Homogenization of Linear Transport Equations with Oscillatory Vector Fields

Hou, Thomas Y.; Xue, Xin

doi:10.1137/0152003

Cited by 51 publications

(49 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Combined with DiPerna's method for reducing measure-valued solutions of conservation laws to Dirac masses 18], it allows us to rigorously homogenize nonlinear transport equations, and nonlinear hyperbolic equations with oscillating forcing terms [19], [20]. In the case of linear hyperbolic equations, two-scale convergence has also been applied by Amirat, Hamdache, and Ziani [5] and Hou and Xin [26]. 4.…”

Section: Y)(xy)dxdymentioning

confidence: 99%

Homogenization and Two-Scale Convergence

Allaire¹

1992

SIAM J. Math. Anal.

1,968

2,637

View full text Add to dashboard Cite

Abstract. Following an idea of G. Nguetseng, the author defines a notion of "two-scale" 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-type theorem (i.e., which permits, in some cases, replacing a sequence by its "two-scale" 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 so-called energy method of Tartar [10], [40] for details). This. method is very simple and powerful, but unfortunately is formal since, a priori, the ansatz (0.3) does not hold true. Thus, the two-scale asymptotic expansion method is used only to guess the form of the homogenized operator L, and other arguments are needed to prove the convergence of the sequence u to u. To this end, the more general and powerful method is the so-called energy method of Tartar [42]. Loosely speaking, it amounts to multiplying equation (0.1) by special test functions (built with the solutions of the cell equation), and passing to the limit as e-0. Although products of weakly convergent sequences are involved, we can actually pass to the limit thanks to some "compensated compactness" phenomenon due to the particular choice of test functions.Despite its frequent success in the homogenization of many different types of equations, this way of proceeding is not entirely satisfactory. It involves two different steps, the formal derivation of the cell and homogenized equation, and the energy method, which have very little in common. In some cases, it is difficult to work out the energy method (the construction of adequate test functions could be especially tricky). The energy method does not take full advantage of the periodic structure of the problem (in particular, it uses very little information gained with the two-scale asymptotic expansion). The latter point is not surprising since the energy method was not conceived by Tartar for periodic problems, but rather in the more general (and more difficult) context of H-convergence. Thus, there is room for a more efficient homogenization method, dedicated to partial differential equations with periodically oscillating coefficients. The purpose of the present paper is to provide such a method that we call two-scale convergence method.This new method relies on the following theorem, which was first proved by Nguetseng [36].

show abstract

Section: Y)(xy)dxdymentioning

confidence: 99%

Homogenization and Two-Scale Convergence

Allaire¹

1992

SIAM J. Math. Anal.

1,968

2,637

View full text Add to dashboard Cite

show abstract

“…[1]: u(x, t) converges pointwise to a limit, v(x, t), which solves the homogenized equation In the two-dimensional case, however, the problem of homogenization becomes more intricate. The effective equations depend sensitively on the topological structure and ergodicity of the flow on T 2 generated by the vector field a, see [3,4] and the references therein. Hou and Xin studied in [4] the homogenization of transport equations of the form (1.1) with divergence-free vector fields, ∇ x · a = 0, as a model problem for the incompressible Euler equations with oscillatory data,…”

mentioning

confidence: 99%

Homogenization of Two-Dimensional Linear Flows with Integral Invariance

Tassa¹

1997

SIAM J. Appl. Math.

View full text Add to dashboard Cite

show abstract

“…For deriving upscaled equations, we will first homogenize (6.1) along the streamlines, and then to homogenize across the streamlines. The homogenization along the streamlines can be done following Bourgeat and Mikelic [17] or following Hou and Xin [68] and E [36]. The latter uses two-scale convergence theory and we refer to [95] for the results on homogenization of (6.1) using two-scale convergence theory.…”

Section: Generalizations Of Msfem and Some Remarksmentioning

confidence: 99%

Multiscale Computations for Flow and Transport in Porous Media

Hou

2009

Series in Contemporary Applied Mathematics

Self Cite

View full text Add to dashboard Cite

Abstract. Many problems of fundamental and practical importance have multiple scale solutions. The direct numerical solution of multiple scale problems is difficult to obtain even with modern supercomputers. The major difficulty of direct solutions is the scale of computation. The ratio between the largest scale and the smallest scale could be as large as 10 5 in each space dimension. From an engineering perspective, it is often sufficient to predict the macroscopic properties of the multiple-scale systems, such as the effective conductivity, elastic moduli, permeability, and eddy diffusivity. Therefore, it is desirable to develop a method that captures the small scale effect on the large scales, but does not require resolving all the small scale features. This paper reviews some of the recent advances in developing systematic multiscale methods with particular emphasis on multiscale finite element methods with applications to flow and transport in heterogeneous porous media.

show abstract

Homogenization of Linear Transport Equations with Oscillatory Vector Fields

Cited by 51 publications

References 6 publications

Homogenization and Two-Scale Convergence

Homogenization and Two-Scale Convergence

Homogenization of Two-Dimensional Linear Flows with Integral Invariance

Multiscale Computations for Flow and Transport in Porous Media

Contact Info

Product

Resources

About