行政院國家科學委員會專題研究計畫 成果報告
破裂介質中的熱傳導方程式
研究成果報告(精簡版)
計 畫 類 別 ： 個別型
計 畫 編 號 ： NSC 98-2115-M-009-011-
執 行 期 間 ： 98 年 08 月 01 日至 99 年 07 月 31 日
執 行 單 位 ： 國立交通大學應用數學系（所）
計 畫 主 持 人 ： 葉立明
報 告 附 件 ： 出席國際會議研究心得報告及發表論文
處 理 方 式 ： 本計畫可公開查詢
中 華 民 國 99 年 10 月 29 日
背景及目的
很多的化學工廠的廢水經由水井排放到地底，地面的有毒廢棄物也經由雨水進入地底，核能電
廠儲存在地底下的核廢棄物由於時間的關係造成容器腐蝕、或是由於地層變動造成容器破裂，放射
性物質也因此進入地下水。這些都造成日常飲用水的不安全。這些問題不是只發生在臺灣，世界其
它國家也有同樣的情形。歐美一些國家都有專責機構負責這些污染源的清除工作。身體內的血液在
血管中流動的問題，地底下的碳氫化合物的抽取等以上都屬於多相流問題[6,10,12,40]。因此，了
解多孔介質中的多相流的變化對解決很多實際問題是重要的。
這是個一年期的計劃。我們計劃探討熱傳導在破裂多孔介質中的宏觀模式。地底下的縫隙結
構與地質性質是十分複雜的，然而流體在地底下的運動卻與它們息息相關，譬如流體運動的方向、
快慢、time-scale 等[17,18,19,20,21,25,26, 27,30, 31,33]。破裂多孔介質是一種特殊的地質
結構，在這種介質中流體的運動往往呈現出兩種不同的 time scale 的現象。污染源擴散時平流與
對流現象則常發生在污染源的附近或較遠處。熱傳導現象(一種 transport equation)則對應到地
下污染源在地底擴散時的平流與對流問題。熱傳導在破裂多孔介質中自然也有流速快慢不同的區
域。描述多孔介質中的可混合與不可混合流體的運動方程式都包含有 transport equation。換言
之，了解熱傳導現象在破裂多孔介質中的的數學模式是要進一步了解污染源在地底擴散情形的基
礎。
探討在破裂多孔介質中的多相流的問題不是一個新的問題。但到目前為止討論的方式主要
仍是數值模擬加上一些平均的方法，數值模擬的缺點是需要很多的計算時間且是 case by case
的討論。至於用數學分析的討論方式則是最近才開始。數學文獻中關於破裂多孔介質中的多相流
的問題大多只單討論 flow equation[51,52]的部份。之前我們的計劃也討論 flow equation (一
種橢圓形方程式) 在破裂多孔介質中的情形。我們希望借助之前的經驗來幫助了解 transport
equation
在破裂多孔介質中的情形。底下是我們考慮的抛物線微分方程式。
3
is a domain and
=
∪
f
m. Denote absolute permeability by
K in
fand
k in
m, phase pressure by
P in
and
fp
in
m, and external source by
F
Q ,
in
and
fq ,
f
in
m. The equations are
K
P
Q
F
P
t
(
)
in
f
( T
0
,
)
,
k
p
q
f
p
t
(
)
in
m
( T
0
,
)
P
Q
n
k
p
q
n
K
)
(
)
(
on
m
( T
0
,
)
,
p
P
on
m
( T
0
,
)
P
P
in
f0
p
p
in
mwith periodic boundary condition on
.我們想考慮當
變動時，
P 的何種 norm 與
的變化無關?
除了抛物線微分方程式，我們也考慮橢圓形方程式在 perforated domain 的數值近似解
的誤差估計問題。
研究方法、進行步驟及執行進度。
這是一個抛物線微分方程式，我們以底下的方式進行: 一. 利用半群理論(semigroup
theory) 我們得到時間方面的解的均勻 Holder norm 的估計。 二.利用 three-step
compactness method 我們得到空間方面的解的均勻 Holder norm 的估計。三.為了估計不
連續截面的變化，我們用到 pseudodifferential operator [40]及 boundary integral
method[14]。在函數空間方面則利用 Besov space and Holder space [2,37,41]。由於考
慮的抛物線微分方程式是在破裂多孔介質中，介質基本上是不連續的，因此在空間方面的
解的 Holder norm 只在局部較平滑的區域得到，在不平滑的區域則沒有均勻 Holder norm 的
估計。
關於橢圓形方程式，我們則利用 three-step compactness method 與 Taylor
expansion 得到 數值近似解的誤差估計。
（四）成果與自評。
一、得到不均勻抛物線微分方程式的解的 Holder norm 的均勻估計結果[58]。有了此結
再利用 two-scale method 我們可以很容易的導出熱傳導在破裂多孔介質中的宏觀模
式。
二、得到橢圓形方程式在 perforated domain 的數值近似解的誤差估計[59]。
References
[1] E. Acerbi, V. Chiado Piat, G. Dal Maso, and D. Percivale
An extension theorem from
connected sets, and homogenization in general periodic domains.
Nonlinear Analysis,
18 (1992) 481-496.
[2] R. A. Adams Sobolev Spaces.
Academic Press, 1975.
[3] Gregoire Allaire
Homogenization and two-scale convergence.
SIAM I. Math. Anal.,
23 (1992) 1482-1518.
porous media.
Annali Scu. norm. sup. Pisa Cl. Sci. 12(4) (1985) 335-392.
[5] H. W. Alt and S. Luckhaus Quasilinear elliptic--parabolic differential equations.
Math. Z. 183 (1983) 311-341.
[6] S.N. Antontsev, A. V. Kazhikhov, and V.N. Monakhov
Boundary Value Problems in
Mechanics in Nonhomogeneous Fluids. Elsevier, {1990}.
[7] T. Arbogast
The existence of weak solutions to single porosity and simple
dual-porosity models of two-phase incompressible flow.
Nonlinear Analysis 19(11)
(1992) 1009-1031.
[8] T. Arbogast, J. Douglas, and U. Hornung
Derivation of the double porosity model of
single phase flow via homogenization theory,
SIAM J. Math. Anal., 21 (1990) 823-836.
[9] T. Arbogast, J. Douglas, and P. J. Paes~Leme Two models for the waterflooding of
naturally fractured reservoirs.
Proceedings, Tenth SPE Symposium on Reservoir
Simulation, SPE 18425, Society of Petroleum Engineers, Dallas, Texas (1989).
[10] Bear J., Buchlin J.M.
Modelling and Applications of Transpot Phenomena in Porous
Media. Kluwer Academic Publishers, Boston, London, 1991.
[11] Alain Bourgeat, Stephan Luckhaus, and Andro Mikelic
Convergence of the
homogenization process for a double-porosity model of immiscible two--phase flow.
SIAM J. Math. Anal. 27(6) (1996) 1520--1543.
[12] Chavent G., Jaffr'e J.
Mathematical Models and Finite Elements for Reservoir
Simulation. North-Holland, Amsterdam, 1986.
[13] Z. Chen Degenerate two-phase incompressible flow 1. existence, uniqueness and
regularity of a weak solution.
Journal of Differential Equations 171 (2001) 203-232.
[14] G. Chen, J. Zhou
Boundary Element Methods . Academic Press, 1992.
[15] G. W. Clark, R.E. Showalter Two-scale convergence of a model for flow in a partially
fissured medium,
Electronic Journal of Differential Equations, 1999(2) (1999) 1-20.
[16] E. DiBenedetto
Degenerate Parabolic Equations. Springer-Verlag, 1993.
[17] J. Jr. Douglas, J. L. Hensley, and T. Arbogast
A dual--porosity model for
waterflooding in naturally fractured reservoirs.
Computer Methods in Applied
Mechanics and Engineering 87 (1991) 157-174.
[18] Douglas J. Jr., Hensley J. L., Arbogast T., Paes~Leme P. J., Nunes N. P.Medium and
tall block models for immiscible displacement in naturally fractured reservoirs. To
appear in the Proceedings of the Symposium on Numerical Analysis,
Polytechnical
University of Madrid, May 1990.
[19] Jim Douglas Jr., Frederico Furtado, Felipe Pereira, L.M. Yeh Numerical Methods for
Transport-Dominated Flows inHeterogeneous Porous Media.
Computational Methods in
Water Resources XII, 1, 469-476, 1998.
[20] Douglas J. Jr., Pereira F., Yeh L. M. A parallelizable characteristic scheme for two
phase flow 1: single porosity models.
Computational and Applied Mathematics, 14, no
1, 1995.
[21] Jim Douglas Jr., Felipe Pereira, L.M. Yeh
A parallel method for two-phase flows in
naturally fractured reservoirs.
Computational Geosciences, 1 no. 3-4, 333—368, 1997.
[22] Jim Douglas Jr., Felipe Pereira, L.M. Yeh A Locally Conservative Eulerian-Lagrangian
Numerical Method and its Application to Nonlinear Transport in Porous Media.
Computational Geosciences, 4 no. 1, 1--40, 2000.
[23] Jim Douglas Jr., Felipe Pereira, L.M. Yeh A Locally Conservative Eulerian-Lagrangian
Method for Flow in a Porous Medium of a Mixture of Two Components Having
Different Densities.
Numerical Treatment of Multiphase Flows in Porous Media
(Zhangxin Chen, Richard E. Ewing, and Z.-C. Shi ed.), Lecture Notes in Physics, 552,
138--155, 2000.
[24] Cesar Almeida,Jim Douglas Jr., Felipe Pereira, L.C. Roman, L. M. Yeh
Algorithmic
Aspects of a Locally Conservative Eulerian-Lagrangian Method for Transport-dominated
Diffusion System.
Fluid Flow and Transport in Porous Media: Mathematical and
Numerical Treatment, Contemporary Mathematics, 295, 37-48, 2002.
[25] Douglas J. Jr., Pereira F., Yeh L. M., Paes Leme P. J. A massively parallel iterative
numerical algorithm for immiscible flow in naturally fractured reservoirs,
volume 114
of International Series of Numerical Mathematics (J. Douglas, Jr., and U. Hornung,
eds.). 1993.
[26] Douglas J. Jr., Pereira F., Yeh L. M., Paes Leme P. J. Domain decomposition for
immiscible displacement in single porosity systems,
volume 164 of Lecture Notes in
Pure and Applied Mathematics (M. Krizek, P. Neittaanmaki, and R. Stenberg, eds.). 1994.
[27] Douglas J. Jr.,Hensley J. L.,Yeh L. M.et-al. A derivation for Darcy's Law for miscible
fluid and a revised model for miscible displacement in porous media.
Computational
Methods in Water Resources, 1X, Vol.2, 1992.
[28] Fabrie P., Langlais M. Mathematical analysis of miscible displacement in porous media.
SIAM I. Math. Analy., 23 no. 6, 1992.
[29] Gilbarg D., Trudinger N. S.
Elliptic Partial Differential Equations of Second Order.
Springer-Verlag, Berlin, second edition, 1983.
[30] James R. Gilman and Hossein Kazemi Improvements in simulation of naturally fractured
reservoirs.
Soc. Petroleum Engr. J. 23 (1983) 695--707.
[31] Glimm J., Lindquist B., Pereira F., Peierls R. The fractal hypothesis and anomalous
diffusion.
Matem\'atica Aplicada e Computacional, 11, 1992.
[32] Jianguo Huang, Jun Zhou. Some new a priori estimates for second order elliptic and
parabolic interface problems,
Journal of Differential Equations, 184 (2002) 570--586.
[33] H. Kazemi,
L.S. Merrill, Jr., K.L. Porterfield, and P.R. Zeman. Numerical simulation
of water-oil flow in naturally fractured reservoirs.
Soc. Petroleum Engr. J. (1976)
317--326.
[34] D. Kroener and S. Luckhaus. Flow of oil and water in a porous medium.
J. diff. Eqns.
55 (1984) 276--288.
[35] O. A. Ladyzenskaja, V.A. Solonnikov, N.N. Ural'tzeva.
Linear and Quasilinear
Equations of Parabolic Type. Transl. Math. Mono. Vol.23 AMS, Providence, RI, 1968.
[36] Yan Yan Li, Michael Vogelius. Gradient estimates for solutions to divergence form
elliptic equations with discontinuous coefficients,
Arch. Rational Mech. Anal.,
153(2000) 91--151.
[37] S. M. Nikol'skii.
Approximation of Functions of Several Variables and Imbedding
Theorems. Springer-Verlag, Berlin, 1975.
[38] Nochetto R. H. Error estimate for multidimensional sigular parabolic problems.
Japan
J. Appl. Math., 4, 1987.
[39] Peaceman D. W.
Fundamentals of Numerical Reservoir Simulation. Elsevier, New York,
[40] Michael E. Taylor.
Psudodifferential Operators. Princeton University, 1981.
[41] Hans Triebel.
Interpolation Theory, Function Spaces, Differential Operators. North
Holland, 1978.
[42] L. M. Yeh. Convergence of A Dual--Porosity Model for Two--phase Flow in Fractured
Reservoirs.
Mathematical Methods in Applied Science, 23, 777--802, 2000.
[43] Douglas J. Jr., Pereira F.,L. M. Yeh. Relations between phase mobilities and capillary
pressures of two--phase flows in fractured media.
Fluid Flow and Transport in Porous
Media: Mathematical and Numerical Treatment, Contemporary Mathematics, 295, 159--171,
2002.
[44] L. M. Yeh. On two-phase flows in fractured media.
Mathematical Models and Methods
in Applied Science, 12, No. 8, 1075--1107, 2002.
[45] L. M. Yeh. Homogenization of two-phase flows in fractured media.
Mathematical Models
and Methods in Applied Science, 11(16) 2006.
[46] L. M. Yeh. Holder continuity for two-phase flows in porous media.
Mathematical Methods
in Applied Science, 2006.
[47] L. M. Yeh. A moderate-sized block model for two-phase flows in fractured porous media.
submitted.
[48] Jim Douglas, L.M. Yeh. Two-component miscible displacement in fractured media.
submitted.
[49] L.M. Yeh. Tall block models for two-phase flows in fractured
porous media.
submitted.
[50] Robert Burridge, Joseph B. Keller. Poroelasticity equations derived from
microstructure, J. Accoust. Soc. Am. 70(4) 1140-1146, 1981.
[51] Jacob Rubinstein, S. Torquato. Flow in random porous media: mathematical formulation,
variational principles, and rigorous bounds. J. Fluid Mech. 206, 25-46, 1989.
[52] A. Yu. Beliaev, S.M. Kozlov, Darcy equation for random porous media, Comm. Pure App.
Math. XLIX 1-34, 1996
[53] V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of differential operators and
integral functionals, Springer-Verlag, 1991.
[54] L.M. Yeh. A tall block model for miscible displacement in fractured media. preprint.
[55] L.M. Yeh. Elliptic equations in highly heterogeneous porous media. conditionally
accept.
[56] L.M. Yeh. A priori estimate for non-uniform elliptic equations. preprint.
[57] L.M. Yeh. Uniform
pL
estimate for elliptic equations in perforated domains.
preprint.
[58] L.M. Yeh. H\"older estimate for non-uniform parabolic equations in highly
heterogeneous media. preprint.
[59] L.M. Yeh. Pointwise error estimate for elliptic equations in perforated domains.
preprint.
October 28, 2010
H¨older estimate for non-uniform parabolic equations in highly heterogeneous media
Li-Ming Yeh
Department of Applied Mathematics
National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C. [email protected]
Uniform bound for the solutions of non-uniform parabolic equations in highly heteroge-neous media is concerned. The space domains are periodic as well as consist of a con-nected high permeability sub-region and a disconcon-nected matrix block sub-region with low permeability. Let ǫ denote the size ratio of matrix blocks to the whole domain and let the permeability ratio of the matrix block sub-region to the connected high permeability sub-region be of the order ǫ2. It is proved that the H¨older norm of the non-uniform
parabolic solutions in connected sub-region is bounded uniformly in ǫ.
Keywords: Highly heterogeneous media, pseudo-differential operator, paramatrix, strict solution, infinitesimal generator, numerical range.
AMS Subject Classification: 35K10, 35K20, 35M13
1. Introduction
Uniform H¨older estimate for the solutions of non-uniform parabolic equations in highly heterogeneous media is presented. The equations have many applications in multiphase flows in porous media, the stress in composite materials, and so on (see [3, 12, 17] and references therein). The domain Ω ⊂ Rn(n ≥ 2) has boundary ∂Ω as
well as contains a connected high permeability sub-region and a disconnected matrix block sub-region with low permeability. Let Y ≡ [0, 1]nbe a cell consisting of a
sub-domain Ymcompletely surrounded by another connected sub-domain Yf(≡ Y \Ym),
ǫ ∈ (0, 1), and Ω(2ǫ) ≡ {x ∈ Ω : dist(x, ∂Ω) > 2ǫ}. The disconnected sub-region is Ωǫ
m ≡ {x : x ∈ ǫ(Ym+ j) ⊂ Ω(2ǫ) for j ∈ Zn}, the connected sub-region is
Ωǫ
f ≡ Ω \ Ωǫm, and the boundary of Ωǫm is represented by ∂Ωǫm. The non-uniform
parabolic equations in [0, T ] × Ω are ∂tUǫ− ∇ · (Λǫ∇Uǫ) = Fǫ in (0, T ] × Ω, Uǫ= 0 on (0, T ] × ∂Ω, Uǫ= Uǫ,0 in {0} × Ω, (1.1) where Λǫ ≡ ( Kǫ in Ωǫf, ǫ2k ǫ in Ωǫm,
and Kǫ and kǫ are positive smooth functions in Ω.
Since ǫ ∈ (0, 1), equations (1.1) are non-uniform parabolic equations with
October 28, 2010
2 H¨older estimate
tinuous coefficients. In [20], existence of solution in W2,1
p ([0, T ]×Ω) space for uniform
parabolic equations with discontinuous coefficients can be found. For non-uniform parabolic equations with smooth coefficients, existence of solution in C2,α([0, T ]×Ω)
space was studied in [13]. It is also known that if Fǫ, Uǫ,0 are smooth, a piecewise
regular solution of (1.1) exists uniquely for each ǫ and, by energy method, the H1
norm of the parabolic solution in the connected high permeability sub-region is bounded uniformly in ǫ [12, 16]. Many studies of the uniform estimate in ǫ for the elliptic equations in highly heterogeneous media had been done [4, 12, 15, 17, 19, 23], but little for parabolic equations. Existence of piecewise regular solutions for elliptic diffraction equations in Hilbert space was considered in [12, 15]. Uniform Lipschitz estimate in ǫ for Laplace equation in perforated domains was given in [23], and uniform Lpestimate in ǫ of the same problem was considered in [19].
Lip-schitz estimate for uniform elliptic equations was studied in [17]. Uniform H¨older and Lipschitz estimates in ǫ for uniform elliptic equations in periodic domains were obtained in [4]. This work is to present uniform H¨older estimate in ǫ for the solutions of the non-uniform parabolic equations with discontinuous coefficients. Permeabil-ity fields (that is, Λǫ) are not periodic and are allowed to have large deviation. It is
proved that the H¨older norm of the non-uniform parabolic solutions in connected sub-region is bounded uniformly in ǫ.
2. Notation and main result
Let Lp (resp.Hk, Wk,p) denote complex Sobolev space with norm k · k
Lp (resp.
k · kHk,k · kWk,p), C0∞ be the set containing all infinite differentiable functions with
compact support, and Cσ (resp. C1,σ) denote H¨older space with norm k · k
Cσ (resp.
k · kC1,σ) for σ ∈ (0, 1], k ≥ −1, and p ∈ [1, ∞] [11]. [ϕ]Cσ (resp. [ϕ]C1,σ) denotes
the H¨older semi-norms of ϕ (resp. ∇ϕ). If ϕ is a complex function, ϕ denotes its complex conjugate. If B1and B2are two Banach spaces, L(B1, B2) is the set of all
bounded linear maps from B1to B2with norm k · kL(B1,B2). For any Banach space
B, define kϕ1, ϕ2, · · · , ϕmkB≡ kϕ1kB+ kϕ2kB+ · · ·+ kϕmkB, denote its dual space
by B′
, and denote the pairing between B and its dual space B′
by h·, ·iB,B′. The
function spaces L∞
(I; B), C(I; B), Cσ(I; B) for σ ∈ (0, 1] and an interval I ⊂ R are
defined as those in [18]. Br(x) represents a ball centered at x with radius r. For any
domain D, D is the closure of D, D/r ≡ {x : rx ∈ D}, |D| is the volume of D, and XDis the characteristic function on D. For any ϕ ∈ L1(B
r(x) ∩ Ω), we define (ϕ)x,r≡ − Z Br(x)∩Ω ϕ(y)dy ≡ 1 |Br(x) ∩ Ω| Z Br(x)∩Ω ϕ(y)dy. For any p ∈ (1, ∞) and ǫ > 0, we define
Aǫϕ ≡ −∇ · (Λ ǫ∇ϕ), Bp(Aǫ) ≡ϕ ∈ W01,p(Ω) : ϕ ∈ W2,p(Ωǫf) ∪ W2,p(Ωǫm), Kǫ∇ϕ · ~nǫ|∂Ωǫ m = ǫ 2k ǫ∇ϕ · ~nǫ|∂Ωǫ m ,
October 28, 2010
H¨older estimate 3
where ~nǫis a normal vector on ∂Ωǫ
m. Bp(Aǫ) with norm kϕkBp(Aǫ)≡ kA
ǫϕk Lp(Ω)is
a normed space. M(α, β; D) ≡ {ϕ : D → R|ϕ ∈ L∞
(D), 0 < α ≤ ϕ ≤ β}. For any σ ∈ (0, 1) and δ, α, β > 0, we assume
A1. Ω and Ymare smooth domains,
A2. Kǫ, kǫ∈ M(α, β; Ω) and
kKǫ(ǫx) − αǫ,jkW1,∞((Y
f+j)∩Ω/ǫ)+ kkǫ(ǫx) − αǫ,jkW1,∞((Ym+j)∩Ω/ǫ)≤ cαǫ,j
where j ∈ Zn, α
ǫ,j depend on ǫ, j, and c is small and depending on Yf,
A3. Fǫ ∈ Cσ([0, T ]; Ln+δ(Ω)), −AǫUǫ,0+ Fǫ(0, x) ∈ Bn+δ(−Aǫ), and Uǫ,0 ∈
Bn+δ(−Aǫ).
The main result is:
Theorem 2.1. Under A1-A3, the solution of (1.1) satisfies kUǫkC1([0,T ];Ln+δ(Ω))+ kUǫkC([0,T ];Cµ(Ωǫ f))+ sup j∈Zn ǫ(Ym +j)⊂Ωǫm ǫkUǫkC([0,T ];Cµ(ǫ(Ym+j))) ≤ c kUǫ,0kBn+δ(Aǫ)+ kFǫkCσ([0,T ];Ln+δ(Ω)), (2.1)
where δ > 0, µ, σ ∈ (0, 1), µ is a constant depending on n, δ, α, β, Yf, Ω, and c is a
constant independent of ǫ. Moreover, there is a ν ∈ (0, µ) such that kUǫkCν([0,T ]×Ωǫ
f)≤ c kUǫ,0kBn+δ(Aǫ)+ kFǫkCσ([0,T ];Ln+δ(Ω)), (2.2)
where c is a constant independent of ǫ.
References
1. E. Acerbi, V. Chiado Piat, G. Dal Maso, and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis 18 (1992) 481–496.
2. R. A. Adams, Sobolev Spaces (second edition, Academic Press, 2003).
3. Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal. 23 (1992) 1482–1518.
4. Marco Avellaneda, Fang-Hua Lin, Compactness methods in the theory of homogeniza-tion, Communications on Pure and Applied Mathematics Vol. XI (1987) 803–847. 5. M. Briane, A. Damlamian, P. Donato, H-convergence for perforated domains,
Non-linear partial differential equations and their applications. College de France Seminar, Vol. XIII (Paris, 1994/1996), Pitman Research Notes in Mathematics Series 391 (1998) 62–100.
6. G. Chen and J. Zhou, Boundary Element Methods (Academic Press, 1992). 7. Doina Cioranescu and Patrizia Donato, An Introduction to Homogenization
(Oxford, 1999).
8. John B. Conway, A course in functional analysis, (Springer-Verlag, 1985). 9. L. Escauriaza, E.B. Fabes, G. Verchota, On a regularity theorem for weak solutions to
transmission problems with internal Lipschitz boundaries, Proceedings of the American Mathematical Society 115(4) (1992) 1069–1076.
10. M. Giaquinta, Multiple integrals in the calculus of variations, (Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983).
October 28, 2010
4 H¨older estimate
11. D. Gilbarg, N. S. Trudinger Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, second edition, 1983.
12. Jianguo Huang, Jun Zou, Some new a priori estimates for second-order elliptic and parabolic interface problems, Journal of Differential Equations 184 (2002) 570–586. 13. A. V. Ivanov, Quasilinear degenerate and nonuniformly elliptic and parabolic
equations of second order , American Mathematical Society, Providence, RI, 1984. 14. V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential
Opera-tors and Integral Functions, (Springer-Verlag, 1994).
15. O. A. Ladyzhenskaya, Nina N. Ural’tseva Elliptic and Quasilinear Elliptic Equations. Academic Press, 1968.
16. O. A. Ladyzenskaja, V. A. Solonnikov, N. N. Ural’ceva Linear and Quasi-linear Equa-tions of Parabolic Type . Providence, RI : American Mathematical Society, 1968. 17. Yan Yan Li, Michael Vogelius, Gradient estimates for solutions to divergence form
el-liptic equations with discontinuous coefficients, Arch. Rational Mech. Anal. 153 (2000) 91–151.
18. Alessandra Lunardi, Analytic semigroups and optimal regularity in parabolic problems (Basel : Birkhauser, 1995).
19. Nader Masmoudi, Some uniform elliptic estimates in porous media, C. R. Acad. Sci. Paris Ser. I 339 (2004) 849–854.
20. A. Maugeri, Dian Palagachev, Lubomira G. Softova, Elliptic and parabolic equa-tions with discontinuous coefficients, Wiley-VCH, Berlin ; New York, 2000. 21. A. Pazy, Semigroups of linear operators and applications to partial
differ-entical equations (New York : Springer-Verlag, 1983).
22. Thomas Runst, Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations (Berlin ; New York : Walter de Gruyter, 1996).
23. Ben Schweizer, Uniform estimates in two periodic homogenization problems, Commu-nications on Pure and Applied Mathematics Vol.LIII (2000) 1153–1176.
24. Michael E. Taylor, Pseudodifferential Operators (Princeton University Press, 1981).
25. Vidar Thom´ee, Galerkin finite element methods for parabolic problems (Berlin : Springer-Verlag, 1997).
October 28, 2010
Pointwise error estimate for elliptic equations in perforated domains
Li-Ming Yeh
Department of Applied Mathematics
National Chiao Tung University, Hsinchu, 30050, Taiwan, R.O.C. [email protected]
Numerical approximation for the solutions of elliptic equations in perforated domains is concerned. Let ǫ denote the size ratio of the holes of some perforated domains to their whole domains. As ǫ closes to 0, the elliptic solutions in perforated domains approach a solution of some homogenized equation. So it is expected that the numerical approxima-tion of the soluapproxima-tion of the homogenized equaapproxima-tion is a good approximaapproxima-tion for the elliptic solutions in perforated domains when ǫ is small. In this work, the L∞
estimate and the Lipschitz estimate for the difference between the elliptic solutions in perforated domains and the numerical approximation of the homogenized solution are derived. Higher order estimate in Lipschitz norm for the difference between the elliptic solutions in perforated domains and the homogenized solution is also derived.
Keywords: elliptic solution, perforated domain, homogenized solution.
AMS Subject Classification: 65N12, 65N15, 65N22
1. Introduction
Pointwise error estimate for the numerical approximation of the solutions of elliptic equations in perforated domains is presented. Let Ω ⊂ Rn (n = 2 or 3) be a smooth
domain with boundary ∂Ω, Y ≡ [0, 1]n consist of a sub-domain Y
m completely
surrounded by another connected sub-domain Yf (≡ Y \ Ym), ǫ ∈ (0, 1), Ω(2ǫ) ≡
{x ∈ Ω : dist(x, ∂Ω) > 2ǫ}, Ωǫm ≡ {x : x ∈ ǫ(Ym+ j) ⊂ Ω(2ǫ) for j ∈ Zn} with
boundary ∂Ωǫ
m, and Ωǫf ≡ Ω \ Ωǫm be a connected region. The equations in the
perforated domain Ωǫ f are −∇ · (Kǫ∇Uǫ) + λUǫ= F in Ωǫf, Kǫ∇Uǫ· ~nǫ= 0 on ∂Ωǫm, Uǫ= 0 on ∂Ω, (1.1)
where λ ≥ 0, Kǫ(x) = K(xǫ), K is a positive periodic function in Rn with period
Y , and ~nǫ is the unit normal vector on ∂Ωǫ
m. When ǫ is small, direct numerical
simulation of the solution of (1.1) can be very expensive. It is known that if F ∈ L2(Ω), the H1 solution of (1.1) exists uniquely and satisfies
kUǫkH1(Ωǫ
f)≤ ckF kL2(Ω),
October 28, 2010
2 Error estimate
where c is a constant independent of ǫ [9]. By compactness principle [2], there exists a function U ∈ H1(Ω) such that the solution Uǫof (1.1) satisfies
Kǫ∇UǫXΩǫ f → K
∗
∇U in L2(Ω) weakly as ǫ → 0, (1.2)
where XΩǫ
f is the characteristic function on Ω
ǫ fand K
∗
is a constant positive definite matrix depending on K, Yf (explicit form of K∗is in (2.2) below). The function U
in (1.2) satisfies (
−∇ · (K∗
∇U ) + λ|Yf|U = |Yf|F in Ω,
U = 0 on ∂Ω, (1.3) where |Yf| is the volume of Yf. Therefore, it is expected that the numerical
approx-imation for the solution of (1.3) is a good approxapprox-imation for the solution of (1.1), especially when ǫ is small. The error estimate between the numerical solution and the analytic solution of (1.3) had been extensively studied (see [4, 7, 10, 15] to name a few). So we shall focus on the error estimate for the solutions of (1.1) and (1.3).
By homogenization theory, solutions of elliptic equations in periodic domains in general converge to a solution of some homogenized elliptic equation with con-vergence rate ǫ in L2 norm and with convergence rate √ǫ in H1 norm as ǫ closes
to 0 (see [3, 11, 14] and references therein). In [5, 13], higher order asymptotic expansion for the solutions of elliptic equations in perforated domains was given. Higher order convergence rate for the solution of (1.1) for λ = 0 case was derived in Hilbert spaces [3, 6, 14]. Different from the literatures mentioned above in which L2 space was considered, we present pointwise error estimate for the solutions of
(1.1) and (1.3) for λ ≥ 0. More precisely, the L∞
error estimate with convergence rate ǫ for the solutions of (1.1) and (1.3) is proved. For equation (1.1) with λ > 0 and with periodic boundary condition case, W1,∞ error estimate with convergence
rate ǫ is also derived. In particular, for equation (1.1) with λ = 0 and with periodic boundary condition case, higher order approximation in Lipschitz norm is obtained as well.
2. Notation and main results
Denote by Ck,α the H¨older space with norm k · k
Ck,α, by [g]Ck,α the H¨older
semi-norm of g, and by Lp (resp. Hs, Ws,p) the Sobolev space with norm k · k
Lp (resp.
k · kHs, k · kWs,p) for k ≥ 0, α ∈ [0, 1], s ≥ 1, and p ∈ [1, ∞] (see [9]). For any
Banach space B, we define kg1, g2, · · · , gkkB≡ kg1kB+ kg2kB+ · · · + kgkkB. Br(x)
is a ball centered at x with radius r. For any domain D, D is the closure of D, D/r ≡ {x : rx ∈ D}, |D| is the volume of D, and XD is the characteristic function
on D. For any g ∈ L1(Ω), (g)x,r≡ − Z Br(x)∩Ω g(y)dy ≡ 1 |Br(x) ∩ Ω| Z Br(x)∩Ω g(y)dy.
October 28, 2010
Error estimate 3
Define Zm≡ ∪j∈Zn(Ym+ j) with boundary ∂Zm, Zf ≡ Rn\ Zm, Zmǫ ≡ ǫZm with
boundary ∂Zmǫ , and Zfǫ≡ Rn\ Zmǫ . For D ∈ {Rn, Zf, Zfǫ}, we define
Wpers,p(D) ≡ {g ∈ Wlocs,p(D) : g is a periodic function in D with period [0, 1]n}
with norm kgkWpers,p(D) ≡ kgkWs,p(D∩Y ) for s ≥ 1 and p ∈ [1, ∞]. Similar definition
for Lpper(D), Hpers (D), Cperk,α(D) when D ∈ {Rn, Zf, Zfǫ}, k ≥ 0, α ∈ [0, 1], s ≥ 1,
and p ∈ [1, ∞]. If G(x) = g(ǫx), g ∈ C0,α
per(Zfǫ) for α ∈ (0, 1), define |||g|||C0,αper(Zfǫ) ≡
kGkC0,α(Z f).
For each i = 1, · · · , n, we find X(i)(y) ∈ H1
per(Zf) satisfying, in cell Yf,
−∇ · (K(∇X(i)+ ~e i)) = 0 in Yf, K(∇X(i)+ ~e i) · ~ny = 0 on ∂Ym, R YfX (i)dy = 0, (2.1)
where ~ny denotes the unit normal vector on ∂Ymand ~ei is a unit vector in the i-th
coordinate direction for i = 1, · · · , n. For any ν > 0, we define X(i)ν (x) ≡ νX(i)(xν),
X ≡ (X(1), · · · , X(n)), and Xν ≡ (X(1)ν , · · · , X(n)ν ). Denote by Ξ a n × n matrix
function whose (i, j) component is ∂yiX
(j) and define
K∗≡ Z
Yf
K(y)(I + Ξ(y))dy, (2.2) where I is the identity matrix. By [2] and remark in page 90 [11], K∗
is a constant symmetric positive definite matrix. For i1, i2= 1, · · · , n, find X(i1,i2)(y) ∈ Hper1 (Zf)
satisfying, in cell Yf, ∇ · (K∇X(i1,i2)) + ∂ i1(KX (i2)) = −K(δ i1,i2+ ∂i1X (i2)) +K ∗ i1,i2 |Yf| in Yf, K(∇X(i1,i2)· ~n y+ X(i2)nyi1) = 0 on ∂Ym, R YfX (i1,i2)dy = 0. (2.3) Here δi1,i2 ≡ ( 1 if i1= i2, 0 if i16= i2, K∗ i1,i2 is the (i1, i2) component of K ∗ in (2.2), and nyi1 is the i1 component of ~ny. Similarly define X(i1,i2,...,iℓ)∈ Hper1 (Zf) for ℓ ≥ 3,
ij ∈ {1, · · · , n}, and j ∈ {1, 2, · · · , ℓ} as, in cell Yf,
∇ · (K∇X(i1,...,iℓ)) + ∂ i1(KX (i2,...,iℓ)) = −K(δi1,i2X (i3,...,iℓ)+ ∂ i1X (i2,...,iℓ)) in Y f, K ∇X(i1,...,iℓ)· ~n y+ X(i2,...,iℓ)nyi1 = 0 on ∂Ym, R YfX (i1,...,iℓ)dy = 0. (2.4)
By energy method and Lax-Milgram theorem [9], X(i1,...,iℓ)for ℓ ≥ 1 in (2.1), (2.3),
and (2.4) are solvable uniquely. By Lemma 6.29 [9], if K ∈ C1,α
per(Zf) for α ∈ (0, 1),
then
kX(i1,...,iℓ)k
C2,α(Z
October 28, 2010
4 Error estimate
2.1. L∞
error estimate We shall assume
A1. Ω and Ym are bounded C1,α domains,
A2. K ∈ C1,α
per(Zf) is a positive function,
A3. F ∈ W1,n+δ(Ω),
where α ∈ (µ, 1), µ ≡ n+δδ , n ∈ {2, 3}, and δ ∈ (0, 3). We recall an extension result in [1].
Lemma 2.1. For 1 ≤ p < ∞, there is a constant c(Yf, p) and a linear continuous
extension operator Πǫ: W1,p(Ωǫf) → W1,p(Ω) such that (1) If ϕ ∈ W1,p(Ωǫf), then
Πǫϕ = ϕ in Ωǫf almost everywhere, kΠǫϕkLp(Ω)≤ c(Yf, p)kϕkLp(Ωǫ f), k∇ΠǫϕkLp(Ω)≤ c(Yf, p)k∇ϕkLp(Ωǫ f), kΠǫϕkL∞(Ω)≤ c(Yf, p)kϕkL∞(Ωǫ f) if ϕ ∈ L ∞ (Ωǫ f), Πǫϕ = ζ in Ω if ϕ = ζ|Ωǫ
f for some linear function ζ in Ω.
(2) If ζ(x) ≡ ϕ(rx) in B1(z) ∩ Ωǫf/r for any z ∈ Rn and any constant r > ǫ, there
is a θ ∈ (0, 1) so that Πǫ/rζ(x) = (Πǫϕ)(rx) in Bθ(z) ∩ Ω/r
Lemma 2.1 also holds if Ωǫ
f (resp. Ω) is replaced by Zfǫ (resp. Rn). For the
solutions of (1.1) and (1.3), we have the following error estimate:
Lemma 2.2. Under A1–A3, there is a constant ǫ0 such that the solutions of (1.1)
and (1.3) satisfy, for any ǫ ∈ (0, 1) and λ ∈ [0, ǫ0),
kUǫ− U kL∞(Ωǫ
f)≤ cǫkF kW1,n+δ(Ω),
where c is a constant independent of ǫ, λ.
We now describe the families of subspaces to be used to approximate the solution of (1.3). Let {Fh : 0 < h ≤ 1} be a family of subdivisions of Ω into disjoint,
non-empty, connected, open sets τ ∈ Fh of diameter not greater than h (a subdivision
means Ω = ∪τ ∈Fhτ ). Then we assume
A4. {Fh: 0 < h ≤ 1} is a quasi-uniform (see page 5 [10]) family of subdivisions
of Ω and {Sh : 0 < h ≤ 1} is a family of linear spaces of functions on Ω
such that, for each ϕ ∈ Sh and τ ∈ Fh, we have ϕ|
τ ∈ P, where P is a
fixed finite dimensional space of polynomials independent of h, τ , and ϕ. A5. For some non-negative integer m not greater than n and for some integer
k greater than m, there is a linear mapping Jhfor each h ∈ (0, 1] of
Dh≡ {ζ ∈ Cm(Ω) : ζ|∂Ω= 0} ∪ {ζϕ : ζ ∈ Cm(Ω), ϕ ∈ Sh}
into Sh such that
October 28, 2010
Error estimate 5
Furthermore, for any integer ℓ and any p ∈ [1, ∞] such that max{1, m} + n ≤ ℓ ≤ k if p = 1, max{1, m} + n/p < ℓ ≤ k if 1 < p < ∞, m < ℓ ≤ k if p = ∞, it follows that (ii) 1 P j=0 hj+n/pkDj(ζ − Jhζ)k L∞(τ )≤ cℓ,phℓkζkWℓ,p(τ )
for all τ ∈ Fh and ζ ∈ Dh∩ Wℓ,p(τ ), where c
ℓ,pis independent of ζ, τ, h.
A6. For all h ∈ (0, 1], Sh⊂ W1,∞(Ω).
Many finite element spaces satisfying A4–A6 have been constructed, see [7, 10] and references therein. Inverse inequalities (see Theorem 3.2.6 [7]) hold in these finite element spaces. Define a bilinear form Lh
γ on Sh× Sh as Lh γ(ϕ, ζ) ≡ Z Ω K∗∇ϕ∇ζ dx + Z Ω λ|Yf|ϕζ dx − Z ∂Ω K∗ϕ∇ζ · ~ndσ − Z ∂Ω K∗ζ∇ϕ · ~n dσ + γh−1Z ∂Ω ϕζ dσ
where ϕ, ζ ∈ Sh, ~n is the unit outward normal vector on ∂Ω, K∗
is that in (2.2), and γ is some positive number. Now we find Uh∈ Sh such that
Lhγ(Uh, ζ) =
Z
Ω
|Yf|F ζdx for all ζ ∈ Sh. (2.6)
Here Uh is the numerical approximation of the solution of (1.3). By Lax-Milgram
theorem [9], (2.6) is solvable uniquely. Let us recall Theorem 3.1 [10].
Theorem 2.1. Suppose A4–A6 hold, U solves (1.3), Uh solves (2.6), and n = 2, 3. Let U ∈ Ws,∞(Ω) with max{m, 1} < s ≤ k, where m and k are the parameters in
A5 and k ≥ 3. Then there are constants γ1< ∞ and h1> 0 such that, for γ ≥ γ1
and 0 < h ≤ h1,
kU − UhkL∞(Ω)≤ chskU kWs,∞(Ω), (2.7)
where c is independent of h and U .
By A3 and Theorem 9.15 [9], we know U ∈ W2,∞(Ω). Lemma 2.2 and Theorem
2.1 imply that the maximum norm estimate of Uǫ− Uh(that is, difference between
the solution of (1.1) and the solution of (2.6)) satisfies kUǫ− UhkL∞(Ωǫ f)≤ kUǫ− U kL ∞(Ωǫ f)+ kU − U hk L∞(Ωǫ f) ≤ c(ǫ + h2)kF kW1,n+δ(Ω). (2.8)
where c is a constant independent of ǫ, h. In other words, if Lemma 2.2 holds, we have the following results:
October 28, 2010
6 Error estimate
Theorem 2.2. Under A1–A6, there exist positive constants ǫ0, h1, γ1 such that
when ǫ ∈ (0, 1), λ ∈ [0, ǫ0), h ∈ (0, h1), and γ > γ1, the solution of (1.1) and the
numerical approximation in (2.6) satisfy the L∞
error estimate (2.8).
2.2. Lipschitz error estimate
In this subsection, functions considered are periodic and have period [0, 1]n for
n ∈ {2, 3}. If λ > 0 and F ∈ L2
per(Rn), we find Uǫ∈ Hper1 (Zfǫ) satisfying
(
−∇ · (Kǫ∇Uǫ) + λUǫ= F in Zfǫ,
Kǫ∇Uǫ· ~nǫ= 0 on ∂Zmǫ.
(2.9)
By Lax-Milgram theorem [9], (2.9) is solvable uniquely in H1
per(Zfǫ) and the solution
satisfies kUǫkH1
per(Zfǫ) ≤ ckF kL2per(Rn), where c is a constant independent of ǫ. By
compactness principle [2] and Lemma 2.1, there is a function U ∈ H1
per(Rn) such
that the solution Uǫ of (2.9) satisfies
( ΠǫUǫ→ U in Hper1 (Rn) weakly Kǫ∇UǫXZǫ f → K ∗ ∇U in L2 per(Rn) weakly as ǫ → 0 and U satisfies −∇ · (K∗∇U ) + λ|Yf|U = |Yf|F in Rn. (2.10)
Here |Yf| is the volume of Yf and K∗ is the positive definite matrix in (2.2).
Lemma 2.3. Under the following conditions A1′. δ ∈ (0, 3), α > 0, and Y
m is a C1,α domain,
A2 . K ∈ C1,α
per(Zf) is a positive function,
A3′
. F ∈ W2,n+δ per (Rn),
there is a constant ǫ0 < 1 such that the solutions of (2.9) and (2.10) satisfy, for
any ǫ ∈ (0, 1) and λ ∈ (0, ǫ0),
k∇Uǫ(x) − (I + ∇X(x/ǫ))∇U (x)kL∞(Zǫ
f)≤ cǫkF kWper2,n+δ(Rn), (2.11)
where c is a constant independent of ǫ.
Next we describe the finite element spaces to approximate the solution of (2.10). Let {Fh : 0 < h < 1} be a family of subdivisions of Rn into disjoint,
non-empty, connected, open sets τ ∈ Fh of diameter not greater than h (a subdivision
means Rn = ∪
τ ∈Fhτ ) and each subdivision is periodic with period [0, 1]n (that is,
τ + j ∈ Fhfor any τ ∈ Fh and j ∈ Zn). For r ≥ 2, Sh
r(Rn) is a family of finite
di-mensional subspace of W1,∞
per (Rn). If D ⊂ Rn, then Srh(D) (resp. Wper1,∞(D)) denotes
the restriction of functions in Sh
r(Rn) (resp. Wper1,∞(Rn)) to D. Let kh + di< di+1
October 28, 2010
Error estimate 7
A7. If t ∈ {0, 1}, t ≤ ℓ ≤ r, 1 ≤ p ≤ ∞, then for each ϕ ∈ Wℓ,p(B
d2) there exists a ζ ∈ Sh r(Bd2) such that kϕ − ζkWt,p(B d1)≤ ch ℓ−tkϕk Wℓ,p(B d2). If n < p ≤ ∞, kϕ − ζkW1,∞(B d1)≤ ch r−1−n/pkϕk Wr,p(B d2).
Furthermore if ϕ vanishes outside of Bd0, ζ vanishes outside of Bd1. The
constant c is independent of h, ϕ, ζ, Bd1, and Bd2.
A8. If ζ ∈ Sh
r(Rn), then for t ∈ {0, 1} and ℓ ≥ 0 is an integer and 1 ≤ q ≤ p ≤ ∞,
kζkWt,p(B
d1)≤ ch
−(n/q−n/p)−t−ℓkζk W−ℓ,q(B
d2).
The constant c is independent of h, ζ, Bd1, and Bd2.
A9. Let ϕ ∈ C∞
0 (Bd1), then for each ζ ∈ S
h
r(Bd3) there exists an η ∈ S
h r(Bd3),
vanishing outside of Bd2, such that for some integer γ > 0
kϕζ − ηkH1(B
d3)≤ chkϕkWγ,∞(Bd1)kζkH1(Bd3).
Furthermore, if ϕ = 1 on Bd0, then η = ζ on Bd−1 and
kϕζ − ηkH1(B
d2)≤ chkϕkWγ,∞(Bd1)kζkH1(Bd2\Bd0).
Here c is independent of ϕ, ζ, η, h, Bd0, Bd1, Bd2.
A10. Let x0 ∈ Rn and d ≥ kh. The transformation y = x−xd 0 takes Bd(x0)
into a domain B1(x0) and Srh(Bd(x0)) into a function space ˆSrh/d(B1(x0)).
Then ˆSrh/d(B1(x0)) satisfies A7-A9 with h replaced by h/d. The constants
in A7-A9 remain unchanged, in particular independent of d.
Finite element spaces satisfying A7-A10 can be found in [15] and reference therein. We find Uh∈ Sh r(Rn) such that Z [0,1]n K∗∇Uh∇ζdx +Z [0,1]n λ|Yf|Uhζdx = Z [0,1]n |Yf|F ζdx, (2.12) where ζ ∈ Sh r(Rn). By A3 ′
and Theorem 9.11 [9], the solution of (2.10) satisfies U ∈ W4,n+δ
per (Rn). Theorem 3.1 [15] implies
Theorem 2.3. If A7-A10 hold, U ∈ W1,∞
per (Rn) solves (2.10), and Uh ∈ S4h(Rn)
solves (2.12), then there exists a constant c independent of U, Uh, h such that
kU − UhkW1,∞
per(Rn)≤ ch
3−n/(n+δ)kF k
Wper2,n+δ(Rn). (2.13)
Lemma 2.3, Theorem 2.3, and (2.5) imply that the Lipschitz norm estimate of Uǫ− Uh(that is, difference between the solution of (2.9) and the solution of (2.12))
satisfies
k∇Uǫ− (I + ∇Xǫ)∇UhkL∞(Zǫ
f)≤ c(ǫ + h
3−n/(n+δ))kF k
W2,n+δ([0,1]n), (2.14)
October 28, 2010
8 Error estimate
Theorem 2.4. Assume A1′
, A2, A3′
, A7–A10. There exist positive constants ǫ0, h1
such that if ǫ ∈ (0, 1), λ ∈ (0, ǫ0), and h ∈ (0, h1), the solution of (2.9) and the
approximation in (2.12) satisfy the Lipschitz error estimate (2.14).
2.3. Higher order Lipschitz estimate Besides L∞
and Lipschitz error estimates, we also have higher order Lipschitz esti-mate. Functions in this subsection are periodic and have period [0, 1]nfor n ∈ {2, 3}.
If F ∈ L2 per(Rn) satisfies R [0,1]n∩Zǫ fF dx = 0, we find Uǫ∈ H 1
per(Zfǫ) such that
−∇ · (Kǫ∇Uǫ) = F in Zfǫ, Kǫ∇Uǫ· ~nǫ= 0 on ∂Zmǫ, R [0,1]n∩Zǫ fUǫdx = 0. (2.15)
By Lax-Milgram theorem [9], (2.15) is solvable uniquely in H1
per(Zfǫ) and the
so-lution satisfies kUǫkH1
per(Zfǫ)≤ ckF kL2per(Rn), where c is a constant independent of
ǫ. By compactness principle [2] and Lemma 2.1, there is a function U ∈ H1 per(Rn)
such that the solution Uǫof (2.15) satisfies
( ΠǫUǫ→ U in Hper1 (Rn) weakly Kǫ∇UǫXZǫ f → K ∗ ∇U in L2 per(Rn) weakly as ǫ → 0 and U satisfies ( −∇ · (K∗ ∇U ) = |Yf|F in Rn, R [0,1]nU dx = 0. (2.16) Here |Yf| is the volume of Yf and K∗ is the positive definite matrix in (2.2).
Theorem 2.5. Under A1′, A2, and
A3′′
. F ∈ Wk,n+δ
per (Rn) for k ≥ 2,
the solutions of (2.15) and (2.16) satisfy, for any ǫ ∈ (0, 1), k∇ϕǫkL∞(Zǫ
f)≤ cǫ
k−1kF k
Wperk,n+δ(Rn).
where c is a constant independent of ǫ and ϕǫ(x) ≡ Uǫ(x) − U (x) − k X ℓ=1 ǫℓ n X i1,...,iℓ=1 X(i1,...,iℓ)(x ǫ)∂i1,...,iℓU (x). References
1. E. Acerbi, V. Chiado Piat, G. Dal Maso, and D. Percivale, An extension theorem from connected sets, and homogenization in general periodic domains, Nonlinear Analysis 18 (1992) 481–496.
2. Gregoire Allaire, Homogenization and two-scale convergence, SIAM I. Math. Anal. 23 (1992) 1482–1518.
October 28, 2010
Error estimate 9
3. N. Bakhvalov and G. Panasenko, Homogenisation : averaging processes in pe-riodic media : mathematical problems in the mechanics of composite ma-terials (Kluwer Academic Publishers, 1989).
4. Susanne C. Brenner, L. Ridgway Scott, The mathematical theory of finite ele-ment methods (Springer, 2008).
5. Alain Bensoussan, Jacques-Louis Lions, George Papanicolaou, Asymptotic analysis for periodic structures (Elsevier North-Holland, 1978).
6. Li-Qun Cao, Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perfo-rated domains, Numerische Mathematik 103 no. 1 (2006) 11–45.
7. Philippe G. Ciarlet, The finite element method for elliptic problems (Amster-dam : North-Holland, 1978).
8. M. Giaquinta, Multiple integrals in the calculus of variations, (Study 105, Annals of Math. Studies, Princeton Univ. Press., 1983).
9. D. Gilbarg, N. S. Trudinger Elliptic Partial Differential Equations of Second Order. Springer-Verlag, Berlin, second edition, 1983.
10. C. I. Goldstein, L. R. Scott, Optimal maximum norm error estimates for some finite element methods for treating the Dirichlet problem , Calcolo 20 1 (1983) 1–52. 11. V.V. Jikov, S.M. Kozlov, O.A. Oleinik, Homogenization of Differential
Opera-tors and Integral Functions, (Springer-Verlag, 1994).
12. O. A. Ladyzhenskaya, Nina N. Ural’tseva Elliptic and Quasilinear Elliptic Equations. Academic Press, 1968.
13. J.L. Lions, Asymptotic expansions in perforated media with a periodic structure, The ROcky Mountain J. Math. 10(1) (1980) 125–144.
14. O.A. Oleinik, A.S. Shamaev, G.A. Tosifan, Mathematical Problems in Elasticity and Homogenization (North-Holland, Amsterdam, 1992).
15. Alfred Schatz, Pointwise error estimates and asymptotic error expansion inequalities for the finite element method on irregular grids: Part I. Global estimates, Mathematics of Computation Vol. 67 No. 223 (1998) 877–899.
