error estimates for full discretization of a model for ostwald ripening

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Error Estimates for Full Discretization ofa Model for Ostwald RipeningPatrícia Nunes da Silva a & José Luiz Boldrini ba Departamento de Análise , IME, UERJ , Rio de Janeiro, Brazilb State University of Campinas – IMECC , Campinas, BrazilPublished online: 17 Sep 2008.

To cite this article: Patrícia Nunes da Silva & José Luiz Boldrini (2008) Error Estimates for FullDiscretization of a Model for Ostwald Ripening, Numerical Functional Analysis and Optimization,29:7-8, 905-926, DOI: 10.1080/01630560802295922

To link to this article: http://dx.doi.org/10.1080/01630560802295922

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Numerical Functional Analysis and Optimization, 29(7–8):905–926, 2008Copyright © Taylor & Francis Group, LLCISSN: 0163-0563 print/1532-2467 onlineDOI: 10.1080/01630560802295922

ERROR ESTIMATES FOR FULL DISCRETIZATIONOF A MODEL FOR OSTWALD RIPENING

Patrícia Nunes da Silva1 and José Luiz Boldrini2

1Departamento de Análise, IME, UERJ, Rio de Janeiro, Brazil2State University of Campinas – IMECC, Campinas, Brazil

� We analyze certain finite element schemes for a family of systems consisting of a Cahn–Hilliard equation coupled with several Allen–Cahn type equations, which are related to a modelproposed by Fan and Chen for the evolution of Ostwald ripening in two-phase material systems.We obtain error bounds both for a semidiscrete (in time) scheme and a fully discrete scheme.

Keywords Allen–Cahn equation; Cahn–Hilliard equation; Error bounds; Fulldiscretization; Phase transitions; Ostwald ripening.

AMS Subject Classification 65M15; 35K55.

1. INTRODUCTION

Ostwald ripening is a phenomenon observed in a wide variety of two-phase material systems in which there is coarsening of one phase dispersedin the matrix of another. Because of its practical importance, this processhas been extensively studied in several degrees of generality. In particularfor Ostwald ripening of anisotropic crystals, Fan et al. [1] presented amodel taking into account both the evolution of the compositional fieldand of the crystallographic orientations. Silva and Boldrini [2] provedexistence and uniqueness of solution for a family of systems consisting ofCahn–Hilliard and several Allen–Cahn type equations closely related to themodel proposed by Fan et al. [1]. In [2], such results were obtained byanalyzing a fully discrete finite element scheme based on the backwardEuler method; the existence and uniqueness of solutions of such discrete

Address correspondence to Patrícia Nunes da Silva, Departamento de Análise, IME, UERJ, RuaSão Francisco Xavier, 524, Sala 6016, Bloco D, CEP 20550-013, Rio de Janeiro, RJ, Brazil; E-mail:[email protected]

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906 P. Nunes da Silva and J. L. Boldrini

problems were proved, as well as their convergence to a solution of theoriginal system.

In the current paper, our objective is to obtain error bounds forthe fully discrete scheme presented by Silva and Boldrini [2]; in theprocess of derivation of the error estimates, we also get error bounds for asemidiscrete (in time) version of the scheme.

The family of models we are interested in is constituted of thefollowing Cahn–Hilliard and Allen–Cahn equations:

ct = �w, (1.1)

w = D ��c� − �c�c� , (1.2)

[�i]t = −Li

[��i� − �i��i

], i = 1, � � � , p, (1.3)

subject to the initial conditions

c(x , 0) = c0(x), x ∈ �, (1.4)

�i(x , 0) = �i0(x), x ∈ �, i = 1, � � � , p, (1.5)

as well as the following boundary conditions

�c�n

= �w�n

= 0 on ��, (1.6)

��i

�n= 0 on ��, for i = 1, � � � , p� (1.7)

Here, � is the physical region where the Ostwald ripening process isoccurring; c(x , t), for t ∈ [0,T ], 0 < T < +∞, x ∈ �, is the compositionalfield (fraction of the soluto with respect to the mixture); �i(x , t), for i =1, � � � , p, are the crystallographic orientations fields; and D, �c , Li , �i arepositive constants related to the material properties.

It is assumed that the local free energy � has the following form:

� (c , �1, � � � , �p) = −A2(c − cm)2 + B

4(c − cm)4 + D

4(c − c)4 + D

4(c − c)4

− �

p∑i=1

g (c , �i) + �

4

p∑i=1

�4i +p∑

i=1

p∑i �=j=1

ij f (�i , �j), (1.8)

where A, B, D, D, �, �, ij , i �= j = 1, � � � , p are positive constants relatedto the material properties, c and c are the solubilities or equilibriumconcentrations for the matrix phase and second phase, respectively, andcm = (c + c)/2.

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Error Analysis of an Ostwald Ripening Model 907

The functions f and g are assumed to satisfy the following properties:

|f (a, b) − f (u, v) + � f (u, v) · (u − a, v − b)|≤ F1(u − a)2 + F2(v − b)2(≤max�F1, F2�|(u, v) − (a, b)|2) (1.9)

and

|g (a, b) − g (u, v) + �g (u, v) · (u − a, v − b)| ≤ G1(u − a)2 + G2(v − b)2,(1.10)

for all (u, v), (a, b) ∈ �2 and fixed constants F1, F2,G1,G2 ≥ 0.We remark that the previous assumptions on f and g imply that the

difference between f (a, b) and g (a, b) and their Taylor polynomials ofdegree one at (u, v), respectively, are bounded up to a multiplicativefixed constant by the square of the Euclidean distance between (u, v)and (a, b). We also remark that the local free energy � is assumed tohave the previously described form to comply with a requirement of Chenet al. [3–6] that � should have 2p degenerate minima at the equilibriumconcentration c to distinguish the 2p different orientations of the second-phase grains in space.

This article is organized as follows: In Section 2, we state sometechnical hypotheses and fix the notations. Section 3 is devoted to thederivation of some a priori estimates for the solution of (1.1)–(1.7), whichwill be necessary to get the error bounds. In Section 4, we consider asemidiscrete (in time) scheme for the Cahn–Hilliard/Allen–Cahn system(1.1)–(1.3) and obtain error bounds for such semidiscretization (seeTheorem 4.3). Finally, in Section 5, we prove our main results; that is, weobtain error bounds for the fully discrete scheme associated to (1.1)–(1.3).Our approach is similar to that used by Feng and Prohl [7] for the Cahn–Hilliard equation and by Feng and Prohl [8] for the Allen–Cahn equation.

2. TECHNICAL HYPOTHESES

Throughout this paper, we assume that � is a bounded domain in Rd ,1 ≤ d ≤ 3, with C 2,1 boundary. Standard notation is used for the requiredfunctional spaces; in particular, (·, ·) and | · | are respectively the innerproduct and the norm in L2(�). We denote by f̄ the mean value of f in� of a given f ∈ L1(�). The duality pairing between H 1(�) and its dual isdenoted by 〈·, ·〉.

To simplify the exposition, in the following we will consider onlytwo orientations field variables (p = 2); the presented results arestraightforward extended to any number of such variables.

As Copetti and Elliott [9], we consider the finite element problemcorresponding with the weak formulation of (1.1)–(1.7) defined as follows.

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Find �(Cm ,W m ,�m1 ),�

m2 )�

Mm=1 ∈ [Sh]4 such that

(dtCm , �) + (�W m ,��) = 0, (2.1)

(W m , v) = �cD(�Cm ,�v) + D(�c� (Cm ,�m1 ,�

m2 ), v), (2.2)

(dt�m1 , �1) + �1L1(��

m1 ,��1) + L1(��1� (Cm ,�m

1 ,�m2 ), �1) = 0, (2.3)

(dt�m2 , �2) + �2L2(��

m2 ,��2) + Li(��2� (Cm ,�m

1 ,�m2 ), �2) = 0, (2.4)

for all �, v, �1, �2 ∈ Sh and suitable starting values (C 0,�01,�

02).

Here, Sh denotes the P1 conforming finite element space defined on aquasi-uniform triangulation Th of � by

Sh = �v ∈ C(�); v|K ∈ P1(K ), ∀K ∈ Th�

and dtum = um − um−1

k.

We define the L2(�)-projection Qh : L2(�) → Sh by

(Qhv − v, �) = 0, ∀� ∈ Sh ,

and the elliptic projection Ph : H 1(�) → Sh by

(�[Phv − v],��) = 0, ∀� ∈ Sh , (2.5)

(Phv − v, 1) = 0� (2.6)

We also introduce the following spaces:

◦Sh= �v ∈ Sh , (v, 1) = 0� and L2

0(�) = �v ∈ L2(�), (v, 1) = 0�

and define the discrete inverse Laplace operator: −�−1h : L2

0(�) → ◦Sh such

that

(�(−�−1h v),��) = (v, �), ∀� ∈ Sh �

We assume that the initial conditions c0, �10, �20 ∈ H 3(�) and are suchthat

J (c0, �10, �20) = �c

2|�c0|2 + �1

2|��10|2 + �2

2|��20|2 +

∫�

� (c0, �10, �20)dx ≤ C ,

(2.7)

lims→0+ |�[�i]t(s)| ≤ C , (2.8)

|�iLi��i0 − Li(��i� (c0, �10, �20)| ≤ C , (2.9)

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where i = 1, 2. The derived initial condition w0 satisfies

‖w0‖H l = ‖�cD�c0 + D�c� (c0, �10, �20)‖H l ≤ C � (2.10)

We denote

m0 = 1|�|

∫�

c0(x)dx � (2.11)

Also, we suppose that f and g in (1.8) are C 2 and satisfy (1.9) and (1.10).

3. A PRIORI ESTIMATES FOR THE SOLUTION OF (1.1)–(1.7)

In this section, we derive energy estimates in several functional spacesfor the solution (c ,w, �1, �2) of the Cahn–Hilliard/Allen–Cahn system(1.1)–(1.7) for given (c0, �10, �20) ∈ [H 3(�)]3. We define for r ≥ 0

H −r (�) = (H r (�))∗, H −r0 (�) = �w ∈ H −r (�); 〈w, 1〉r = 0�,

where 〈·, ·〉r stands for the dual product between H −r (�) and H r (�); wedenote L2

0(�) = H 00 (�).

Given v ∈ L20(�), let v1 = −�−1v ∈ H 1(�) ∩ L2

0(�) be the solution of

−�v1 = v, in �

�v1�n

= 0, on ��

and define �− 12 v as

�− 12 v = �v1 = −��−1v�

Lemma 3.1. Suppose c0, �10, �20 ∈ H 3(�) satisfy (2.7)–(2.11). Then thefollowing estimates hold for the solution (c ,w, �1, �2) of (1.1)–(1.7):

1. 1|�|∫�c dx = m0, ∀t ≥ 0�

2. ess sup[0,∞)

{�c2 |�c |2 + �1

2 |��1|2 + �22 |��2|2 + |� (c , �1, �2)|L1

}+∫∞

0 |�w(s)|2L2 ds + ∫∞0

[|[�1]t(s)|2 + |[�2]t(s)|2]ds ≤ J (c0, �10, �20) + C �

3. ess sup[0,∞)

{�c2 |�c |2 + �1

2 |��1|2 + �22 |��2|2 + |� (c , �1, �2)|L1

}+∫∞

0 ‖ct(s)‖2H−1ds + ∫∞

0

[|[�1]t(s)|2 + |[�2]t(s)|2]ds ≤ J (c0, �10, �20) + C �

4. ess sup[0,∞)

{|�−1ct |2 + |[�1]t |2 + |[�2]t |2}+ ∫∞

0

[|ct(s)|2L2 + |�[�1]t(s)|2L2+|�[�2]t(s)|2L2

]ds ≤ C �

5.∫∞0 ‖�−1ctt‖2

H−1 ds + ∫∞0

[|[�1]tt |2 + |[�2]tt |2]ds ≤ C.

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Proof. 1. The assertion follows immediately from the weak formulation

of system (1.1)–(1.3). We have just to choose v = 1 in

ddt(c , v) + (�w,�v) = 0�

2. This assertion is proved using the fact that

ddtJ (c , �1, �2) = − 1

D|�w|2 − 1

L1|[�1]t |2 − 1

L2|[�2]t |2,

where

J (c , �1, �2) = �c

2|�c |2 + �1

2|��1|2 + �2

2|��2|2 +

∫�

� (c , �1, �2)dx , ∀t ≥ 0,

and (1.9) and (1.10).

3. To prove items 2 and 3, we formally1 differentiate (1.1)–(1.3) intime to obtain:

ctt = �wt , (3.1)

wt = D[�cc� ct + ��1c� (�1)t + ��2c� (�2)t − �c�ct

], (3.2)

[�i]tt = −Li

[�c�i� ct + ��1�i� [�1]t + ��2�i� [�2]t − �i�[�i]t

]� (3.3)

Next, we test (3.1) with �−2ct , (3.2) with −�−1ct , (3.3) with [�i]t and useYoung’s inequality to get

12ddt

|�−1ct |2 + �cD|ct |2 = D(�cc� ct ,�−1ct) +2∑

i=1

D(��i c� [�i]t ,�−1ct)

≤ D|�cc� (c , �1, �2)|L3 |ct ||�−1ct |L6

+2∑

i=1

D|��i c� (c , �1, �2)|L3 |[�i]t ||�−1ct |L6 , (3.4)

12ddt

|[�i]t |2 + �iLi |�[�i]t |2 = −Li(�c�i� ct + ��1�i� [�1]t + ��2�i� [�2]t , [�i]t)

1The estimates obtained by these formal computations can be rigorously proved by consideringsuitable spectral Galerkin approximations cm and �im (that is, by using the eigenfunctions of theLaplacian as a basis) for which the formal computations can be justified; the estimates then followby taking the limit and using the lower semicontinuity.

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≤ Li

[|�c�i� (c , �1, �2)|L3 |ct ||[�i]t |L6

+2∑

j=1

|��j �i� (c , �1, �2)|L3 |[�j ]t ||[�i]t |L6

]� (3.5)

To deal with the terms on the right-hand side of the above equations, weuse the immersion of H 1(�) in L6(�) and the estimates:

|D2� (c , �1, �2)|L3(�) ≤ (C + |c |2L6 + |�1|2L6 + |�2|2L6)

and

|�−1ct |2L6 ≤ |�− 12 ct |2(= |�w|2)�

From item 2 of Lemma 3.1, after adding (3.4) and (3.5), we obtain

12ddt

(|�−1ct |2 + |[�1]t |2 + |[�2]t |2)+ �cD

2|ct |2 + �1L1

2|�[�1]t |2 + �2L2

2|�[�2]t |2

≤ C [|�w|2 + |[�1]t |2 + |[�2]t |2]�The above estimates, Gronwall’s lemma, and an integration over [0,∞)then give the result.

4. Multiplying (3.1) by −�−3ctt , we obtain

|�− 32 ctt |2 = −(ctt ,�−3ctt) = −(�wt ,�−3ctt) = (wt ,�−2ctt)�

Next, we multiply (3.2) by −�−2ctt to get

−(wt ,�−2ctt) = −D(��c� · (ct , (�1)t , (�2)t),�−2ctt

)+ �cD2

ddt

∣∣∣�− 12 ct∣∣∣2 �

Finally, multiplying (3.3) by [�i]tt leads to

|[�i]tt |2 = −Li

(��i� · (ct , [�1]t , [�2]t), [�i]tt

)− �iLi

2ddt

|�[�i]t |2�

Combining the previous three equalities, after some computations, we get

|�− 32 ctt |2 + �cD

2ddt

(|�− 1

2 ct |2)

+2∑

i=1

(|[�i]tt |2 + �iLi

2ddt

(|�[�i]t |2))

≤ CD

[|�cc� · (c , �1, �2)|2L∞|ct |2 +

2∑i=1

|��i c� (c , �1, �2)|2L∞|[�i]t |2]

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+C2∑

i=1

Li

|�c�i� (c , �1, �2)|2L∞|ct |2 +

2∑j=1

|��j �i� (c , �1, �2)|2L∞|(�i)t |2

+ 12|�− 3

2 ctt |2 + �cD2

|�− 12 ct |2 +

2∑i=1

(12|[�i]tt |2 + �iLi

2|�[�i]t |2

)�

This last inequality together with items (2) and (4) imply the assertion. �

4. ERROR ANALYSIS FOR A SEMIDISCRETE (IN TIME)APPROXIMATION

We start with a weak formulation (when p = 2) of (1.1)–(1.7): Find(c(t),w(t), �1(t), �2(t)) ∈ [H 1(�)]4 such that for almost all t ∈ (0,T )

(ct , �) + (�w,��) = 0, (4.1)

(w, v) = �cD(�c ,�v) + D(�c� (c , �1, �2), v), (4.2)

([�i]t , �) + �iLi(��i ,��) + Li(��i� (c , �1, �2), �) = 0, (4.3)

c(x , 0) = c0(x), x ∈ �, (4.4)

�i(x , 0) = �i0(x), x ∈ �, i = 1, 2 (4.5)

where �, v, � ∈ H 1(�)A semidiscrete mixed formulation via implicit Euler method on

the time mesh Jk = �tm�Mm=0 for (4.1)–(4.5) reads: Find (cm ,wm , �m1 , �m2 ) ∈

[H 1(�)]4 such that for every 1 ≤ m ≤ M

(dt cm , �) + (�wm ,��) = 0, (4.6)

(wm , v) = �cD(�cm ,�v) + D(�c� (cm , �m1 , �m2 ), v), (4.7)

(dt�mi , �i) + �iLi(��mi ,��i) + Li(��i� (cm , �m1 , �

m2 ), �i) = 0, (4.8)

with c0 = c0, �0i = �i0, i = 1, 2 and where �, v, � ∈ H 1(�). Here Jk = �tm�Mm=0

is a uniform partition of [0,T ] of size k = TM . Also, dtum = um−um−1

k .

Lemma 4.1. For

k < min{

4�cD(A + 4�G1)2

,1

2L[�G2 + ̄(F1 + F2)]}, (4.9)

and c0, �10, �20 ∈ H 3(�), the solution of the scheme (4.6)–(4.8) satisfies thefollowing estimates

1. 1|�|∫�cm dx = m0.

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2. max0≤m≤M

{|�cm |2 + |��m1 |2 + |��m2 |2 + |� (cm , �m1 , �m2 )|L1

}+k∑M

m=1

{|�wm |2+‖dt cm‖2

H−1 + k|�dt cm |2}+ k∑M

m=1

{|dt�m1 |2 + k|�dt�m1 |2|dt�m2 |2 + k|�dt�m2 |2}≤ C(c0, �01, �

02).

3. max0≤m≤M

{|�−1dt cm | + |dt�m1 |2 + |dt�m2 |2}+ k∑M

m=1

[|dt cm |2 + |�dt�m1 |2 +|�dt�m2 |2] ≤ C.

4. max0≤m≤M

{|�cm |2 + |��m1 |2 + |��m2 |2} ≤ C.5. max0≤m≤M |��−1dt cm |2 + k

∑Mm=1 |�dt cm |2 ≤ C.

Proof. 1. The proof of item 1 is trivial (set � = 1 in (4.6)).

2. To prove item 2, we argue as in Lemma 1.1 of [2] (see Chapter 1).Choosing (�, v, �i) = (wm , dt cm , dt�mi ) in (4.6)–(4.8) leads to

1D

|�wm |2 + �c (�cm ,�dt cm) +2∑

i=1

[1Li

|dt�mi |2 + �i(��mi ,�dt�

mi )

]

= −�� (cm , �m1 , �m2 ) · (dt cm , dt�m1 , dt�m2 )� (4.10)

To estimate the right-hand side terms, we consider the convex function[� + H ](cm , �m1 , �m2 ), where

H (cm , �m1 , �m2 ) = A

2(cm − cm)2 + �

2∑i=1

g (cm , �mi ) −2∑

i=1

2∑i �=j=1

ij f (�mi , �mj )

and by using (1.9) and (1.10), we obtain

−�� (cm , �m1 , �m2 ) · (dt cm , dt�m1 , dt�m2 )

≤ 1k[� (cm−1, �m−1

1 , �m−12 ) − � (cm , �m1 , �

m2 )] + k(A + 4�G1)

2(dt cm)2

+ k2∑

i=1

��G2 + ̄(F1 + F2)� (dt�mi )2�

Next, we observe that

(�cm ,�dt cm) = [|�cm |2 − |�cm−1|2]2k

+ k2|�dt cm |2 = 1

2dt [|�cm |2] + k

2|�dt cm |2�

Therefore, (4.10) becomes

kD

|�wm |2 + k2�c2

|�dt cm |2 + k�c2

dt [|�cm |2]

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≤ Ckm∑r=1

[‖dt c r‖2

H−1 +2∑

i=1

|dt�ri |2]

+ 12|�−1dt c0| + 1

2

2∑i=1

|dt�0i |2�

Item 2, (4.13), (2.10), and (2.9) then imply that the right-hand side isbounded.

4. We test (4.6)–(4.8)with (cm ,�cm ,��mi ),

(dt cm ,�−1�cm) = (wm ,�cm) = −�cD|�cm |2 + D(�c� (cm , �m1 , �m2 ),�c

m),

�iLi |��mi |2 = (dt�mi ,��mi ) + Li(��i� (cm , �m1 , �

m2 ),��

mi )�

Next, (1.9), (1.10), and item 2 imply that

|�c� (cm , �m1 , �m2 )|2 +

2∑i=1

|��i� (cm , �m1 , �m2 )|2 ≤ C

[1 + ‖cm‖2

H 1 +2∑

i=1

‖�mi ‖2H 1

],

and we have

|(dt cm , cm)| ≤ |�−1dt cm ||�cm |�

Thus, we obtain

�cD|�cm |2 +2∑

i=1

�iLi |��mi |2

≤ C

[1 + ‖cm‖2

H 1 + |�−1dt cm |2 +2∑

i=1

(‖�mi ‖2H 1 + |dt�mi |2)

],

and the assertion follows from items 2 and 3.

5. Testing (4.11)–(4.12) with (wmt , dt c

m), and using Young’sinequality give

(dt��−1dt cm ,��−1dt cm) = −�cD|�dt cm |2 − D(dt�c� (cm , �m1 , �m2 ), dt c

m),

12dt |��−1dt cm |2 + k

2|��−1d2

t cm |2 + �cD|�dt cm |2

≤ C [|dt cm |2] + C(1 + ‖cm‖2

H 1 + ‖�m1 ‖2H 1 + ‖�m2 ‖2

H 1

)× [|dt cm |2L6 + |dt�m1 |2 + |dt�m2 |2] �

By adding these inequalities over m using items 2 and 3, we get

|��−1dt cm |2 + km∑r=1

[k|��−1d2

t cr |2 + 2�cD|�dt c r |2

]

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≤ Ckm∑r=1

[|dt c r |2 +

2∑i=1

|dt�ri |2]

+ 12|��−1dt c0|2

≤ C + 12‖�−1dt c0‖2

H−1 ≤ C + 12‖dt c0‖2

H−1 �

We then estimate ‖dt c0‖2H−1 using (2.10) and (4.13), and the result follows

from the last inequality. �

4.1. Error Estimates for the Scheme (4.6)–(4.8)

In the proof of the error estimates, we will need the following discreteversion of Gronwall’s lemma: (see Lemma 7.1, p. 626 in Elliott andLarsson [10]).

Lemma 4.2. Let 0 ≤ �l ≤ R and 0 ≤ tl ≤ T , l = 1, 2, � � � ,

�l ≤ A1t−1+1l + A2t

−1+2l + Bk

l∑m=1

t−1+l−m �m , 0 < tl ≤ T ,

for some constants A1,A2,B ≥ 0, 1, 2, > 0, then there are constants C1 =C1(R ,B, ) and C2 = C2(B,T , 1, 2, ) such that for k ≤ C1,

�l ≤ C2(A1t−1+1l + A2t

−1+2l ), 0 < tl ≤ T �

Theorem 4.3. Let �(cm ,wm , �m1 , �m2 )�

Mm=0 solve (4.6)–(4.8) on a uniform mesh

Jk = �tm�Mm=0 of mesh size k, and c0, �10, �20 ∈ H 3(�). Suppose (2.11)–(2.9) holdand that

k < min{

4�cD(A + 4�G1)2

,1

2L[�G2 + ̄(F1 + F2)]}�

Then there exists a positive constant C = C(c0, �10, �20,T ,�) such that thesolution of (4.6)–(4.8) satisfies the following error estimate

max0≤m≤M

‖c(tm) − cm‖H−1 +(k

M∑m=1

{k‖dt [c(tm) − cm]‖2

H−1 + |�[c(tm) − cm]|2}) 1

2

+2∑

i=1

[max0≤m≤M

|�i(tm) − �mi | +(k

M∑m=1

�k|dt [�i(tm) − �mi ]|2

+ |�[�i(tm) − �mi ]|2�) 1

2]

≤ Ck�

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Proof. Let em = c(tm) − cm ∈ L20(�), pm = w(tm) − wm and qm

i = �i(tm) −�mi denote the error functions. We denote

� (tm) = � (c(tm), �1(tm)) and � (m) = � (cm , �m1 , �m2 )�

Respectively subtracting (4.6)–(4.8) from (4.1)–(4.3), we obtain the errorequations

(dt em , �) + (�pm ,��) = (�(ctt ;m), �),

(pm , v) = �cD(�em ,�v) + D(�c� (tm)) − �c� (m), v),

(dtqmi , �i) + �iLi(�qm

i ,��i) + Li(��i� (tm) − ��i� (m), �i) = (�([�i]tt ;m), �i),

which are valid for (�, v, �i) ∈ [H 1(�)]3, and where

�(u;m) = −1k

∫ tm

tm−1

(s − tm−1)u(s)ds�

We take (�, v, �) = (−�−1em , em , qmi ) and use

(�dtum ,�um) = 12dt |um |2 + k

2|dtum |2, (�pm ,−��−1em) = (pm , em)

and

(dt em ,−�−1em) = (−��−1dt em ,−��−1em),

to find

12D

dt |�− 12 em |2 + k

2D|�− 1

2 dt em |2 + �c |�em |2

+2∑

i=1

(12Li

dt |qmi |2 + k

2Li|dtqm

i |2 + �i |�qmi |2)

+ (�[� (tm) − � (m)] · (em , qm1 , q

m2 ), 1)

= 1D(−�−1�(ctt ;m), em) +

2∑i=1

1Li(�([�i]tt ;m), qm

i )� (4.14)

From item 5 of Lemma 3.1, it follows that

kM∑

m=1

‖�−1�(ctt ;m)‖2H−1 + k

2∑i=1

M∑m=1

|�([�i]tt ;m)|2

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≤ 1k

M∑m=1

[∫ tm

tm

(s − tm)2ds] [∫ tm

tm

‖�−1ctt(s)‖2H−1ds

]

+ 1k

2∑i=1

M∑m=1

[∫ tm

tm

(s − tm)2ds] [∫ tm

tm

|[�i]tt(s)|2ds]

≤ Ck� (4.15)

To control the last term on the left side of (4.14), we use the convexityof the fourth order terms on the definition of � , (1.9) and (1.10), toconclude that there is a constant C > 0 such that

(�[� (tm) − � (m)] · (em , qm1 , q

m2 ), 1) ≥ −A|em |2 − C

[|em |2 +

2∑i=1

|qmi |2]�

(4.16)

Estimates (4.14) and (4.16) give

12D

dt |�− 12 em |2 + k

2D|�− 1

2 dt em |2 + �c

2|�em |2

+2∑

i=1

[12Li

dt |qmi |2 + k

2Li|dtqm

i |2 + �i |�qmi |2]

≤ C

[|em |2 +

2∑i=1

|qmi |2]

+ CD

‖�−1�(ctt ;m)‖2H−1 +

2∑i=1

CLi

|�([�i]tt ;m)|2�(4.17)

The first term on the right-hand side can be bounded as

C |em |2 ≤ C |�− 12 em |2 + �c

4|�em |2� (4.18)

Then, from (4.15) and (4.18), after adding (4.17) over m from 0 to l(≤ M − 1) and taking into account that e0 = q0

i = 0, we obtain that

12D

|�− 12 em |2 + k

m∑r=1

{k2D

|�− 12 dt e r |2 + �c

4|�e r |2

}

+2∑

i=1

12Li

|qmi |2 + k

2∑i=1

m∑r=1

{k2Li

|dtq ri |2 + �i |�qr

i |2}

≤ Ckm∑r=1

[|�− 1

2 e r |2 +2∑

i=1

|qri |2]

+ Ck�

We conclude the proof by using the discrete Gronwall’s inequalitygiven in Lemma 4.2 (with A1 = Ck,A2 = 0 and 1 = 2 = = 1) to obtain

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12D

|�− 12 em |2 + k

m∑r=1

{k2D

|�− 12 dt e r |2 + �c

4|�e r |2

}

+2∑

i=1

12Li

|qmi |2 + k

2∑i=1

m∑r=1

{k2Li

|dtq ri |2 + �i |�qr

i |2}

≤ Ck� �

5. ERROR ANALYSIS FOR THE FULLY DISCRETE SCHEME

In this section, we prove our main result:

Theorem 5.1. Let �(Cm ,W m ,�m1 ,�

m2 )�

Mm=0 solve (2.1)–(2.4) on a uniform

mesh Jk and a quasi-uniform triangulation Th of �. Let the assumptions ofTheorem 4.3 hold. Assume the mesh and initial values satisfy the followingconstraints

1. k < min{

4�cD(A+4�G1)

2 ,1

2L[�G2+ ̄(F1+F2)]

}�

2. (C 0, 1) = (c0, 1) and ‖C 0 − c0‖H−1 ≤ Ch3‖c0‖H 2 �3. |�0

i − �i0| ≤ Ch2‖�i0‖H 2 �

Then the solution of (2.1)–(2.4) satisfies the error estimates

1.

max0≤m≤M

‖c(tm) − Cm‖H−1 +(k

M∑m=1

k‖dt [c(tm) − Cm]‖2H−1

) 12

+(

M∑m=1

k|c(tm) − Cm |2) 1

2

+2∑

i=1

max

0≤m≤M|�i(tm) − �m

i | +(k

M∑m=1

k|dt [�i(tm) − �mi ]|2

) 12

≤ C(h4 + k2

) 12 �

2. (k

M∑m=1

{|�[c(tm) − Cm]|2 +

2∑i=1

|�[�i(tm) − �mi ]|2

}) 12

≤ C(h2 + k2

) 12 �

Proof. We will proceed in several steps.

1. Let Em = cm − Cm , Gm = wm − W m , and F mi = �mi − �m

i and alsodenote

� (cm , �m1 , �m2 ) = � (m), � (Cm ,�m

1 ,�m2 ) = � (mm),

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� (Phcm ,Ph�m1 ,Ph�

m2 ) = � (Phm)�

We subtract (2.1)–(2.4) from (4.6)–(4.8) to get the following errorequations

(dtEm , �) + (�Gm ,��), (5.1)

(Gm , v) = �cD(�Em ,�v) + D(�c� (m) − �c� (mm), v), (5.2)

(dtF mi , �) + �iLi(�F m

i ,��) + Li(��i� (m) − ��i� (mm), �) = 0, (5.3)

for all �, v, � ∈ Sh . We introduce the decompositions Em = Bm + �m , Gm =�m + �m , and F m

i = Hmi + �m

i , where

Bm = cm − Phcm , �m = Phcm − Cm ,

�m = wm − Phwm , �m = Phwm − W m ,

Hmi = �mi − Ph�

mi , �m

i = Ph�mi − �m

i �

Then, from definition of Ph in (2.5)–(2.6), we can rewrite (5.1)–(5.3) asfollows

(dt�m , �) + (��m ,��) = −(dtBm , �), (5.4)

(�m , v) + (�m , v) − D(�c� (m) − �c� (Phm), v)

= �cD(��m ,�v) + D(�c� (Phm) − �c� (mm), v), (5.5)

(dt�mi , �) + �iLi(��

mi ,��) + Li(��i� (Phm) − ��i� (mm), �)

= −(dtH mi , �) − Li(��i� (m) − ��i� (Phm), �)� (5.6)

Because Em ,�m ∈ L20(�), for 0 ≤ m ≤ M , by setting (�, v, �) =

(−�−1h �m ,�m ,�m

i ) in (5.4)–(5.6) and by adding over m from 1 to l (≤M )and also adding the resulting equations, we arrive at

12D

|��−1h �l |2 +

2∑i=1

12Li

|�li |2 + k2

2

l∑m=1

[1D

|��−1h dt�m |2 +

2∑i=1

1Li

|dt�li |2]

+ kl∑

m=1

[�c |��m |2 + (�c� (Phm) − �c� (mm),�m)

]

+ k2∑

i=1

l∑m=1

[�i |��m |2 + (��i� (Phm) − ��i� (mm),�m

i )]

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= kl∑

m=1

[− 1D(dtBm ,−�−1

h �m) + 1D(�m ,�m) −

2∑i=1

1Li(dtH m

i ,�mi )

]

+ kl∑

m=1

[(�c� (m) − �c� (Phm),�m) +

2∑i=1

(��i� (m) − ��i� (Phm),�mi )

]

+ 12D

|��−1h �0|2� (5.7)

The first sum of the right-hand side can be bounded as

kl∑

m=1

[− 1D(dtBm ,−�−1

h �m) + 1D(�m ,�m) −

2∑i=1

1Li(dtH m

i ,�mi )

]

≤ Ckl∑

m=1

[‖dtBm‖2

H−1 + ‖�m‖2H−1 +

2∑i=1

‖dtH mi ‖2

H−1

]

+ kl∑

m=1

[12D

|��−1h �m |2 + �c

6|��m |2 +

2∑i=1

min{�i

4,12Li

}‖�m

i ‖2H 1

]

≤ Ckl∑

m=1

[‖dtBm‖2

H−1 + ‖�m‖2H−1 +

2∑i=1

‖dtH mi ‖2

H−1

]

+ kl∑

m=1

[12D

|��−1h �m |2 + �c

6|��m |2 +

2∑i=1

(�i

4|��m

i |2 + 12Li

|�mi |2)]

�

(5.8)

Next, by using

(a − b)3 − (c − b)3 = (a − c)[(a − b)2 + (a − b)(c − b) + (c − b)2],|��uf (a, b)|∞ + |��v f (a, b)|∞ + |��ug (a, b)|∞ + |��vg (a, b)|∞ ≤ C ,

we can estimate

(�c� (m) − �c� (Phm),�m) +2∑

i=1

(��i� (m) − ��i� (Phm),�mi )

≤ C2∑

i=1

|�m |2 + C(1 + |cm |2L6 + |Phcm |2L6)|Bm |2 + �c

12CP|�m |2L6

+2∑

i=1

[|�m

i |2 + C(1 + |�mi |2L6 + |Ph�mi |2L6)||Hm

i |2 + �i

2C|�m

i |2L6

]

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≤ C [|��−1h �m |2 + |Bm |2 + �c

6|��m |2 +

2∑i=1

[|�m

i |2 + |Hmi |2] + �i

4|� �m

i |2]�

(5.9)

To control the two terms on the left side of (5.7), which depend on � , weuse the convexity of the fourth order terms on the definition of � , (1.9)and (1.10), to conclude that

(�c� (Phm) − �c� (mm),�m) +2∑

i=1

(��i� (Phm) − ��i� (mm),�mi )

≥ −C

[|�m |2 +

2∑i=1

|�mi |2]

≥ −C [|��−1h �m |2 +

2∑i=1

|�mi |2] − �c

6|��m |2�

(5.10)

By substituting (5.8)–(5.10) in (5.7), we arrive at

12D

|��−1h �l |2 +

2∑i=1

12Li

|�li |2 + k2

2

l∑m=1

[1D

|��−1h dt�m |2d +

2∑i=1

1Li

|dt�li |2]

+ k2

l∑m=1

[�c |��m |2 +

2∑i=1

�i |��m |2]

≤ Ckl∑

m=1

[|��−1h �m |2 + |�m |2]

+Ckl∑

m=1

[‖dtBm‖2

H−1 + ‖�m‖2H−1 + |Bm |2 +

2∑i=1

(‖dtH mi ‖2

H−1 + |Hmi |2)

]

+ 12D

|��−1h �0|2� (5.11)

Next, we make use of the following approximation properties in H −1 andL2 of the elliptic projection Ph , for � = 0, 1

|cm − Phcm | ≤ Ch2−�‖cm‖H 2−� ,

‖cm − Phcm‖H−1 ≤ Ch3−�‖cm‖H 2−� ,

‖dt(cm − Phcm)‖H−1 ≤ Ch3−�|dt cm |H 2−� ,

‖wm − Phwm‖H−1 ≤ Ch3−�‖wm‖H 2−� (≤ Ch3−�‖dt cm‖H 1−�),

|�mi − Ph�mi | ≤ Ch2−�‖�mi ‖H 2−� ,

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|dt [�mi − Ph�mi ]| ≤ Ch2−�‖dt�mi ‖H 2−� ,

to bound the right-hand side of (5.11) by

Ch4kl∑

m=1

[‖dt cm‖2

H 1 + |dt cm |2 + ‖cm‖2H 2 +

2∑i=1

(‖dt�mi ‖2H 1 + ‖�mi ‖2

H 2 + ‖c0‖2H 2

)]�

Next, thanks to items 3 and 5 of Lemma 4.1, and the hypothesis on item 2of Theorem 5.1, we have

kl∑

m=1

[‖dt cm‖2

H 2 + ‖dt cm‖2H 1 + ‖cm‖2

H 2 + ‖c0‖2H 2

+2∑

i=1

(‖dt�mi ‖2H 2 + ‖�mi ‖2

H 2

) ] ≤ C �

Arguing as in Theorem 4.3, we obtain

12D

|��−1h �l |2 +

2∑i=1

12Li

|�li |2 + k

2

l∑m=1

{k

[1D

|��−1h dt�m |2d +

2∑i=1

1Li

|dt�li |2]

+ �c |��m |2 +2∑

i=1

�i |��m |2}

≤ Ch4�

Finally, the assertions of the theorem follow by applying the triangleinequality on Em = Bm − �m , Gm = �m − �m , and F m

i = Hmi − �m

i , toobtain

max0≤m≤M

‖c(tm) − Cm‖H−1 +(k

M∑m=1

k‖dt [c(tm) − Cm]‖2H−1

) 12

+(

M∑m=1

k|c(tm) − Cm |2) 1

2

+2∑

i=1

max

0≤m≤M|�i(tm) − �m

i | +(k

M∑m=1

k|dt [�i(tm) − �mi ]|2

) 12

≤ max0≤m≤M

‖c(tm) − cm‖H−1 + max0≤m≤M

‖Bm‖H−1 + max0≤m≤M

‖�m‖H−1

+(k

M∑m=1

k‖dt [c(tm) − cm]‖2H−1

) 12

+(k

M∑m=1

k‖dt [Bm]‖2H−1

) 12

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+(k

M∑m=1

k‖dt [�m]‖2H−1

) 12

+(

M∑m=1

k|c(tm) − cm |2) 1

2

+(

M∑m=1

k|Bm |2) 1

2

+(

M∑m=1

k|�m |2) 1

2

+2∑

i=1

[max0≤m≤M

|�i(tm) − �mi | + max

0≤m≤M|Hm

i | + max0≤m≤M

|�mi |]

+2∑

i=1

(k M∑

m=1

k|dt [�i(tm) − �mi ]|2) 1

2

+(k

M∑m=1

k|dtH mi |2) 1

2

+(k

M∑m=1

k|dt�mi |2) 1

2

≤ C(h4 + k2

) 12 , (from Theorem 4.3)

(k

M∑m=1

{|�[c(tm) − Cm]|2 +

2∑i=1

|�[�i(tm) − �mi ]|2

}) 12

≤(k

M∑m=1

{|�[c(tm) − cm]|2 +

2∑i=1

|�[�i(tm) − �mi ]|2}) 1

2

+(k

M∑m=1

{|�Bm |2 +

2∑i=1

|�Hmi |2}) 1

2

+(k

M∑m=1

{|��m |2 +

2∑i=1

|��mi |2}) 1

2

≤ C(h2 + k2

) 12 � (from Theorem 4.3) �

REFERENCES

1. D. Fan, L.-Q. Chen, S. Chen, and P. W. Voorhees (1998). Phase field formulations for modelingthe Ostwald ripening in two-phase systems. Comp. Mater. Sci. 9:329–336.

2. P.N. Silva and J.L. Boldrini (2003). Modelos do Tipo Campo de Fases em Processos de Cristalização,Ph.D. Thesis, IMECC/UNICAMP.

3. L.-Q. Chen and D. Fan (1996). Computer simulation for a coupled grain growth and Ostwaldripening—application to Al2O3–Zr2 two-phase systems. J. Am. Ceram. Soc. 79:1163–1168.

4. L.-Q. Chen and D. Fan (1997). Computer simulation of grain growth using a continuum filedmodel. Acta Mater. 45:611–622.

5. L.-Q. Chen and D. Fan (1997). Diffusion-controlled grain growth in two-phase solids. Acta Mater.45:3297–3310.

6. L.-Q. Chen, D. Fan, and C. Geng (1997). Computer simulation of topological evolution in 2-dgrain growth using a continuum diffusion-interface filed model. Acta Mater. 45:1115–1126.

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7. X. Feng and A. Prohl (2005). Numerical analysis of the Cahn–Hilliard equation andapproximation for the Hele–Shaw problem. Interfaces and Free Boundaries 7:1–28.

8. X. Feng and A. Prohl (2003). Numerical analysis of the Allen–Cahn equation andapproximation for the mean curvature flows. Numer. Math. 94:33–65.

9. M.I.M. Copetti and C.M. Elliott (1992). Numerical analysis of the Cahn–Hilliard equation witha logarithmic free energy. Numer. Math. 63:39–65.

10. C.M. Elliott and S. Larsson (1992). Error estimates with smooth and nonsmooth data for afinite element method for the Cahn–Hilliard equation. Math. Comp. 58:603–630.

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error estimates for full discretization of a model for ostwald ripening

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