Parallel domain decomposition procedures of improved D–D type for parabolic problems

Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

Contents lists available at ScienceDirect

Journal of Computational and Applied Mathematics journal homepage: www.elsevier.com/locate/cam

Parallel domain decomposition procedures of improved D–D type for parabolic problemsI Danping Yang ∗ Department of Mathematics, East China Normal University, Shanghai, 200241, PR China

article

info

Article history: Received 19 November 2008 Received in revised form 8 November 2009 MSC: 65N12 65N15 65N30 65N55 Keywords: Parallel algorithm Domain decomposition Parabolic problem A priori error estimate

abstract Two parallel domain decomposition procedures for solving initial-boundary value problems of parabolic partial differential equations are proposed. One is the extended D–D type algorithm, which extends the explicit/implicit conservative Galerkin domain decomposition procedures, given in [5], from a rectangle domain and its decomposition that consisted of a stripe of sub-rectangles into a general domain and its general decomposition with a net-like structure. An almost optimal error estimate, without the factor H −1/2 given in Dawson–Dupont’s error estimate, is proved. Another is the parallel domain decomposition algorithm of improved D–D type, in which an additional term is introduced to produce an approximation of an optimal error accuracy in L2 -norm. © 2009 Elsevier B.V. All rights reserved.

1. Introduction Domain decomposition and parallel computation have become powerful tools for solving a large scale PDE systems. Domain decomposition methods decompose a large or complicated domain into several small or regular sub-domains with overlapping or non-overlapping, on which subproblems may be solved by the use of effective or fast algorithms. Recently, a lot of algorithms based on overlapping and non-overlapping domain decompositions have been proposed to solve parabolic problems, for example, see [1–22,26,27]. In [5], Dawson and Dupont proposed some explicit/implicit conservative Galerkin domain decomposition procedures to solve parabolic problems and also proved the a priori error estimates. We call these procedures D–D type algorithms. Their numerical experiments showed an almost optimal error accuracy in L2 -norm. Since the earliest work in [5], D–D type algorithms have been developed and applied to many applications in Engineering. But, in the a priori error estimate in [5], there was a loss of H −1/2 , i.e., error estimates were not optimal in L2 -norm. The authors said that they did not know how to improve it. On the other hand, D–D type algorithms only were designed for a problem defined a rectangle domain and its domain decomposition consisted of a stripe of sub-rectangles, in which the inner boundaries of sub-domains are parallel to the same coordinate. So these procedures are not suitable to problems defined on a domain of a general shape and of a general domain decomposition without a chain structure. The purpose of this article is to propose parallel domain decomposition algorithms on a general domain and its general decompositions. One is the extended D–D type algorithm. Particularly, the D–D procedure is a special case of the extended D–D type algorithm. Our analysis shows that an approximation given by the extended D–D type algorithm has an almost optimal error accuracy in L2 -norm. As a consequence, we prove a new error estimate of Dawson–Dupont’s domain

I This work was supported by the National Basic Research Program of PR China under the grant 2005CB321703.

∗

Tel.: +86 21 54345180; fax: +86 21 54342609. E-mail address: [email protected].

0377-0427/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.cam.2009.11.024

2780

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

decomposition procedure, in which there is no a loss of H −1/2 order. Numerical results in [5] have confirmed these theoretical analyses. Furthermore, in order to obtain an approximation of an optimal error accuracy, we propose the improved D–D type domain decomposition algorithm. The organization of the article is as follows. In Section 2, we propose two D–D type domain decomposition procedures. One is the extended D–D type algorithm, which is designed for a problem defined on a general domain and its decomposition without a chain structure assumption. Another is the improved D–D type algorithm, in which an additional term is introduced to improve approximating accuracy of numerical solutions. Section 3 is devoted to an analysis of L2 -norm error estimate for the extended and improved D–D type domain decomposition procedures described in Section 2. We prove that the extended D–D type domain decomposition algorithm reaches to accuracy of an almost optimal order in L2 -norm. Furthermore, we prove that the improved D–D type domain decomposition algorithm has an optimal error estimate in L2 -norm. 2. D–D type domain decomposition algorithms Let Ω be a bounded domain in R2 , with a boundary ∂ Ω . We consider the initial-boundary value problem of parabolic partial differential equation of the form:

∂u − ∇ · (a∇ u) + bu = f , in Ω × (0, T ]; ∂t ∂u (b) = 0, on ∂ Ω × (0, T ]; ∂n (c) u = u0 , in Ω , at t = 0; (a)

(2.1)

where n is the unit vector outward normal to ∂ Ω , coefficients a = a(x) with 0 < a0 ≤ a(x) ≤ a1 , b = b(x), f = f (x, t ) and u0 = u0 (x) are some given real-valued functions. In [5], Dawson and Dupont presented the explicit/implicit conservative Galerkin domain decomposition procedures. Their methods were designed for a rectangle domain and a special decomposition consisted of several rectangle sub-domains whose sides are parallel to the same coordinate so as to be of a chain-like structure. In this section, we will extend D–D algorithms to general cases of multi-sub-domains decompositions with net structures. Throughout the article, we use the standard definition and denotation of Sobolev spaces and their norms as in [25], such as L2 (G), W m,p (G) for 1 ≤ p ≤ ∞ and H m (G) = W m,2 (G) defined in a domain G associated with norms k · kL2 (G) , k · kW m,p (G) and k · kH m (G) . Let ( · , · )Σ be the inner product in L2 (Σ ),

(ψ, ρ)Σ =

Z Σ

ψρ dΣ .

In the case of Σ = Ω , we omit the subscript such as (ψ, ρ) = (ψ, ρ)Ω . J We consider a domain decomposition of many sub-domains. Divide Ω into many non-overlapping sub-domains, {Ωj }j=1 ,

SJ

SJ

¯ = j =1 Ω ¯ j . Set Γj = ∂ Ωj \ ∂ Ω and Γ = j=1 Γj , which is the set of inner boundaries of sub-domains. Here we such that Ω do not require the domain and its decomposition to be rectangles and to be of a chain structure. In order to apply D–D type algorithms, we assume that (H1) Inner-boundaries Γj of Ωj , 1 ≤ j ≤ J, are piecewise linear. (H2) There exists a constant H0 > 0 such that Gj = {y : y = x − τ nΓj , x ∈ Γj , 0 ≤ τ ≤ H0 } ⊂ Ωj , where nΓj is the unit vector outward normal to Γj for 1 ≤ j ≤ J. Under Condition (H1), it is convenient to construct finite element spaces on sub-domains. As well Condition (H2) is not a severe constraint, which only requires that inner boundaries of sub-domains have included angles larger than or equal to π/2 at their intersection points. For j = 1, 2, . . . · · · , J, let Tj h be a family of triangulation of Ωj and Mhj be a family of piecewise linear finite element spaces defined on Tj h having the approximate property that,

n

inf

vh ∈Mhj

Set T h =

SJ

o ku − vh kL2 (Ωj ) + hk∇(u − vh )kL2 (Ωj ) ≤ Ch2 kukH 2 (Ωj ) .

j =1

(2.2)

Tj h . Let Mh be the subspace of L2 (Ω ) such that vh ∈ Mh , if and only if vh |Ωj ∈ Mhj for each 1 ≤ j ≤ J. For any

point at Γ , assign nΓ as the given unit vector normal to Γ . Along the direction nΓ , a function v in Mh has a well-defined jump [v] across Γ :

[v](x) = κi lim v(x − snΓi ) + κj lim v(x − snΓj ), s→0+

s→0+

∀ x ∈ Γi

\

Γj ,

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2781

where lims→0+ v(x − snΓi ) is the inner limit of the function along the outward normal direction nΓi , and

κi = (nΓi · nΓ ) =



1, −1,

as nΓi = nΓ ; as nΓi = −nΓ .

For each 0 < H ≤ H0 , define the function as 1 + τ, ϕ(τ ) = 1 − τ , 0,

(

−1 ≤ τ ≤ 0; 0 ≤ τ ≤ 1; |τ | > 1

(2.3)

τ ∈ R1 .

(2.4)

and

φ(τ ) =

1 H

ϕ

τ  H

,

Integrating by part, H

Z

φ(τ )∇ u(x + τ nΓ ) · nΓ dτ =

H

Z

−H

−H

φ(τ )u0τ (x + τ nΓ )dτ = −

Z

H

φ 0 (τ )u(x + τ nΓ )dτ , −H

we define an approximate of a normal direction derivative as follows: B(ψ)(x) = −

H

Z

φ 0 (τ )ψ(x + τ nΓ )dτ ,

x ∈ Γi

\

Γj , 1 ≤ i < j ≤ J .

(2.5)

−H

It is clear that B(ψ) is the weighted averaging of the normal direction derivative and with the weighted function φ , which is an approximation of δ -function. The condition (H2) insures that B(ψ) is of the certain definition. Noting that

−

J X

∇ · (a∇ u), v

j=1




Ωj

=

J X

a∇ u, ∇v

j =1




Ωj

−

J X

a∇ u · nΓj , v




j =1

Γj

and J X

a∇ u · nΓj , v




j=1

Γj

=

X

a∇ u · nΓ , [v]

i



Γi ∩Γj

= a∇ u · nΓ , [v]




Γ

and defining the bilinear form D( · , · ): D(ψ, ρ) =

J Z X j =1

Ωj

a∇ψ · ∇ρ dx,

(2.6)

we have






∂t u, vh + D u, vh + bu, vh − a∇ u · nΓ , [vh ] Γ = f , vh ,

∀ vh ∈ Mh .

(2.7)

From (2.7), we derive the extended D–D type domain decomposition algorithm in Section 2.1 and the improved D–D type domain decomposition algorithm in Section 2.2, respectively. 2.1. Extended D–D type domain decomposition algorithm Let 0 = t 0 < t 1 < · · · < t N = T be a time partition, ∆t n = t n − t n−1 , ∆t = max1≤n≤N ∆t n , un (x, y) = u(x, y, t n ) and ¯∂t U n = (U n − U n−1 )/∆t n . Replacing with (2.5) the direction derivative on the inner boundary Γ , the extended D–D type domain decomposition algorithm reads: Extended domain decomposition procedure (EDDP) Give an initial approximation, U 0 ∈ Mh , of u0 . Seek U n ∈ Mh such that






∂¯t U n , V + D U n , V + bU n , V − aB(U n−1 ), [V ] Γ = f n , V ,

∀ V ∈ Mh

for n = 1, 2, . . . , N. J Once U n−1 is given, the procedure (2.8) could be performed in sub-domains {Ωj }j=1 in parallel. Equivalent parallel algorithm of EDDP Give an initial approximation, U 0 ∈ Mh , of u0 . Seek U n ∈ Mh , for n = 1, . . . , N, in two steps:

(2.8)

2782

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

Step 1. Seek Ujn ∈ Mhj , for 1 ≤ j ≤ J, in parallel, such that Ujn , V




Ωj

+ ∆t n D Ujn , V




Ωj

+ ∆t n bUjn , V




Ωj

= U n−1 + ∆t n f n , V




Ωj

+ ∆t n aB(U n−1 ), κj V




Γj

,

∀ V ∈ Mhj .

(2.9)

Step 2. Define U n (x) = Ujn (x),

x ∈ Ωj , 1 ≤ j ≤ J .

(2.10)

To obtain error estimate of the algorithm (EDDP), we introduce an elliptic projection, W ∈ Mh , of the solution u as follows: D u( · , t ) − W ( · , t ), V + κ u( · , t ) − W ( · , t ), V = 0,







∀ V ∈ Mh , 0 ≤ t ≤ T ,

(2.11)

√

where κ ≥ 63k akW 1,∞ + 1. The parameter κ will be used in the analysis of the convergence. Denote the error function of the projection by

η = u − W.

(2.12)

For the algorithm (EDDP), we have the following convergence theorem. Theorem 2.1. Assume that the solution u is sufficiently smooth. Let {U n }Nn=1 be the solution of the algorithm (EDDP) and U 0 ∈ Mh be the interpolation or L2 -projection of u0 . The following a priori error estimate

n

5

max kun − U n kL2 (Ω ) ≤ C kηkL2 ([0,T ],L∞ (Ω )) + kηt kL2 ([0,T ],L∞ (Ω )) + ∆t + h2 + H 2

o

(2.13)

1≤n≤N

holds, provided

∆t ≤

H2 4a1

.

(2.14)

The proof of Theorem 2.1 is put into Section 3.2. Here we will prove that (2.13) is almost optimal. It is clear that the auxiliary problem (2.11) is equivalent to D W, V




Ωj

+ κ W, V




Ωj

∂u + (κ − b)u, V = f− ∂t

! Ωj

∂u + a ,V ∂ nΓj

! ,

∀ V ∈ Mhj , 1 ≤ j ≤ J .

(2.15)

Γj

These are some standard finite element equations. Theorem 2.1 shows that the L2 -norm error estimate reaches the same order accuracy as the L∞ -norm error of the standard finite element method for elliptic problems. In the case of the linear finite element space, from [23,24,17,18], we see the following L∞ -error estimate:

n o kηkL∞ (Ω ) + kηt kL∞ (Ω ) ≤ Ch2 | ln h| kukW 2,∞ (Ω ) + kut kW 2,∞ (Ω ) .

(2.16)

As a consequent, we conclude that the error estimate (2.13) is almost optimal. Remark. It is clear that the algorithm S given in [5] is the particular case of the algorithm (EDDP), in which a ≡ 1, Ω = [0, 1] × [0, 1] and Ω = Ω1 Ω2 where Ω1 = [0, 12 ] × [0, 1] and Ω2 = [ 21 , 1] × [0, 1]. Dawson and Dupont proved the following error estimate:

( max ku − U kL2 (Ω ) ≤ C ∆t + H n

1≤n≤N

n

5 2

)

T

Z +

kηt kdt + H

−1/2

kηkL∞ (Ω ×(0,T ]) .

(2.17)

0

In the error estimate (2.17), there is a loss of accuracy of H −1/2 order. But their numerical experiments showed an accuracy of higher order than this estimate. Theorem 2.1 shows there is no a loss of H −1/2 -order, which were supported by numerical experiments in [5]. 2.2. Improved D–D type domain decomposition algorithm Theorem 2.1 shows that the approximate accuracy of the solution of (EDDP) is almost optimal in L2 -norm, which depends upon L∞ -norm error estimate. To obtain an algorithm of an optimal order accuracy, we propose the improved D–D type domain decomposition procedure, which produces an approximation with an optimal L2 -norm error accuracy independent of L∞ -norm error estimates so that it could be applied to other problems without good L∞ -norm error estimates. Improved Domain Decomposition Procedure (IDDP) Give an initial approximation, U 0 ∈ Mh , of u0 .

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2783

Seek U n ∈ Mh such that







∂¯t U n , V + D U n , V + bU n , V − aB(U n−1 ), [V ] Γ − aB(V ), [U n−1 ] Γ = f n , V , ∀ V ∈ Mh (2.18) for n = 1, 2, . . . , N. In (IDDP), only an additional term, −(aB(V ), [U n−1 ])Γ , is introduced. Once U n−1 is given, this procedure could also be J performed in sub-domains {Ωj }j=1 in parallel. For the algorithm (IDDP), we have the following convergence theorem. Theorem 2.2. Assume that the solution u is sufficiently smooth. Let {U n }Nn=1 be the solution of the algorithm (IDDP) and that U 0 ∈ Mh be the interpolation or L2 -projection of u0 . Then there exists a constant C independent of h, ∆t and H such that

n

5

max kun − U n kL2 (Ω ) ≤ C ∆t + h2 + H 2

1≤n≤N

o

,

(2.19)

provided

∆t ≤

r !

H2 4a1

2

1−

.

3

(2.20)

The proof of Theorem 2.2 is put into Section 3.3. Theorem 2.2 shows the optimal error accuracy in L2 -norm without the factor | ln h|. 3. Convergence analysis In this section, we prove convergence theorems given in Section 2. We first give some lemmas in the Section 3.1, which will be used to prove the convergence results. Then Theorems 2.1 and 2.2 are proved in Sections 3.2 and 3.3, respectively. 3.1. Some lemmas Similar results have been given in [5], but here we give some finer conclusions. First of all, we have some results about B(ψ). Lemma 3.1. For each ψ ∈ Mh , there holds the following inequality aB(ψ), B(ψ)

≤ 23−2/p a1 H −2−2/p |Γ |1−2/p kψk2Lp (Ω ) ,




Γ

2 ≤ p ≤ ∞,

(3.1)

where |Γ | is the length of Γ . Proof. It follows from the definition of B(ψ) that for each x ∈ Γi

Z |B(ψ)(x)| =

H

2 Z φ (τ )ψ(x + τ nΓ )dτ ≤ 22−2/p H −2−2/p

−H

!2/p

H

|ψ(x + τ nΓ )| dτ

0

2

which leads to (3.1).

Γj , (1 ≤ i < j ≤ J )

T

p

,

∀ 1 ≤ p ≤ ∞,

−H



Lemma 3.2. For each ψ ∈ Mh , there holds the equality that for any x ∈ Γ B(ψ)(x) = −

1 H

[ψ](x) +

H

Z

φ(τ )∇ψ(x + τ nΓ ) · nΓ dτ +

0

Z

φ(τ )∇ψ(x + τ nΓ ) · nΓ dτ .

(3.2)

−H

0

RH

Proof. By using φ(−H ) = φ(H ) = 0, −H φ dτ = 1 and integrating by part, we have

Z

H

φ 0 (τ )ψ(x + τ nΓ )dτ =

− −H

1



lim ψ(x + τ nΓ ) − lim ψ(x + τ nΓ )

H τ →0+

Z +

τ →0−

H

φ(τ )∇ψ(x + τ nΓ ) · nΓ dτ +

Z

0

φ(τ )∇ψ(x + τ nΓ ) · nΓ dτ . −H

0

Noting that

n

o

lim ψ(x + τ nΓ ) − lim ψ(x + τ nΓ ) = − (nΓ , nΓj ) lim ψ(x − τ nΓj ) + (nΓ , nΓi ) lim ψ(x − τ nΓi )

τ →0+

τ →0−

τ →0+

= −[ψ](x), we derive (3.2).



∀ x ∈ Γi

τ →0+

\

Γj , 1 ≤ i, j ≤ J ,

2784

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

As consequences of Lemma 3.2, we have the following results Lemma 3.3. For each ψ ∈ Mh , there holds the following inequality:

kB(ψ)kL2 (Ω ) ≤ CH

− 12

H

−1

2 L2 (Γ )

J X

+

k[ψ]k

! 21 k∇ψk

,

2 L2 (Ωj )

∀ ψ ∈ Mh

(3.3)

j=1

and

k[ψ]kL2 (Γ ) ≤ H −1/2 kψkL2 (Ω ) + H 1−1/p k∇ψkLp (Ω ) ,

∀ 2 ≤ p ≤ ∞.

(3.4)

Proof. The inequality (3.3) follows directly from Lemma 3.2. It follows from Lemma 3.2 again that 1 H

[ψ](x) = −B(ψ)(x) +

H

Z

φ(τ )ψ 0 (x + τ nΓ )dτ

−H

such that

k[ψ]kL2 (Γ )

Z

≤ H kB(ψ)kL2 (Γ ) +

H

!

φ(τ )ψ (x + τ nΓ )dτ

2

.

0

−H

L (Γ )

Noting (3.1) and

Z Z Γ

H

2 Z φ(τ )ψ (x + τ nΓ )dτ dsx ≤

−H

H

Z

0

Γ

≤ CH

p0

φ(τ ) dτ

!2/p0 Z

−H −2/p

!2/p

H

|ψ (x + τ nΓ )| dτ 0

p

dsx

−H

Z

Z Γ

!2/p

H

dsx ≤ CH −2/p k∇ψk2Lp (Ω ) ,

|ψ (x + τ nΓ )| dτ 0

p

−H

we have

k[ψ]kL2 (Γ ) ≤ H −1/2 kψkL2 (Γ ) + H 1−1/p k∇ψkLp (Ω ) . This ends the proof of Lemma 3.3.



Lemma 3.4. Let G be a domain containing the inner boundary Γ such that supp{φ} ⊂ G for small 0 < H ≤ H0 . If ψ is smooth in G, we have estimates:

kB(ψ) − ∇ψ · nΓ kL2 (Γ ) ≤ CH kψkW 2,∞ (G)

(3.5)

kB(ψ) − ∇ψ · nΓ kL2 (Γ ) ≤ CH 2 kψkW 3,∞ (G) .

(3.6)

and

Proof. Let g (τ ) = ψ(x + τ nΓ ). By integration by part, we have H

Z

φ 0 (τ )g (τ )dτ = −φψ |H−H +

−

H

Z

−H

φ(τ )g 0 (τ )dτ −H

= g 0 (0) +

H

Z

φ(τ )(g 0 (τ ) − g 0 (0))dx = g 0 (0) + −H

H

Z

φ(τ ) −H

τ

Z

g 00 (s)dsdτ ,

0

RH

where we have used φ(−H ) = φ(H ) = 0, −H φ dx = 1. This leads to B(ψ)(x) = ∇ψ(x) · nΓ +

H

Z

φ(τ )

τ

Z

−H

0

∂ 2ψ (x + snΓ )dsdτ . ∂ n2Γ

(3.7)

So (3.5) holds. Furthermore, we have

Z

H

φ(τ ) −H

τ

Z

g 00 (s)dsdτ = g 00 (0)

Z

H

φ(τ )τ dτ +

−H

0

Z

H

φ(τ )

= −H

Z

H

φ(τ ) −H

Z 0

τ

g 000 (ξ )dξ dsdτ = 0

(g 00 (s) − g 00 (0))dsdτ

0

s

Z

τ

Z

Z

H

φ(τ ) −H

τ

Z 0

(τ − s)g 000 (s)dsdτ ,

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

RH

2785

RH

where we used −H φ(τ )p(τ )dτ = p(0) for each polynomial of degree at most one such that −H φ(τ )τ dτ = 0. This means that H

Z

B(ψ)(x) = ∇ψ(x) · nΓ +

φ(τ )

τ

Z

−H

(τ − s)

0

∂ 3ψ (x + snΓ )dsdτ . ∂ n3Γ

(3.6) is derived. The proof of Lemma 3.4 is complete.

(3.8)



Lemma 3.5. For each β > 0, the following inequality

− (aB(ψ), a[ψ])Γ ≥ 1 −

11 24

! β

1 H

J 5 X

(aψ, ψ)Γ −

6β j=1

(a∇ψ, ∇ψ)Ωj − C1 kψk2L2 (Ω )

(3.9)

√

holds, where C1 = 63k ak2W 1,∞ (Ω ) β −1 . Proof. It follows from Lemma 3.2 that H

Z p 1p − a(x)B(ψ)(x) = a(x)[ψ](x) − H

p φ(τ ) a(x + τ nΓ )ψ 0 (x + τ nΓ )dτ −H

H

Z −

  p p p φ 0 (τ )( a(x + τ nΓ ) − a(x) ) + φ(τ )( a(x + τ nΓ ) )0 ψ(x + τ nΓ )dτ

−H

≥

1 H

r [ψ](x) − r

−

2 3H

a(x + τ nΓ )|∇ψ(x + τ nΓ )|2 dτ −H

r ! Z

2 √ H

!1/2

H

Z

k akW 1,∞ 1 +

3

!1/2

H

1

ψ (x + τ nΓ )dτ 2

.

−H

Noting the fact that integrations near the intersections of inner boundaries may be calculated repeatedly two times at most under Condition (H2), we derive

−(aB(ψ), [ψ])Γ ≥

! β 1 2 1 − √ (1 + δ) (a[ψ], [ψ])Γ − √

6β j=1

H

6

√ − Noting 2.4 ≤

6 √

δβ

J X

(a∇ψ, ∇ψ)Ωj

!2

1

(3.10)

kψk2L2 (Ω ) .

k ak2W 1,∞ (Ω ) 1 + √

3

√

6 ≤ 2.5 and taking δ = 0.1, we derive

−(aB(ψ), a[ψ])Γ ≥ 1 −

11 24

! β

1 H

(aψ, ψ)Γ −

J 5 X

6β j=1

We obtain (3.9). The proof of Lemma 3.5 is complete.

(a∇ψ, ∇ψ)Ωj − C1 kψk2L2 (Ω ) .



In next subsections, we prove Theorems 2.1 and 2.2, respectively. 3.2. Proof of Theorem 2.1

e ∈ Mh , of the solution In this subsection, we prove Theorem 2.1. To this end, we introduce another elliptic projection, W u as follows: e, v + κ u − W e , v − aB(u − W e ), [v] D u−W





For a function ψ with restrictions in

SJ

j =1




Γ

= 0,

By taking β = 1 in Lemma 3.5, we see 6

(3.11)

H 1 (Ωj ), define a norm




|||ψ|||2 = a∇ψ, ∇ψ + ψ, ψ + H −1 a[ψ], [ψ] Γ . 1

∀ v ∈ Mh , 0 < t ≤ T .




|||v|||2 ≤ D v, v + κ v, v − aB(v), [v] Γ ,

∀ v ∈ Mh .

(3.12)

2786

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

It follows from Lax–Milgram theorem that the project problem (3.11) has a unique solution. Let

e n, ξ n = un − W

e n. θn = Un − W

(3.13)

We estimate ξ n and θ n respectively in next lemmas. Lemma 3.6. {θ n }Nn=0 satisfies the following estimate

( n 2 L2 (Ω )

max kθ k 1≤n≤N

≤C

N X

∆t

n



k∂¯t ξ k

n 2 L2 (Ω )

n 2 L2 (Ω )

+ kξ k



) + ∆t + H 2

5

,

(3.14)

n=1

provided

∆t ≤

H2 4a1

.

(3.15)

Proof. First of all, we see that u satisfies




∂ un ∂¯t un , V + D un , V + bun , V − a , [V ] ∂ nΓ

!
= f n + ρn, V ,

∀ v ∈ Mh ,

(3.16)

Γ

where the time truncation term ρ n satisfies M X

n 2 ∆t n L2 (Ω )

kρ k

≤ ∆t

2

T

Z 0

n =1

kutt k2L2 (Ω ) dt .

(3.17)

e that It follows from (3.16) and the definition of W



e n, V + D W e n , V + bW e n , V − aB(W e n ), [V ] ∂¯t W Γ    
∂ un = f n + (κ + b)ξ n + ρ n − ∂¯t ξ n , V − a B(un ) − , [V ] , ∂ nΓ Γ

∀ V ∈ Mh .

(3.18)

(2.8) and (3.18) give

!  ∂ un n − B(u ) , [V ] ∂ nΓ Γ
+ aB(ξ n − ξ n−1 ), [V ])Γ − (aB(un − un−1 ), [V ] Γ , ∀ V ∈ Mh . (3.19)




∂¯t θ n , V + D θ n , V ) − (aB(θ n−1 ), [V ] Γ = ∂¯t ξ n − ρ n − (κ + b)ξ n − bθ n , V − a



Taking V = θ n in (3.19), we have

(∂¯t θ n , θ n ) + D(θ n , θ n ) − (aB(θ n ), [θ n ])Γ = −(aB(θ n − θ n−1 ), [θ n ])Γ + (∂¯t ξ n − ρ n − (κ + b)ξ n − bθ n , θ n )   n   ∂u n n − a − B(u ) , [θ ] + (aB(ξ n − ξ n−1 ), [θ n ])Γ − (aB(un − un−1 ), [θ n ])Γ . ∂ nΓ Γ

(3.20)

Noting that

(∂¯t θ n , θ n ) =

1 2∆t n



kθ n k2L2 (Ω ) − kθ n−1 k2L2 (Ω ) + kθ n − θ n−1 k2L2 (Ω )



and substituting this equality into (3.20) and then summing it over n, we get

kθ n k2L2 (Ω ) +

n  X
 kθ k − θ k−1 k2L2 (Ω ) + 2∆t k D(θ k , θ k ) − (aB(θ k ), [θ k ])Γ k=1

= kθ 0 k2L2 (Ω ) + 2

n X

 ∆t k −(aB(θ k − θ k−1 ), [θ k ])Γ + (∂t ξ k − ρ k − (κ + b)ξ k − bθ n , θ k )

k=1

   ∂ uk − a − B(uk ) , [θ k ] + (aB(ξ k − ξ k−1 ), [θ k ])Γ − (aB(uk − uk−1 ), [θ k ])Γ . ∂ nΓ Γ  

(3.21)

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2787

For the third term on the left-hand side of (3.21), from Lemma 3.5, we see 2 D(θ k , θ k ) − (aB(θ k ), [θ k ])Γ




≥

J 1X

13

3 j =1

12H

(a∇θ k , ∇θ k )Ωj +

(a[θ k ], [θ k ])Γ − C kθ k k2L2 (Ω ) .

We estimate the terms on the right-hand side of (3.21) one by one. It follows from Lemma 3.4 that

n n X X p √ k k k−1 k 2 ∆t (aB(θ − θ ), [θ ])Γ ≤ 2 ∆t k 4a1 H −3/2 kθ k − θ k−1 kL2 (Ω ) k a [θ k ]kL2 (Γ ) k=1 k =1 n n X X ∆t k (a[θ k ], [θ k ])Γ , ≤ 4a1 H −2 ∆t kθ k − θ k−1 k2L2 (Ω ) + H −1 k=1 n X

(3.22)

k=1

n X p √ ∆t k |(aB(ξ k − ξ k−1 ), [θ k ])Γ | ≤ ∆t 2a1 H −3/2 ∆t k k∂t ξ k kL2 (Ω ) k a [θ k ]kL2 (Γ )

k=1

k=1

≤ C ∆t H 2

−2

n X

∆t k k∂¯t ξ k k2L2 (Ω ) +

k=1

1

H −1

72

n X

∆t k (a[θ k ], [θ k ])Γ

(3.23)

k=1

and n X

∆t k |(aB(uk − uk−1 ), [θ k ])Γ | ≤ C ∆tH −1/2

k=1

n X

√ ∆t k k∂¯t uk kH 1 (Ω ) k a [θ k ]kL2 (Γ )

k=1 n

≤ C ∆t 2

X

∆t k k∂¯t uk k2H 1 (Ω ) +

k=1

1 72

n X

H −1

∆t k (a[θ k ], [θ k ])Γ .

(3.24)

k=1

From (3.6), we see that

k

∂u 2 k k

− B ( u )

2 ≤ CH ku kW 3,∞ (G)

∂n Γ L (Γ ) such that n X k=1

  k   ∂u k k ∆t a − B(u ) , [θ ] ∂n k

Γ

Γ

n X √ ≤ CH 2 ∆t k kuk kW 3,∞ (G) k a [θ k ]kL2 (Γ ) k=1

≤ CH 5

n X

∆t k kuk k2W 3,∞ (G) +

k=1

1 72

H −1

n X

∆t k (a[θ k ], [θ k ])Γ .

k=1

Substituting these estimates into (3.21), we have n 2 L2 (Ω )

kθ k

+

n X

kθ − θ k

k−1 2 L2 (Ω )

k

k=1

J X 1 + ∆t (a∇θ k , ∇θ k )Ωj + (a[θ k ], [θ k ])Γ

1

!

k

3

24H

j =1

n

≤ kθ 0 k2L2 (Ω ) + 4a1 ∆tH −2

X

kθ k − θ k−1 k2L2 (Ω )

k=1

( +C

n X

∆t

k



k 2 L2 (Ω )

kθ k

+ k∂¯t ξ k k2L2 (Ω ) + kξ k k2L2 (Ω )



) + ∆t + H 2

5

.

(3.25)

k=1

If 0 < 4a1 ∆tH −2 ≤ 1, then

( n 2 L2 (Ω )

kθ k

≤C

n X

∆t

k



k 2 L2 (Ω )

kθ k

+ k∂¯t ξ k

k 2 L2 (Ω )

k 2 L2 (Ω )

+ kξ k



) + ∆t + H 2

5

.

(3.26)

k=1

Applying discrete Bellman’s inequality to (3.26) leads to (3.14). The proof of Lemma 3.6 ends.



Next, we estimate the bound of ξ . n

Lemma 3.7. There holds the a priori error estimate:

e ]kL2 (Γ ) ≤ C kηkL∞ (Ω ) . k[W − W

(3.27)

2788

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

Proof. We prove

e ]||| ≤ CH − 12 kηkL∞ (Ω ) |||[W − W

(3.28)

such that (3.27) holds. Since

e, V + κ W − W e , V − aB(W − W e ), [V ] D W −W








Γ


= D W − u, V + κ W − u, V − aB(W − u), [V ] Γ
= − aB(W − u), [V ] Γ , ∀ V ∈ Mh .

(3.29)

From Lemma 3.3, we have 2

e ||| ≤ D(W − W e, W − W e ) + κ(W − W e, W − W e ) − (aB(W − W e ), [W − W e ])Γ α|||W − W e ])Γ . = −(aB(W − u), [W − W

(3.30)

It follows from (3.1) that

(aB(u − W ), [W − W e ])Γ ≤ CH −1 ku − W kL∞ (G) k[W − W e ]kL2 (Γ ) . Substituting (3.31) into (3.30) leads to (3.28).

(3.31)



Lemma 3.8. There hold the a priori error estimates:

n o kξ kL2 (Ω ) ≤ C h2 kukH 2 (Ω ) + kηkL∞ (Ω )

(3.32)

o n
kξt kL2 (Ω ) ≤ C h2 kukH 2 (Ω ) + kut kH 2 (Ω ) + kηt kL∞ (Ω ) .

(3.33)

and

Proof. Firstly, we prove

|||ξ ||| ≤ ChkukH 2 (Ω ) .

(3.34)

From the definition (3.11), we have that for each v ∈ Mh 1 6

e − v|||2 + kW e − vk22 e − v, W e − v) + κ(W e − v, W e − v) − (aB(W e − v), [W e − v])Γ |||W ≤ D(W L (Ω ) e − v) + κ(u − v, W e − v) − (aB(u − v), [W e − v])Γ = D(u − v, W  1/2  1/2 e − v|||2 + kW e − vk22 ≤ C k∇(u − v)k2L2 (Ω ) + ku − vk2L2 (Ω ) + H kB(u − v)k2L2 (Γ ) |||W L (Ω )

such that

o 1 ku − vkH 1 (Ω ) + H 2 kB(u − v)kL2 (Γ ) v∈Mh n o ≤ C inf ku − vkH 1 (Ω ) + H −1 ku − vkL2 (Ω ) ≤ ChkukH 2 (Ω ) .

e − v||| ≤ C inf inf |||W

v∈Mh

n

v∈Mh

Hence, we get

|||ξ ||| ≤ inf

v∈Mh

n

o e − v||| + |||u − v||| ≤ ChkukH 2 (Ω ) . |||W

This implies (3.34). Secondly, we consider L2 -norm error estimate. Introduce an auxiliary function ζ ∈ H 1 (Ω ) such that

− ∇ · (a∇ζ ) + κζ = ξ ,

in Ω ;

∂ζ = 0. ∂ nΩ

It is well known that ζ ∈ H 2 (Ω ) and satisfies

kζ kH 2 (Ω ) ≤ C kξ kL2 (Ω ) .

(3.35)

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2789

By using ζ , we have

  ∂ζ , [ξ ] − (aB(ξ ), [ζ ])Γ kξ k2L2 (Ω ) = D(ζ , ξ ) + κ(ζ , ξ ) − a ∂ nΓ Γ   ∂ζ = D(ζ − I h ζ , ξ ) + κ(ζ − I h ζ , ξ ) − (aB(ξ ), [ζ − I h ζ ])Γ − a , [ξ ] , ∂ nΓ Γ

(3.36)

where I h is the interpolating operator from H 1 (Ω ) on to Mh . We bound the terms on the right-hand side of (3.36). It is clear that D(ζ − I h ζ , ξ ) + κ(ζ − I h ζ , ξ ) − (aB(ξ ), [ζ − I h ζ ])Γ ≤ Chkζ kH 2 (Ω ) |||ξ ||| ≤ Ch2 kξ kL2 (Ω ) kukH 2 (Ω ) .

(3.37)

From (3.7) and (3.27), we see that

  a ∂ζ , [ξ ] ∂n Γ

Γ


≤ C kζ kH 2 (Ω ) k[W e − W ]kL2 (Γ ) + k[W − u]kL2 (Γ ) ≤ C kξ kL2 (Ω ) k[η]kL∞ (Γ ) .

(3.38)

From estimates above, we get the a priori error estimate (3.32). Similarly, we can get (3.33). The proof of Lemma 3.8 is complete.  Now we can prove Theorem 2.1. Noting that

kun − U n kL2 (Ω ) ≤ kξ n kL2 (Ω ) + kθ n kL2 (Ω ) and applying Lemmas 3.6 and 3.8, we derive (2.13). 3.3. Proof of Theorem 2.2

e ∈ Mh of the To obtain an optimal error estimate of the algorithm (IDDP), we introduce another elliptic projection W solution u as follows: e , v) + κ(u − W e , v) − (aB(u − W e ), [v])Γ − ([u − W e ], aB(v))Γ = 0, D(u − W

∀ v ∈ Mh .

(3.39)

Denote error functions again by

e, ξ =u−W

e n. θn = Un − W

(3.40)

We only need to estimate ξ and θ n , respectively. We firstly estimate θ n . Lemma 3.9. There holds the a priori error estimate: n 2 L2 (Ω )

max kθ k 1≤n≤N

≤C

( N X

∆t

n

n 2 L2 (Ω )

kξ k

+ k∂t ξ k

n 2 L2 (Ω )

)
k 2 4 2 5 ¯ + H ∆t k[∂t ξ ]kL2 (Γ ) + h + ∆t + H , k

(3.41)

n =1

provided that

∆t ≤

H2 4a1

r ! 1−

2 3

.

Proof. It follows from (3.39) that

e n , v) + D(W e n , v) + (bW e n , v) − (aB(W e n ), [v])Γ − (aB(v), [W e n ])Γ (∂¯t W     ∂ un , [v] , ∀ v ∈ Mh . = (f n + (κ + b)ξ n + ρ n − ∂¯t ξ n , v) − a B(un ) − ∂ nΓ Γ

(3.42)

(2.8) and (3.42) give

(∂¯t θ n , v) + D(θ n , v) + κ(θ n , v) − (aB(θ n ), [v])Γ − (aB(v), [θ n ])Γ = (∂¯t ξ n − ρ n − (κ + b)ξ n + (κ − b)θ n , v) − (aB(θ n − θ n−1 ), [v])Γ   n   ∂u − (aB(v), [θ n − θ n−1 ])Γ − a − B(un ) , [v] + (aB(ξ n − ξ n−1 ), [v])Γ ∂ nΓ Γ + (aB(v), [ξ n − ξ n−1 ])Γ − (aB(un − un−1 ), [v])Γ ,

∀ v ∈ Mh .

(3.43)

2790

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

Taking v = θ n − θ n−1 = ∂¯t θ n ∆t n in (3.43), we have

(∂¯t θ n , ∂¯t θ n )∆t n + D(θ n , θ n − θ n−1 ) + κ(θ n , θ n − θ n−1 ) − (aB(θ n ), [θ n − θ n−1 ])Γ − (aB(θ n − θ n−1 ), [θ n ])Γ = (∂¯t ξ n − ρ n − (κ + b)ξ n + (κ − b)θ n , θ n − θ n−1 ) − 2(aB(θ n − θ n−1 ), [θ n − θ n−1 ])Γ     n ∂u n n n−1 − B(u ) , [θ − θ ] + (aB(ξ n − ξ n−1 ), [θ n − θ n−1 ])Γ − a ∂ nΓ Γ + (aB(θ n − θ n−1 ), [ξ n − ξ n−1 ])Γ − (aB(un − un−1 ), [θ n − θ n−1 ])Γ .

(3.44)

Noting that D(θ n , θ n − θ n−1 ) =

 1 D(θ n , θ n ) − D(θ n−1 , θ n−1 ) + D(θ n − θ n−1 , θ n − θ n−1 ) 2

and

(aB(θ n ), [θ n − θ n−1 ])Γ + (aB(θ n − θ n−1 ), [θ n ])Γ = (aB(θ n ), [θ n ])Γ − (aB(θ n−1 ), [θ n−1 ])Γ + (aB(θ n − θ n−1 ), [θ n − θ n−1 ])Γ , we have 1n (∂¯t θ n , ∂¯t θ n )∆t n + D(θ n , θ n ) + κ(θ n , θ n ) − 2(aB(θ n ), [θ n ])Γ 2  o + (∆t n )2 D(∂¯t θ n , ∂¯t θ n ) + κ(∂¯t θ n , ∂¯t θ n ) − (aB(∂¯t θ n ), [∂¯t θ n ])Γ

=

 1 D(θ n−1 , θ n−1 ) + κ(θ n−1 , θ n−1 ) − 2(aB(θ n−1 ), [θ n−1 ])Γ 2  3 + ∆t n (∂¯t ξ n − ρ n − (κ + b)ξ n + (κ − b)θ n , ∂¯t θ n ) − ∆t n (aB(∂¯t θ n ), [∂¯t θ n ])Γ 2   n   ∂u − a − B(un ) , [∂¯t θ n ] + ∆t n (aB(∂¯t ξ n ), [∂¯t θ n ])Γ ∂ nΓ Γ  + ∆t n (aB(∂¯t θ n ), [∂¯t ξ n ])Γ − ∆t n (aB(∂¯t un ), [∂¯t θ n ])Γ .

(3.45)

Summing (3.45) over n, we get n X 1 (∂¯t θ k , ∂¯t θ k )∆t k + D(θ n , θ n ) + κ(θ n , θ n ) − 2(aB(θ n ), [θ n ])Γ

2

k=1 n

+



 X (∆t k )2 D(∂¯t θ k , ∂¯t θ k ) + κ(∂¯t θ k , ∂¯t θ k ) − (aB(∂¯t θ k ), [∂¯t θ k ])Γ k=1

 1 = D(θ 0 , θ 0 ) + κ(θ 0 , θ 0 ) − 2(aB(θ 0 ), [θ 0 ])Γ 2 +

n X

 3∆t k ∆t k (∂¯t ξ k − ρ k − (κ + b)ξ k + (κ − b)θ k , ∂¯t θ k ) − (aB(∂¯t θ k ), [∂¯t θ k ])Γ 2

k=1

  ∂u − a − B(uk ) , [∂¯t θ k ] + ∆t k (aB(∂¯t ξ k ), [∂¯t θ k ])Γ ∂ nΓ Γ  k k k k + ∆t (aB(∂¯t θ ), [∂¯t ξ ])Γ − ∆t (aB(∂¯t uk ), [∂¯t θ k ])Γ .  

k

It follows from Lemma 3.5 for β =

(θ n , θ n ) +

89 144

5 6

(3.46)

that

H −1 (a[θ n ], [θ n ])Γ ≤ D(θ n , θ n ) + κ(θ n , θ n ) − (aB(θ n ), [θ n ])Γ .

(3.47)

We estimate the terms on the right-hand side of (3.46) one by one. From Lemma 3.1, we have the following estimates: 3∆t k 2

√ √ |(aB(∂¯t θ k ), [∂¯t θ k ])Γ | ≤ 3∆t k a1 H 3/2 k∂¯t θ k kL2 (Ω ) k a ∂¯t θ k kL2 (Ω ) ≤

11 −1 81 H (a[∂¯t θ k ], [∂¯t θ k ])Γ + (∆t k )2 a1 H −2 k∂¯t θ k k2L2 (Ω ) , 36 11

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2791

 1 ∂ uk ∂ uk − B(uk ) , [∂¯t θ k ] ≤ H −1 (a[∂¯t θ k ], [∂¯t θ k ])Γ + CH k − B(uk )k2L2 (Ω ) , a ∂ nΓ 1728 ∂ n Γ Γ 

 

1

∆t k |(aB(∂¯t ξ k ), [∂¯t θ k ])Γ | ≤

1728

H −1 (a[∂¯t θ k ], [∂¯t θ k ])Γ + C (∆t k )2 H −2 k∂¯t ξ k k2L2 (Ω ) ,

J o n X (a∇ ∂¯t θ k , ∇ ∂¯t θ k )Ωj + C (∆t k )2 H −1 k[∂¯t ξ k ]k2L2 (Γ ) ∆t k (aB(∂¯t θ k ), [∂¯t ξ k ])Γ ≤  H −1 k[∂¯t θ k ]k2L2 (Γ ) + j =1

and

∆t k (aB(∂¯t uk ), [∂¯t θ k ])Γ ≤

1 1728

H −1 (a[∂¯t θ k ], [∂¯t θ k ])Γ + C (∆t k )2 k∂¯t uk k2W 1,∞ (Ω ) .

If 8a1 ∆tH −2 ≤ 1, then

kθ n k2L2 (Ω ) +

n X

2

∆t k |||∇θ k ||| ≤ C

k=1

n nX

k  X 2 ∆t i |||θ i ||| + kξ k k2L2 (Ω ) ∆t k kθ k k2L2 (Ω ) + i =1

k=1

+ H ∆t k[∂¯t ξ ]k k

k

2 L2 (Γ )

 o + k∂t ξ k k2L2 (Ω ) + h4 + ∆t 2 + H 5 .

Applying the discrete Bellman’s inequality to (3.48), we get (3.41).

(3.48)



Next, we bound ξ k by the following lemmas.

e be defined by (3.39). For each p ≥ 2, there holds the a priori error estimate: Lemma 3.10. Let W n o 1 e ]kL2 (Γ ) ≤ C H − 12 kηkL2 (Ω ) + H 1− p k∇ηkLp (Ω ) . k[W − W

(3.49)

Proof. We prove

n h i o 2 e ||| ≤ C H −1 kηkL2 (Ω ) + hk∇ηkL2 (Ω ) + H 1− p k∇ηkLp (Ω ) |||W − W

(3.50)

such that (3.49) holds. Since

e , v) + κ(W − W e , v) − (aB(W − W e ), [v])Γ − ([W − W e ], aB(v))Γ D(W − W = D(W − u, v) + κ(W − u, v) − (aB(W − u), [v])Γ − ([W − u], aB(v))Γ = −(aB(W − u), [v])Γ − ([W − u], aB(v))Γ , ∀ v ∈ Mh , we have

e |||2 ≤ D(W − W e, W − W e ) + κ(W − W e, W − W e ) − 2(aB(W − W e ), [W − W e ])Γ α|||W − W e ])Γ − ([W − u], aB(W − W e ))Γ . = −(aB(W − u), [W − W

(3.51)

We estimate two terms on the right-hand side of (3.51). It follows from (2.5) that

e ])Γ | ≤ CH − 32 ku − W kL2 (Ω ) k[W − W e ]kL2 (Γ ) . |(aB(W − u), [W − W

(3.52)

On the other hand, it follows from (3.4) that

n o k[W − u]kL2 (Γ ) ≤ C H −1/2 kW − ukL2 (Ω ) + H 1−1/p k∇(W − u)k2Lp (Ω ) . The proof of Lemma 3.10 is complete.



Lemma 3.11. There hold a priori error estimates:

n o kξ kL2 (Ω ) ≤ C h2 kukH 2 (Ω ) + kηkL2 (Ω ) + H 5/4 k∇ηkL4 (Ω )

(3.53)

n o kξt kL2 (Ω ) ≤ C h2 kut kH 2 (Ω ) + kηt kL2 (Ω ) + H 5/4 k∇ηt kL4 (Ω ) .

(3.54)

and

2792

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

Proof. Firstly, we prove

|||ξ ||| ≤ ChkukH 2 (Ω ) .

(3.55)

From the definition of (3.39), we have that for each v ∈ Mh C0 |||ξ |||2 + kξ k2L2 (Ω ) ≤ D(ξ , ξ ) + (ξ , ξ ) + 2(aB(ξ ), [ξ ])Γ

n

= inf

v∈Mhj

o

D(ξ , u − v) + (ξ , u − v) + (aB(ξ ), [u − v])Γ + ([ξ ], aB(u − v))Γ .

(3.56)

Noting that

(aB(ξ ), [u − I h u])Γ ≤ Ch 32 H − 21 kuk

n 3 H2

(Γ )

1

H − 2 k[ξ ]kL2 (Γ ) + k∇ξ kL2 (Ω )

o

(3.57)

and 1

 Z (aB(u − I h u), [ξ ])Γ = H −1 (a[u − I h u], [ξ ])Γ + a

φ(x)(u − I h u)x (x, y)dx, [ξ ] 0

≤



C0 4H

k[ξ ]k2L2 (Γ ) + C h3 H −1 kuk2 3

H 2 (Γ )

 Γ



+ h2 kuk2H 2 (Ω ) ,

(3.58)

we obtain

|||ξ ||| + kξ kL2 (Ω ) ≤ ChkukH 2 (Ω ) .

(3.59)

This implies (3.55). Secondly, we consider L2 -norm error estimate. Introduce an auxiliary function ζ ∈ H 1 (Ω ) such that

− ∇ · (a∇ζ ) + ζ = ξ ,

∂ζ = 0, ∂ nΩ

in Ω ;

on ∂ Ω .

(3.60)

It is well known that ζ ∈ H 2 (Ω ) and satisfies

kζ kH 2 (Ω ) ≤ C kξ kL2 (Ω ) .

(3.61)

By using ζ , we have

kξ k

2 L2 (Ω )

∂ζ , [ξ ] = D(ζ , ξ ) + (ζ , ξ ) + a ∂ nΓ 

 Γ

+ (aB(ξ ), [ζ ])Γ

= D(ζ − I h ζ , ξ ) + (ζ − I h ζ , ξ ) + (aB(ξ ), [ζ − I h ζ ])Γ + (aB(ζ − I h ζ ), [ξ ])Γ     ∂ζ − B(ζ ) , [ξ ] . + a ∂ nΓ Γ

(3.62)

We bound the terms on the right-hand side of (3.62). It is clear that D(ζ − I h ζ , ξ ) + (ζ − I h ζ , ξ ) ≤ Chkζ kH 2 (Ω ) kξ kH 1 (Ω )

(3.63)

and

(aB(ξ ), [ζ − I h ζ ])Γ + (aB(ζ − I h ζ ), [ξ ])Γ  1  1  1 ≤ Ch h 2 H − 2 kζ k 1 + kζ kH 2 (Ω ) H − 2 k[ξ ]kL2 (Γ ) + k∇ξ kL2 (Ω ) H 2 (Ω )

≤ Ch kζ kH 2 (Ω ) kukH 2 (Ω ) . 2

(3.64)

On the other hand, we have

    a ∂ζ − B(ζ ) , [ξ ] ∂n Γ

Γ

    ≤ a ∂ζ − B(ζ ) , [u − W ] ∂n Γ

Γ

    + a ∂ζ − B(ζ ) , [W − W e] ∂n Γ

It follows from (3.49) that

    a ∂ζ − B(ζ ) , [W − W e] ∂n Γ

≤ CH 21 kζ kH 2 (Ω ) k[W − W e ]kL2 (Γ ) Γ  ≤ C kζ kH 2 (Ω ) kηkL2 (Ω ) + H 5/4 k∇ηkL4 (Ω ) .

Γ



(3.65)

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

2793

We get the a priori error estimate

o n kξ kL2 (Ω ) ≤ C h2 kukH 2 (Ω ) + kηkL2 (Ω ) + H 5/4 k∇ηkL4 (Ω ) . This is (3.53). Since for any v ∈ Mh D((u − W )t , v) + ((u − W )t , v) + (aB((u − W )t ), [v])Γ + ([(u − W )t ], aB(v))Γ = 0 and

e )t , v) + ((u − W e )t , v) + (aB((u − W e )t ), [v])Γ + ([(u − W e )t ], aB(v))Γ = 0, D((u − W hence we can similarly get (3.54). The proof of Lemma 3.11 is complete.



Now we can prove Theorem 2.2. Proof of Theorem 2.2. Applying the results in Lemma 3.11, we have

n

4− 4p

n

4− 4p

max kun − U n k2L2 (Ω ) ≤ C H

1≤n≤N

M X

o
∆t k k∇ηk k2Lp (Ω ) + k∇ ∂¯t ηk k2Lp (Ω ) + h4 + ∆t 2 + H 5

k=1

≤C H

T

Z 0

o
k∇ηk2L2 (Ω ) + k∇ηt k2L2 (Ω ) dt + h4 + ∆t 2 + H 5 ,

which means

n

max kun − U n kL2 (Ω ) ≤ C H

2− 2p

1≤n≤N

o (k∇ηkL2 (0,T ;Lp (Ω )) + k∇ηt kL2 (0,T ;Lp (Ω )) ) + h2 + ∆t + H 5/2 .

Taking p = 8/3, we have

n

5

max kun − U n kL2 (Ω ) ≤ C H 4 (k∇ηkL2 (0,T ;L8/3 (Ω )) + k∇ηt kL2 (0,T ;L8/3 (Ω )) ) + h2 + ∆t + H 5/2

o

1≤n≤N

n o ≤ C k∇ηk2L2 (0,T ;L8/3 (Ω )) + k∇ηt k2L2 (0,T ;L8/3 (Ω )) + h2 + ∆t + H 5/2 . The proof of Theorem 2.2 is complete.



Acknowledgements We wish to express our thanks to the referees for their useful suggestions and comments. References [1] H. Blum, S. Lisky, R. Rannacher, A domain splitting algorithm for parabolic problems, Computing 49 (1) (1992) 11–23. [2] R.E. Ewing, R.D. Lazarov, J.E. Pasciak, P.S. Vassilevski, Domain decomposition type iterative techniques for parabolic problems on locally refined grids, SIAM J. Numer. Anal. 30 (6) (1993) 1537–1557. [3] X.C. Cai, Additive Schwarz algorithms for parabolic convection-diffusion equations, Numer. Math. 60 (1) (1991) 41–61. [4] X.C. Cai, Multiplicative Schwarz methods for parabolic problems, SIAM J. Sci. Comput. 15 (3) (1994) 587–603. [5] C.N. Dawson, T.F. Dupont, Explicit/implicit conservative Galerkin domain decomposition procedure to solve parabolic problem, Math. Comp. 58 (195) (1992) 21–34. [6] C.N. Dawson, T.F. Dupont, Explicit/implicit, conservative domain decomposition procedures for parabolic problems based on block-centered finite differences, SIAM J. Numer. Anal. 31 (4) (1994) 1045–1061. [7] C.N. Dawson, Q. Du, T.F. Dupont, A finite difference domain decomposition algorithm for numerical solution of the heat equation, Math. Comput. 57 (195) (1991) 63–71. [8] Q. Du, M. Mu, Z.N. Wu, Efficient parallel algorithms for parabolic problems, SIAM J. Numer. Anal. 39 (5) (2002) 1469–1487. [9] M. Dryja, X.M. Tu, A domain decomposition discretization of parabolic problems, Numer. Math. 107 (4) (2007) 625–640. [10] V. Girault, R. Glowinski, H. López, A domain decomposition and mixed method for a linear parabolic boundary value problem, IMA J. Numer. Anal. 24 (3) (2004) 491–520. [11] E. Giladi, H.B. Keller, Space–time domain decomposition for parabolic problems, Numer. Math. 93 (2) (2002) 279–313. [12] F.K. Hebeker, Yu.A. Kuznetsov, Unsteady convection and convection-diffusion problems via direct overlapping domain decomposition methods, Numer. Methods PDEs. 14 (3) (1998) 387–406. [13] G. Lube, F.C. Otto, H. Müller, A non-overlapping domain decomposition method for parabolic initial-boundary value problems, Computing 64 (1) (2000) 49–68. [14] G.A. Meurant, A domain decomposition method for parabolic problems, Appl. Numer. Math. 8 (4–5) (1991) 427–441. [15] T.P. Mathew, P.L. Polyakov, G. Russo, J. Wang, Domain decomposition operator splittings for the solution of parabolic equations, SIAM J. Sci. Comput. 19 (3) (1998) 912–932. [16] H.X. Rui, D.P. Yang, Schwarz type domain decomposition algorithms for parabolic equations and error estimates, Acta Math. Appl. Sinica 14 (3) (1998) 300–313. [17] A.H. Schatz, L.B. Wahlbin, Maximun norm estimates in the finite element method on plane polygonal domains, I, Math. Comp. 32 (1958) 53–109. [18] A.H. Schatz, L.B. Wahlbin, Maximun norm estimates in the finite element method on plane polygonal domains, II, Math. Comp. 33 (1959) 485–492. [19] X.C. Tai, A space decomposition method for parabolic equations, Numer. Methods PDEs. 14 (1) (1998) 27–46.

2794

D. Yang / Journal of Computational and Applied Mathematics 233 (2010) 2779–2794

[20] Y. Zhuang, X.H. Sun, Stabilized explicit–implicit domain decomposition methods for the numerical solution of parabolic equations, SIAM J. Sci. Comput. 24 (1) (2003) 335–358. [21] J.H. Yang, D.P. Yang, Additive Schwarz methods for parabolic problems, Appl. Math. Comput. 163 (1, 5) (2005) 17–28. [22] Yu.M. Laevskii, S.V. Gololobov, Explicit–implicit domain decomposition methods for solving parabolic equations, Siberian Math. J. 36 (3) (1995) 506–516. [23] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North-Holland Publ., Amsterdam, 1958. [24] J. Nitsche, L∞ -error analysis for finite elements, in: J.R. Whiteman (Ed.), The Mathematics of Finite Elements and Applications III, Academic Press, New York, 1959, pp. 153–186. [25] R.A. Adams, Sobolev Spaces, Academic Press, New York, 1955. [26] P.N. Vabishchevich, Parallel domain decomposition algorithms for parabolic problems, Mat. Model. 9 (5) (1997) 77–86. [27] M.Y. Kim, E.J. Park, J. Park, Mixed finite element domain decomposition for nonlinear parabolic problems, Comput. Math. Appl. 40 (8–9) (2000) 1061–1070.