We present an efficient method for the computation of homogenized coefficients of divergence-form operators with random coefficients. The approach is based on a multiscale representation of the homogenized coefficients. We then implement the method numerically using a finite-element method with hierarchical hybrid grids, which is a semi-implicit method allowing for significant gains in memory usage and execution time. Finally, we demonstrate the efficiency of our approach on two- and three-dimensional examples, for piecewise-constant coefficients with corner discontinuities. For moderate ellipticity contrast and for a precision of a few percentage points, our method allows to compute the homogenized coefficients on a laptop computer in a few seconds, in two dimensions, or in a few minutes, in three dimensions.
It has been observed long ago that inappropriate boundary conditions for approximate cell problems'' can cause important resonant errors'', and initial attempts at bypassing the problem involved the notion of oversampling @cite_5 @cite_11 @cite_45 @cite_57 . A powerful approach has been studied in @cite_32 @cite_42 @cite_7 @cite_26 @cite_40 @cite_31 @cite_28 @cite_16 , based on the introduction of a small zero-order term in the equation. The method we propose here, by combining this idea with a multiscale decomposition, enables to take fuller advantage of this idea. We refer to @cite_39 for a detailed comparison between the single-scale and the multiscale approaches. As is shown in @cite_53 , the benefits of the multiscale approach can be seen even in the setting of periodic coefficient fields, if we operate under the constraint that the lattice of periods is unknown.
@cite_26 @cite_7 @cite_28 @cite_53 @cite_42 @cite_32 @cite_39 @cite_57 @cite_40 @cite_45 @cite_5 @cite_31 @cite_16 @cite_11
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In this paper, we study a multiscale finite element method for solving a class of elliptic problems arising from composite materials and flows in porous media, which contain many spatial scales. The method is designed to efficiently capture the large scale behavior of the solution without resolving all the small scale features. This is accomplished by constructing the multiscale finite element base functions that are adaptive to the local property of the differential operator. Our method is applicable to general multiple-scale problems without restrictive assumptions. The construction of the base functions is fully decoupled from element to element; thus, the method is perfectly parallel and is naturally adapted to massively parallel computers. For the same reason, the method has the ability to handle extremely large degrees of freedom due to highly heterogeneous media, which are intractable by conventional finite element (difference) methods. In contrast to some empirical numerical upscaling methods, the multiscale method is systematic and self- consistent, which makes it easier to analyze. We give a brief analysis of the method, with emphasis on the “resonant sampling” effect. Then, we propose an oversampling technique to remove the resonance effect. We demonstrate the accuracy and efficiency of our method through extensive numerical experiments, which include problems with random coefficients and problems with continuous scales. Parallel implementation and performance of the method are also addressed. Many multiscale methods are based on the idea of extracting macroscopic behavior of solutions by solving an array of microscale models over small domains. A key ingredient in such multiscale methods is the boundary condition and the size of the computational domain over which the microscale problems are solved. This problem is systematically investigated in the present paper in the context of modeling strongly heterogeneous media. Three different boundary conditions are considered: the periodic boundary condition, Dirichlet boundary condition, and the Neumann boundary condition. Each is applied to several benchmark problems: the random checker-board problem, periodic problem with isotropic macroscale behavior, periodic problem with anisotropic macroscale behavior and periodic laminated media. In each case, convergence studies are conducted as the domain size for the microscale problem is changed. Convergence rates as well as the size of fluctuations in the computed effective coefficients are compared for the different formulations. In addition, we will discuss a mixed Dirichlet-Neumann boundary condition that is often used in porous medium modeling. We explain why that leads to unsatisfactory results and how it can be corrected. Also discussed are the different averaging methods used in extracting the effective coefficients. Suppose a discrete-time signal S(t), 0 spl les t<N, is a superposition of atoms taken from a combined time-frequency dictionary made of spike sequences 1 sub t= spl tau and sinusoids exp 2 spl pi iwt N spl radic N. Can one recover, from knowledge of S alone, the precise collection of atoms going to make up S? Because every discrete-time signal can be represented as a superposition of spikes alone, or as a superposition of sinusoids alone, there is no unique way of writing S as a sum of spikes and sinusoids in general. We prove that if S is representable as a highly sparse superposition of atoms from this time-frequency dictionary, then there is only one such highly sparse representation of S, and it can be obtained by solving the convex optimization problem of minimizing the l sup 1 norm of the coefficients among all decompositions. Here "highly sparse" means that N sub t +N sub w < spl radic N 2 where N sub t is the number of time atoms, N sub w is the number of frequency atoms, and N is the length of the discrete-time signal. Underlying this result is a general l sup 1 uncertainty principle which says that if two bases are mutually incoherent, no nonzero signal can have a sparse representation in both bases simultaneously. For the above setting, the bases are sinusoids and spikes, and mutual incoherence is measured in terms of the largest inner product between different basis elements. The uncertainty principle holds for a variety of interesting basis pairs, not just sinusoids and spikes. The results have idealized applications to band-limited approximation with gross errors, to error-correcting encryption, and to separation of uncoordinated sources. Related phenomena hold for functions of a real variable, with basis pairs such as sinusoids and wavelets, and for functions of two variables, with basis pairs such as wavelets and ridgelets. In these settings, if a function f is representable by a sufficiently sparse superposition of terms taken from both bases, then there is only one such sparse representation; it may be obtained by minimum l sup 1 norm atomic decomposition. The condition "sufficiently sparse" becomes a multiscale condition; for example, that the number of wavelets at level j plus the number of sinusoids in the jth dyadic frequency band are together less than a constant times 2 sup j 2 . In recent years a variety of high–order schemes for the numerical solution of conservation laws has been developed. In general, these numerical methods involve expensive flux evaluations in order to resolve discontinuities accurately. But in large parts of the flow domain the solution is smooth. Hence in these regions an unexpensive finite difference scheme suffices. In order to reduce the number of expensive flux evaluations we employ a multiresolution strategy which is similar in spirit to an approach that has been proposed by A. Harten several years ago. Concrete ingredients of this methodology have been described so far essentially for problems in a single space dimension. In order to realize such concepts for problems with several spatial dimensions and boundary fitted meshes essential deviations from previous investigations appear to be necessary though. This concerns handling the more complex interrelations of fluxes across cell interfaces, the derivation of appropriate evolution equations for multiscale representations of cell averages, stability and convergence, quantifying the compression effects by suitable adapted multiscale transformations and last but not least laying grounds for ultimately avoiding the storage of data corresponding to a full global mesh for the highest level of resolution. The objective of this paper is to develop such ingredients for any spatial dimension and block structured meshes obtained as parametric images of Cartesian grids. We conclude with some numerical results for the two–dimensional Euler equations modeling hypersonic flow around a blunt body.