Weinan E. Principles of Multiscale Modeling

Подождите немного. Документ загружается.

Bibliography

[1] A. Abdulle, “The ﬁnite element heterogeneous multiscale method: a computational

strategy for multiscale PDEs,” GAKUTO Int. Ser. Math. Sci. Appl., vol. 31, pp.

135–184, 2009.

[2] A. Abdulle, “On a priori error analysis of fully discrete heterogeneous multiscale

ﬁnite element method,” Multiscale Model. Simul., vol. 4, no 2, pp. 447–459, 2005.

[3] A. Abdulle and A. Nonnenmacher, “Adaptive ﬁnite element heterogeneous multi-

scale method for homogenization problems,” Comput. Methods Appl. Mech. Engrg.,

in press.

[4] U. Andersson, B. Engquist, G. Ledfelt and O. Runborg, “A contribution to wavelet-

based subgrid modeling,” Appl. Comput. Harmon. Anal., vol. 7, pp. 151-164, 1999.

[5] T. Arbogast, “Numerical subgrid upscaling of two-phase ﬂow in porous media,”

Numerical Treatment of Multiphase Flows in Porous Media, Z. Chen et al., eds.,

Lecture Notes in Physics vol. 552, pp. 35–49, Springer, Berlin, 2000.

[6] T. Arbogast and K. J. Boyd, “Subgrid upscaling and mixed multiscale ﬁnite ele-

ments,” SIAM J. Numer. Anal., vol. 44, no. 3, pp. 1150–1171, 2006.

[7] I. Babuˇska, “Homogenization and its applications, mathematical and computa-

tional problems,” In Numerical Solutions of Partial Diﬀerential Equations-III,

(SYNSPADE 1975, College Park MD, May 1975), B. Hubbard eds., pp. 89–116,

Academic Press, New York, 1976.

[8] I. Babuˇska, “Solution of interface by homogenization I, II, III,” SIAM J. Math.

Anal., vol. 7, pp. 603–634, 635–645,1976; vol. 8, pp. 923–937, 1977.

387

388 BIBLIOGRAPHY

[9] I. Babuˇska, “Solution of problems with interface and singularities,” In Mathematical

Aspects of Finite Elements in Partial Diﬀerential Equations. Cal de Boor eds., pp.

213-277, Academic Press, New York–London, 1974.

[10] I. Babuˇska, U. Banerjee and J. Osborn, “Survey of meshless and generalized ﬁ-

nite element methods: A uniﬁed approach,” Acta Numerica, pp. 1–125, Cambridge

University Press, 2003.

[11] I. Babuˇska, G. Caloz and J. Osborn, “Special ﬁnite element method s for a class of

second order elliptic with rough coeﬃcients,” SIAM J. Numer. Anal., vol. 31, pp.

945–981, 1994.

[12] I. Babuˇska and J. E. Osborn, “Finite element methods for the solution of problems

with rough data,” In Singularities and Constructive Methods and Their Treatment,

Springer-Verlag Lecture Notes in Mathematics, vol. 1121, pp. 1–18, P. Grisvard, W.

Wendland, and J. R. Whiteman, eds., Springer-Verlag, Berlin-New York, 1985.

[13] I. Babuˇska and J. E. Osborn, “Generalized ﬁnite element methods: Their perfor-

mance and their relation to mixed methods,” SIA M J. Numer. Anal., vol. 20, pp.

510–536, 1983.

[14] A. Bensoussan, J.-L . Lions and G. C. Papanicolaou, “Asymptotic Analysis for Peri-

odic Structures,” Studies in Mathematics and Its Applications, vol. 5, 1978.

[15] G. Beylkin and M. Brewster, “A multiresolution strategy for numerical homogeniza-

tion,” Appl. Comput. Harmon. Anal., vol. 2, pp. 327–349, 1995.

[16] G. Beylkin, R. Coifman and V. Rokhlin, “Fast wavelet transforms and numerical

analysis,” Comm. Pure Appl. Math., vol. 44, no. 2, pp . 141–183, 1991.

[17] F. Brezzi and L.D. Marini, “Augmented spaces, two-level methods, and stabilizing

subgrids,” Int. J. Numer. Meth. Fluids, vol. 40, pp. 31–46, 2002.

[18] F. Brezzi, D. Marini and E. S¨uli, “Residual-free bubbles for advection-diﬀusion prob-

lems: The general err or analysis,” Numer. Math., vol. 85, pp. 31–47, 2000.

[19] F. Brezzi and A. Russo, “Choosing bubbles for advection-diﬀusion problems,” Math.

Models Methods Appl. Sci., vol. 4, pp. 571–587, 1994.

BIBLIOGRAPHY 389

[20] Z. Chen and T. Y. Hou, “A mixed multiscale ﬁnite element method for elliptic

problems with oscillating coeﬃcients,” Math. Comp., vol. 72, pp. 541–576, 2003.

[21] P.G. Ciarlet, T he Finite Element Method for Elliptic Problems, SIAM, 2002.

[22] J. Z. Cui and L. Q. Cao, “The two-scale analysis method for woven comp osite

materials,” Engineering Computation and Computer Simulation I, pp. 203-212, Zhi-

Hua Zhong eds., Hunan University Press, 1995.

[23] M. Dorobantu and B. Engquist, “Wavelet-based numerical homogenization,” SIAM

J. Numer. Anal., vol. 35, no. 2, pp. 540–559, 1998.

[24] W. E, T. Li and J. Lu, “Localized basis of eigen-subspaces and operator compres-

sion,” PNAS, vol. 107, pp. 1273–1278, 2010.

[25] W. E, P.B. Ming and P.W. Zhang, “Analysis of the heterogeneous multiscale method

for elliptic homogenization problems,” J. Amer. Math. Soc., vol. 18, pp. 121–156,

2005.

[26] W. E and X. Yue, “Heterogeneous multiscale method for locally self-similar prob-

lems,” Comm. Math. Sci., vol. 2, pp. 137–144, 2004.

[27] Y. R. Efendiev, T. Y. Hou and X. Wu, “Convergence of a nonconforming multiscale

ﬁnite element method,” SIAM J. Numer. Anal., vol. 37, pp. 888–910, 2000.

[28] B. Engquist and E. Luo, “Convergence of a multigrid method for elliptic equations

with highly oscillatory coeﬃcients,” SIAM J. Numer. Anal., vol. 34, no. 6, pp. 2254–

2273, 1997.

[29] B. Engquist and O. Runborg, “Wavelet-based numerical homogenization with ap-

plications,” In Multiscale and Multiresolution Methods: Theory and Applications,

vol. 20, pp. 97–148, T.J. Barth, T.F. Chan and R. Haimes, eds.; Lecture Notes in

Computational Science and Engineering, Springer Verlag, Heidelberg, 2001.

[30] J. M. Farmer, “Upscaling: a review,” Int. J. Numer. Meth. Fluids, vol. 40, pp.

63–78, 2002.

[31] J. Fish, “The s-version ﬁnite element method,” Computation and Structure, vol. 43,

pp. 539–547, 1992.

390 BIBLIOGRAPHY

[32] J. Fish and A. Wagiman, “Multiscale ﬁnite element method for a locally nonperiodic

heterogeneous medium,” Computational Mechanics, vol. 12, pp. 164–180, 1993.

[33] M. J. Gander, “Optimized Schwarz methods,” SIAM J. Numer. Anal., vol. 44, pp.

699–731, 2006.

[34] A. Gloria and F . Otto, “An optimal variance estimate in stochastic homogenization

of discrete elliptic equations,” preprint, 2009.

[35] G. H. Golub and C. Van Loan, Matrix Computations, Johns Hopkins University

Press, 3rd ed., 1996.

[36] H. Han, Z. Huang and B. Kellogg, “A tailored ﬁnite point method for a singular

perturbation problem on an unbounded domain,” J. Sci. Comput., vol. 36, pp. 243–

261, 2008.

[37] T. Y. Hou and X.-H. Wu, “A multiscale ﬁnite element method for elliptic problems

in composite materials and porous media,” J. Comput. Phys., vol. 134, pp. 169–189,

1997.

[38] T. Y. Hou, X.-H. Wu and Z. Cai, “Convergence of a multiscale ﬁnite element method

for elliptic problems with rapidly oscillating coeﬃcients,” Math. Comp., vol. 68, pp.

913–943, 1999.

[39] B.D. Hughes, Random Walks and Random Environments, vol. 2, Oxford University

Press, 1996.

[40] T.J.R. Hughes, “Multiscale phenomena: Green’s functions, the Dirichlet-to-

Neumann formulation, subgrid scale models, bubbles and the origin of stablized

methods,” Comput. Methods Appl. Mech. Engrg., vol. 127, pp. 387–401, 1995.

[41] T.J.R. Hughes, G. Feij´oo, L. Mazzei and J.-B. Quincy, “The variational multiscale

method – a paradigm for computational mechanics,” Comput. Methods Appl. Mech.

Engrg., vol. 166, pp. 3–24, 1998.

[42] W. K. Liu, Y. F. Zhang and M. R. Ramirez, “Multiple scale ﬁnite element methods,”

Int J Numer Meth Eng, vol. 32, pp. 969–990, 1991.

BIBLIOGRAPHY 391

[43] P. B. Ming and X. Yue, “Numerical methods for multiscale elliptic problems,” J.

Comput. Phys., vol. 214, pp . 421–445, 2006.

[44] J.D. Moulton, J.E. Dendy and J.M. Hyman, “The black box multigrid numerical

homogenization algorithm,” J. Comput. Phys., vol. 141, pp. 1–29, 1998.

[45] N. Neuss, W. J¨ager, G. Wittum, “Homogenization and multigrid,” Computing, vol.

66, no. 1, pp. 1–26, 2001.

[46] J. Nolen, G. Papanicolaou and O. Pironneau, “A framework for adaptive multiscale

methods for elliptic problems,” Multiscale Model. Simul., vol. 7, pp. 171–196, 2008.

[47] J.T. Oden and K.S. Vemaganti, “Estimation of local modeling error and goal-

oriented adaptive modeling of heterogeneous materials; Part I: Error estimates and

adaptive algorithms,” J. Comput. Phys., vol. 164, pp. 22–47, 2000.

[48] A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Diﬀerenti al

Equations, Oxford University Press, 1999.

[49] E. Ramm, A. Hund and T. Hettich, “A variational multiscale model for compos-

ites with special emphasis on the X-FEM and level sets,” Proc. EURO-C 2006, G.

Meschke, R. de Borst, H. Mang, N. Bicanic eds., 2006.

[50] M. Sahimi, Flow and Transport in Porous Medi a and Fractured Rock, John Wiley,

1995.

[51] G. Sangalli, “Capturing small scales in elliptic problems using a residual-free bubbles

ﬁnite element method,” Multiscale Model. Simul., vol. 1, pp. 485–503, 2003.

[52] C. Schwab and A.-M. Matache, “Two-scale FEM for homogenization problems,”

Mathematical Modellin g and Nume rical Simulation in Continuum Mechanics Conf.

Proc., Yamaguchi, Japan, 2000; Lecture Notes in Comp. Sci. and Engrg., I. Babuska,

et.al (eds.), Springer-Verlag, 2002.

[53] G. Strang and G.J. Fix, An Analysis of the Finite Element Method, Wellesley-

Cambridge, 2nd ed., 2008.

[54] S. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic

Properties, Springer-Verlag, New York, 2002.

392 BIBLIOGRAPHY

[55] X.-Y. Yue, “Residual-free bubble methods for numerical h omogenization of elliptic

problems,” preprint, 2008.

[56] X.-Y. Yue and W. E, “ The local microscale problem in the multiscale modelling

of strongly heterogeneous media: Eﬀect of boundary conditions and cell size,” J.

Comput. Phys., vol. 222, pp . 556–572, 2007.

[57] V. V. Zhikov, S. M. Kozlov and O. A. Oleinik, Homogenization of Diﬀerential Op-

erators and Integral Functionals, Springer-Verlag Berlin Heidelberg, 1994.

Chapter 9

Problems wi th Multiple Time Scales

The last chapter was devoted to an example with disparate spatial scales. In this and

the next chapters, we will focus on problems with disparate time scales. This chapter

is concerned with problems which are “stiﬀ” in a broad sense, i.e. problems for which

the fast processes relax in a short time scale and the slow processes obey an eﬀective

dynamics obtained by averaging the slow component of the original dynamics over the

quasi-equilibrium distribution of the fast processes. Classical stiﬀ ODEs will be consid-

ered as a special case. The next chapter is concerned with the other important class of

problems with disparate time scales, namely, the rare events.

9.1 ODEs with disparate time scales

9.1.1 General setup for limit theorems

Consider the diﬀerential equation

x = f (x, ε) (9.1.1)

We have used an explicit dependence on ε to indicate that the system has disparate time

scales. For example, f may take the form

f(x, ε) = f

(x) +

(x) (9.1.2)

or more generally

f(x, ε) = f

(x, ε) +

(x, ε) (9.1.3)

393

394 CHAPTER 9. PROBLEMS WITH MULTIPLE TIME SCALES

where f

and f

are bounded in ε. Roughly speaking, we have in mind the following

picture: The fast component of the dynamics brings the system to some local equilibrium

states and this relaxation process is faster than the time scale on which the slow variables

change. These local equilibrium states are often called quasi-equilibrium. They are

parametrized by the local values of the slow variables and they inﬂuence the dynamics

of the slow variables. We are interested in capturing the dynamics of the slow variables

without resolving the details of the fast component of the dynamics in (9.1.1).

Let z = ϕ(x). If f takes the form of (9.1.3), then

z = ∇ϕ ·

x = ∇ϕ ·





(9.1.4)

Therefore, to be a slow variable, ϕ should satisfy

∇ϕ · f

= 0 (9.1.5)

In other words, if we deﬁne the “virtual fast dynamics”, as in [7]:

y =

(y, ε) (9.1.6)

then the slow variables are conserved quantities for the virtual fast dynamics. Assume

that z = (z

, z

, ··· , z

) is a complete set of slow variables, i.e. the ergo dic components

for the virtual fast dynamics (9.1.6) are parametrized completely by the values of z.

Denote by µ

(dx) the equilibrium distribution of the virtual fast dynamics, when the

slow variable z is ﬁxed at the value

z. Since

z = ∇ϕ(x) · f

(x, ε) (9.1.7)

The averaging principle discussed in Chapter 2 suggests th at the eﬀective dynamics for

the slow variable z should be described by:

z =

∇ϕ(x) · f

(x, ε)µ

(dx) (9.1.8)

We note that the set of slow variables have to be suﬃciently large such th at the

quasi-equilibrium distribution of the fast dynamics is completely parametrized by z. See

[17] for a more thorough discussion of this and related issues.

9.1. ODES WITH DISPARATE TIME SCALES 395

Example 1.

= f(x, y) (9.1.9)

= −

(y − φ(x)) (9.1.10)

In this case, x is a slow variable. Fix x, the quasi-equilibrium distribution for the fast

variable y is given by a delta function µ

(dy) = δ(y − φ(x))dy. The eﬀective equation

for the slow variable x is given by:

f(x, y)µ

(dy) = f(x, φ(x)) (9.1.11)

Example 2.

= 1 + y (9.1.12)

z (9.1.13)

= −

y (9.1.14)

It is easy to see that the dynamics of y and z exhibit fast oscillations. One choice of

slow variables is (x, r =

+ z

). Given the variables (x, r), the quasi-equilibrium

distribution is a uniform distribution concentrated on the circle of radius r in the y − z

plane, centered at the origin. The eﬀective equation for the slow variables is simply:

= 1 (9.1.15)

= 0 (9.1.16)

Example 3.

= f(x, y) (9.1.17)

= −

(y − φ(x)) +

˙w (9.1.18)

where ˙w is a standard Gaussian white noise. Again, x is one possible choice of the slow

variable. Given the value of x, the quasi-equilibrium distribution is a normal distribution

in y with mean φ(x) and variance 1. The eﬀective dynamics for x is obtained by averaging

f(x, y) with respect to a standard normal distribution in y.

396 CHAPTER 9. PROBLEMS WITH MULTIPLE TIME SCALES

9.1.2 Implicit methods

For a problem of the type (9.1.9), a standard explicit scheme such as the forward

Euler method would require a time step size of O(ε). However, if we are only interested in

capturing accurately the dynamics of the slow variable, not the details of the relaxation

process for the fast variable, as is often the case in applications, then there are more

eﬃcient approaches. The most well-known examples are the implicit methods.

It is more convenient to consider the more general problem of

= F (z, ε) (9.1.19)

The simplest implicit method is the backward Euler scheme:

n+1

− z

∆t

= F (z

n+1

, ε) (9.1.20)

The reason that this is better than the forward Euler scheme

n+1

− z

∆t

= F (z

, ε) (9.1.21)

is that backward Euler has much better stability p roperties. In order to see this diﬀerence,

let us introduce the notion of stability region [21].

Given an ODE solver, consider its application to the following linear problem

= λy (9.1.22)

where λ is a complex constant. The stability region of the ODE solver is deﬁned as the

subset of complex numbers z = λ∆t such that the numerical solution {y

} produced by

the ODE solver is bounded in n, i.e. there exists constant C which is independent of n

such that

| ≤ C (9.1.23)

When the forward Euler method is applied to (9.1.22), we have y

n+1

= (1 + z)y

Hence (9.1.23) is satisﬁed if

|1 + z| ≤ 1 (9.1.24)

This is a unit disk of radius 1 on the complex z-plane centered at z = −1. For backward

Euler, we have y

n+1

= (1 − z)

−1

. Hence (9.1.24) is satisﬁed when

|1 − z| ≥ 1 (9.1.25)