adaptive form of the finite element algorithms for solving problems with no clear scale separation. We present numerical simulations demonstrating the effectiveness and adaptivity of the multiscale method, assess its computational complexity, and discuss the relationship between this framework and other multiscale methods, such as wavelets, multiscale finite element methods, and the use of harmonic coordinates. We prove in detail that the projection based method captures homogenization when there is strong scale separation. Key words. Multiscale finite elements, numerical homogenization, variational multiscale method AMS subject classifications. 65N30, 35J20, 35B27 1. Introduction. In
Multiscale Mixed/Mimetic Methods on CornerPoint Grids
"... Abstract. Multiscale simulation is a promising approach to facilitate direct simulation of large and complex gridmodels for highly heterogeneous petroleum reservoirs. Unlike traditional simulation approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by i ..."
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Cited by 15 (6 self)
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Abstract. Multiscale simulation is a promising approach to facilitate direct simulation of large and complex gridmodels for highly heterogeneous petroleum reservoirs. Unlike traditional simulation approaches based on upscaling/downscaling, multiscale methods seek to solve the full flow problem by incorporating subscale heterogeneities into local discrete approximation spaces. We consider a multiscale formulation based on a hierarchical grid approach, where basis functions with subgrid resolution are computed numerically to correctly and accurately account for subscale variations from an underlying (finescale) geomodel when solving the global flow equations on a coarse grid. By using multiscale basis functions to discretise the global flow equations on a (moderatelysized) coarse grid, one can retain the efficiency of an upscaling method, while at the same time produce detailed and conservative velocity fields on the underlying fine grid. For pressure equations, the multiscale mixed finiteelement method (MsMFEM) has shown to be a particularly versatile approach. In this paper we extend the method to cornerpoint grids, which is the industry standard for modelling complex
Multiscale iterative techniques and adaptive mesh refinement for flow in porous media
, 2002
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Multiscale modeling of acoustic wave propagation
"... Conventional finitedifference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir ..."
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Cited by 5 (3 self)
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Conventional finitedifference methods produce accurate solutions to the acoustic and elastic wave equation for many applications, but they face significant challenges when material properties vary significantly over distances less than the grid size. This challenge is likely to occur in reservoir characterization studies, because important reservoir heterogeneity can be present on scales of several meters to ten meters. Here, we describe a new multiscale finiteelement method for simulating acoustic wave propagation in heterogeneous media that addresses this problem by coupling fine and coarsescale grids. The wave equation is solved on a coarse grid, but it uses basis functions that are generated from the fine grid and allow the representation of the finescale variation of the wavefield on the coarser grid. Time stepping also takes place on the coarse grid, providing further speed gains. Another important property of the method is that the basis functions are only computed once, and time savings are even greater when simulations are repeated for many source locations. We first present validation results for simple test models to demonstrate and quantify potential sources of error. These tests show that the finescale solution can be accurately approximated when the coarse grid applies a discretization up to four times larger than the original fine model. We then apply the multiscale algorithm to simulate a complete 2D seismic survey for a model with strong, finescale scatterers and apply standard migration algorithms to the resulting synthetic seismograms. The results again show small errors. Comparisons to a model that is upscaled by averaging densities on the fine grid show that the multiscale results are more accurate.
NUMERICAL HOMOGENIZATION: SURVEY, NEW RESULTS, AND PERSPECTIVES
, 2012
"... These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problem ..."
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Cited by 5 (1 self)
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These notes give a state of the art of numerical homogenization methods for linear elliptic equations. The guideline of these notes is analysis. Most of the numerical homogenization methods can be seen as (more or less different) discretizations of the same family of continuous approximate problems, which Hconverges to the homogenized problem. Likewise numerical correctors may also be interpreted as approximations of Tartar’s correctors. Hence the convergence analysis of these methods relies on the Hconvergence theory. When one is interested in convergence rates, the story is different. In particular one first needs to make additional structure assumptions on the heterogeneities (say periodicity for instance). In that case, a crucial tool is the spectral interpretation of the corrector equation by Papanicolaou and Varadhan. Spectral analysis does not only allow to obtain convergence rates, but also to devise efficient new approximation methods. For both qualitative and quantitative properties, the development and the analysis of numerical homogenization methods rely on seminal concepts of the homogenization theory. These notes contain some new results.