Results 1  10
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487
HOMOGENIZATION AND TWOSCALE CONVERGENCE
, 1992
"... Following an idea of G. Nguetseng, the author defines a notion of "twoscale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A corrector ..."
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Cited by 185 (11 self)
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Following an idea of G. Nguetseng, the author defines a notion of "twoscale" convergence, which is aimed at a better description of sequences of oscillating functions. Bounded sequences in L2(f) are proven to be relatively compact with respect to this new type of convergence. A correctortype theorem (i.e., which permits, in some cases, replacing a sequence by its "twoscale " limit, up to a strongly convergent remainder in L2(12)) is also established. These results are especially useful for the homogenization of partial differential equations with periodically oscillating coefficients. In particular, a new method for proving the convergence of homogenization processes is proposed, which is an alternative to the socalled energy method of Tartar. The power and simplicity of the twoscale convergence method is demonstrated on several examples, including the homogenization of both linear and nonlinear secondorder elliptic equations.
A Multiscale Finite Element Method For Elliptic Problems In Composite Materials And Porous Media
 JOURNAL OF COMPUTATIONAL PHYSICS
, 1997
"... 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 ..."
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Cited by 162 (23 self)
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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 multiplescale 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 ...
ANALYSIS OF MULTISCALE METHODS
, 2004
"... The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method. ..."
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Cited by 127 (13 self)
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The heterogeneous multiscale method gives a general framework for the analysis of multiscale methods. In this paper, we demonstrate this by applying this framework to two canonical problems: The elliptic problem with multiscale coefficients and the quasicontinuum method.
Frequency content of randomly scattered signals
 PART I, WAVE MOTION
, 1990
"... The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equatio ..."
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Cited by 73 (20 self)
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The statistical properties of acoustic signals reflected by a randomly layered medium are analyzed when a pulsed spherical wave issuing from a point source is incident upon it. The asymptotic analysis of stochastic equations and geometrical acoustics is used to arrive at a set of transport equations that characterize multiply scattered signals observed at the surface of the layered medium. The results of extensive numerical simulations are presented, illustrating the scope of the theory. A number of inverse problems for randomly layered media are also formulated where we
Mixed Finite Elements for Elliptic Problems with Tensor Coefficients as CellCentered Finite Differences
 SIAM J. NUMER. ANAL
, 1997
"... We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor ..."
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Cited by 63 (36 self)
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We present an expanded mixed finite element approximation of second order elliptic problems containing a tensor coefficient. The mixed method is expanded in the sense that three variables are explicitly approximated, namely, the scalar unknown, the negative of its gradient, and its flux (the tensor coefficient times the negative gradient). The resulting linear system is a saddle point problem. In the case of the lowest order RaviartThomas elements on rectangular parallelepipeds, we approximate this expanded mixed method by incorporating certain quadrature rules. This enables us to write the system as a simple, cellcentered finite difference method, requiring the solution of a sparse, positive semideflnite linear system for the scalar unknown. For a general tensor coefficient, the sparsity pattern for the scalar unknown is a nine point stencil in two dimensions, and 19 points in three dimensions. Existing theory shows that the expanded mixed method gives optimal order ap proximations in the L a and HSnorms (and superconvergence is obtained between the Laprojection of the scalar variable and its approximation). We show that these rates of convergence are retained for the finite difference method. If h denotes the maximal mesh spacing, then the optimal rate is O(h). The superconvergence rate O(h ) is obtained for the scalar unknown and rate O(h 3/) for its gradient and flux in certain discrete norms; moreover, the full O(h ) is obtained in the strict interior of the domain. Computational results illustrate these theoretical results.
Particles and fields in fluid turbulence
 Rev. Mod. Phys
, 2001
"... 5E 8F #a`TbO8c=/d/e_Lgfah \ C3;i) 0 C"jS *) k "jSlZX9 ( 2 CQ#WNTmM/e e_L/#9? =/]?\MOEa 7#T^_CD) + The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, ..."
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Cited by 55 (5 self)
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5E 8F #a`TbO8c=/d/e_Lgfah \ C3;i) 0 C"jS *) k "jSlZX9 ( 2 CQ#WNTmM/e e_L/#9? =/]?\MOEa 7#T^_CD) + The understanding of fluid turbulence has considerably progressed in recent years. The application of the methods of statistical mechanics to the description of the motion of fluid particles, i.e. to the Lagrangian dynamics, has led to a new quantitative theory of intermittency in turbulent transport. The first analytical description of anomalous scaling laws in turbulence has been obtained. The underlying physical mechanism reveals the role of statistical integrals of motion in nonequilibrium systems. For turbulent transport, the statistical conservation laws are hidden in the evolution of groups of fluid particles and arise from the competition between the expansion of a group and the change of its geometry. By breaking the scaleinvariance symmetry, the statistically conserved quantities lead to the observed anomalous scaling of transported fields. Lagrangian methods also shed new light on some practical issues, such as mixing and turbulent magnetic dynamo. 1 n4oqpsrutKpsrwv
Convection enhanced diffusion for periodic flows
"... We study the in uence of convection by periodic or cellular ows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by twodimensional, steady, divergencefree, periodic velocity fields. ..."
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Cited by 54 (5 self)
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We study the in uence of convection by periodic or cellular ows on the effective diffusivity of a passive scalar transported by the fluid when the molecular diffusivity is small. The flows are generated by twodimensional, steady, divergencefree, periodic velocity fields.
Improved Time Bounds for NearOptimal Sparse Fourier Representations
 in Proc. SPIE Wavelets XI
, 2003
"... We study the problem of finding a Fourier representation R of B terms for a given discrete signal A of length N . The Fast Fourier Transform (FFT) can find the optimal Nterm representation in O(N log N) time, but our goal is to get sublinear algorithms for B ! N , typically, B N . Suppose kAk2 M ..."
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Cited by 49 (11 self)
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We study the problem of finding a Fourier representation R of B terms for a given discrete signal A of length N . The Fast Fourier Transform (FFT) can find the optimal Nterm representation in O(N log N) time, but our goal is to get sublinear algorithms for B ! N , typically, B N . Suppose kAk2 M kRoptk 2 , where Ropt is the optimal output. The previously best known algorithms output R such that kA \Gamma Rk poly(B; log(1=ffi); log N; log M; 1=ffl): Even though this is sublinear in the input size, the dominating term is the polynomial factor in B which is B . In our experience, this is a limitation in practice. Our main result is a significantly improved algorithm for this problem. Our algorithms output R such that kA \Gamma Rk B \Delta poly(log(1=ffi); log N; log M; 1=ffl): We also obtain improvements for higher dimensional Fourier transforms. We need two crucial ideas to achieve this bound: bulk sampling and estimation for multipoint polynomial evaluation using an unevenlyspaced Fourier tranform, and construction and use of arithmeticprogression independent random variables. Our improved algorithms are likely to find many applications. 1
Convergence Of A Nonconforming Multiscale Finite Element Method
, 2000
"... . 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. 169189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coe#cients. This ..."
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Cited by 48 (10 self)
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. 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. 169189] has been introduced to capture the large scale solutions of elliptic equations with highly oscillatory coe#cients. 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 di#culty also arises in numerical upscaling methods. An oversampling technique has been introduced to alleviate this di#culty [T. Y. Hou and X. H. Wu, J. Comput. Phys., 134 (1997), pp. 169189]. 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....
Analysis of the heterogeneous multiscale method for ordinary differential equations
 Commun. Math. Sci
"... Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution. ..."
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Cited by 46 (4 self)
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Abstract. The heterogeneous multiscale method (HMM) is applied to various parabolic problems with multiscale coefficients. These problems can be either linear or nonlinear. Optimal estimates are proved for the error between the HMM solution and the homogenized solution.