# Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems

Authors: Kening Wang; Shuang Li
Journal: International Journal of Mathematics and Mathematical Sciences
 (2009)

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## Abstract

We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems onQ=Ω×(0,T], where Ω is a bounded domain in ℛd(d≤4) with piecewise smooth boundary. We establish the global two order superconvergence results for the error between the approximate solution and the Ritz projection of the exact solution of our model problem in W1,p(Ω) and Lp(Q) with 2≤p<∞ and the almost two order superconvergence in W1,∞(Ω) and L∞(Q). Results of the p=∞ case are also included in two space dimensions (d=1 or 2). By applying the interpolated postprocessing technique, similar results are also obtained on the error between the interpolation of the approximate solution and the exact solution.


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## Body

## 1. Introduction

Consider the followinginitial boundary value parabolic problem:(1.1)ut+Au=f,inQ=Ω×J,J=(0,T],u(x,t)=0,on∂Ω×J,u(x,0)=u0,inΩ,
where Ω is a bounded domain in ℛd(d≤4) with piecewise smooth boundary ∂Ω, and A is a second-order symmetric positive definite elliptic operator. Coefficients of A,f(x,t) and u0(x) together with their derivatives up to certain order are bounded in order to guarantee our analysis. Note that our assumptions on u do not have any restrictions, since it will be shown that approximate solutions considered below are uniformly close to the exact solution and thus only depend on the data of (1.1) in a neighborhood of u.Superconvergence of finite element methods for parabolic problems has been studied in many works. For example, Thomée [1], Chen and Huang [2] studied superconvergence of the gradient in L2 norm. In 1989, Thomée et al. [3] studied maximum-norm superconvergence of the gradient in piecewise linear finite element approximations of a parabolic problem. An analogous result was also obtained by Chen [4]. Moreover, Li and Wei [5] investigated global strong superconvergence of finite element schemes for a class of Sobolev equations in ℛd(d≥1), and two order superconvergence results are proved in W1,p(Ω) and Lp(Ω) for 2≤p<∞. In particular, Kwak et al. [6] studied superconvergence of a semi-discrete finite element scheme for parabolic problems in ℛ2, in which superconvergence results in W1,p(Ω) and Lp(Ω) are established for 2≤p≤∞.In this paper, we extend superconvergence results obtained in [6]. We derive the two order (2≤p<∞) and the almost two order (p=∞) global superconvergence estimates of U-Rhu in W1,p(Ω) and in Lp(Q), where U is the approximate solution, and Rhu is the Ritz projection of the exact solution of (1.1). In addition to the results in [6], we establish two order superconvergence estimates in Lp norm for piecewise cubic or higher elements. Moreover, results of the p=∞ case are also included in two space dimensions (d=1 or 2). As an application, by employing the interpolated finite element operators (cf. [7, 8]) to the approximate solution U in the rectangular mesh, we obtain the two order global superconvergence of the error between u and the interpolation of U. For a general domain Ω, we can also apply the optimal partition to most rectangular meshes to derive one and a half- order superconvergence.The rest of this paper is organized as follows. Section2 provides some preliminaries. Several useful lemmas are established in Section 3. In Sections 4 and 5, we derive the superconvergence in W1,p(Ω) and Lp(Q) respectively. Finally, an application is presented in Section 6.


## 2. Preliminaries

We denoteWm,p(Ω) and Hm(Ω), m≥0 and 1≤p≤∞,  the Sobolev spaces on Ω associated with the norms
(2.1)∥·∥m,p=∥·∥Wm,p(Ω),∥·∥m=∥·∥Hm(Ω),∥·∥=∥·∥L2(Ω).IfX is a normed space with the norm ∥·∥X and ϕ:J→X, then we define
(2.2)∥ϕ∥Lp(0,t;X)p=∫0t∥ϕ(τ)∥Xpdτ,∥ϕ∥Lp(J;X)p=∫0T∥ϕ(τ)∥Xpdτ,∥ϕ∥L∞(J;X)=esssupt∈J∥ϕ(t)∥X.Moreover, we denote(2.3)(ϕ,ψ)=∫Ωϕψdx,(ϕ,ψ)Q=∫Qϕψdxdt=∫0T(ϕ,ψ)dt,
the inner product in L2(Ω) (or L2(Ω)2) and L2(Q) (or L2(Q)2), respectively. We also use Sobolev spaces Wp2l,l(Q) with norm
(2.4)∥ϕ∥Wp2l,l(Q)=(∫Q∑|α|+2s≤2l|DxαDtsϕ(x,t)|pdxdt)1/p.We useC to denote a generic positive constant independent of h that can take different values at different occurrences.Let𝒯 be a family of quasiuniform triangulation of Ω, and let Sh⊂H01(Ω) be the kth (k≥1) degree finite element space satisfying the following properties (cf. [9, 10]).Lemma 2.1.
For allk≥1,1≤s≤k+1 and 1≤p≤∞, we have
(2.5)infχ∈Sh{∥v-χ∥0,p+h∥v-χ∥1,p}≤Chs∥v∥s,p,∀v∈Ws,p(Ω)∩H01(Ω).Lemma 2.2.
For allχ∈Sh, we have
(2.6)∥χ∥1,p′≤Ch-d/p∥χ∥1,1,2≤p<∞,p′=pp-1,(2.7)∥χ∥0,∞≤Ch-d/p∥χ∥0,p,2≤p<∞.Given a functionu∈Wk+1,p(Ω)∩H01(Ω), we define its Ritz projection Rhu∈Sh that satisfies
(2.8)A(Rhu-u,χ)=0,∀χ∈Sh.
Then we get the following well-known estimate:
(2.9)∥Rhu-u∥0,p+h∥Rhu-u∥1,p≤Chk+1∥u∥k+1,p,1<p<∞.Moreover, by using the duality argument and (2.9), it is true that for v∈Wl,p′(Ω),
(2.10)|(Rhu-u,v)|≤Chk+1+l∥u∥k+1,p∥v∥l,p′,0≤l≤k-1,
where 1<p<∞ and 1/p+1/p′=1.We now turn to the finite element scheme of (1.1).Find a mapU(t):J→Sh such that
(2.11)(Ut,χ)+A(U,χ)=(f(t),χ),χ∈Sh,t∈J,U(0)=U0inΩ,
where U0∈Sh is defined by
(2.12)A(U0,χ)=(f(0),χ)-(Rhut(0),χ),χ∈Sh,
and ut(0)=f(0)-Au0 is given by (1.1).


## 3. Auxiliary Lemmas

To investigate the superconvergence of finite element approaches for parabolic problems, here and throughout the paper, we decompose the error asU-u=(U-Rhu)+(Rhu-u)=ξ+η and estimate ξ in a superconvergent order.We start with the superconvergence of initial value errors.Lemma 3.1.
Letu and U be solutions of (1.1) and (2.11), respectively. Then the following estimates are true:
(3.1)Ut(0)=Rhut(0),(3.2)∥ξ(0)∥1≤{Chk+2∥ut(0)∥k+1,k>1,Ch2∥ut(0)∥2,k=1,(3.3)∥ξ(0)∥0,p≤Chk+3∥ut(0)∥k+1,p,2≤p<∞,k>2.Proof.
From (2.12) and (2.11), we have for all χ∈Sh(3.4)(Rhut(0),χ)=(f(0),χ)-A(U0,χ)=(Ut(0),χ),
and thus (3.1) holds.
By the definition ofξ, (2.11), (2.8), (1.1), and the definition of η, we obtain that for all χ∈Sh,
(3.5)(ξt,χ)+A(ξ,χ)=(Ut,χ)+A(U,χ)-(Rhut,χ)-A(Rhu,χ)=(f,χ)-(Rhut,χ)-A(u,χ)=(ut,χ)-(Rhut,χ)=-(ηt,χ).
Let t=0, and note that ξt(0)=0, we obtain
(3.6)A(ξ(0),χ)=-(ηt(0),χ),χ∈Sh.
Then by taking χ=ξ(0), it follows from (2.10) that
(3.7)∥ξ(0)∥12≤{Chk+2∥ut(0)∥k+1∥ξ(0)∥1,k>1,Ch2∥ut(0)∥2∥ξ(0)∥,k=1,
which implies (3.2).
Finally we turn to the proof of (3.3). To do so, we construct an auxiliary problem. Let Φ∈W01,p′(Ω) satisfy
(3.8)A(v,Φ)=(v,ϕ),v∈H01(Ω),
and hence by the regularity estimate, it holds that
(3.9)∥Φ∥2,p′≤C∥ϕ∥0,p′,
where p′=p/(p-1).
Therefore, it follows from (3.8), (2.8), (3.6), (2.10), (2.9), and (3.9) that for ϕ∈Lp′(Ω),
(3.10)(ξ(0),ϕ)=A(ξ(0),Φ)=A(ξ(0),RhΦ)=(ηt(0),Φ-RhΦ)-(ηt(0),Φ)≤Chk+2∥ut(0)∥k+1,p∥Φ-RhΦ∥1,p′+Chk+3∥ut(0)∥k+1,p∥Φ∥2,p′≤Chk+3∥ut(0)∥k+1,p∥Φ∥2,p′≤Chk+3∥ut(0)∥k+1,p∥ϕ∥0,p′,
which implies (3.3).The following lemma gives superconvergence estimates forξtt and ∇ξ.Lemma 3.2.
Letu and U be solutions of (1.1) and (2.11), respectively. Then for k>1,
(3.11)(∫0t∥ξtt∥2dτ)1/2+∥ξt∥1≤Chk+2[∥utt∥k+1+(∫0t∥uttt∥k+12dτ)1/2],t∈J.Proof.
By differentiating (3.5) in time, we have
(3.12)(ξtt,χ)+A(ξt,χ)=-(ηtt,χ),χ∈Sh.
Choosing χ=ξtt, (3.12) becomes
(3.13)∥ξtt∥2+12ddtA(ξt,ξt)=-(ηtt,ξtt).
Integrating both sides of (3.13) with respect to t and applying the integration by parts argument, we obtain that from (3.1) and (2.10)
(3.14)∫0t∥ξtt∥2dτ+12A(ξt,ξt)=-∫0t(ηtt,ξtt)dτ=-(ηtt,ξt)+∫0t(ηttt,ξt)dτ≤Chk+2∥utt∥k+1∥ξt∥1+Chk+2∫0t∥uttt∥k+1∥ξt∥1dτ≤ε∥ξt∥12+C[h2k+4(∥utt∥k+12+∫0t∥uttt∥k+12dτ)+∫0t∥ξt∥12dτ].
Therefore, the proof is completed by eliminating ε∥ξt∥12 and applying the Gronwall inequality.Furthermore, the result below fork=1 can then be obtained by replacing (2.10) by (2.9) in the proof of Lemma 3.2.Lemma 3.3.
It holds that(3.15)(∫0t∥ξtt∥2dτ)1/2+∥ξt∥1≤Ch2[∥utt∥2+(∫0t∥uttt∥22dτ)1/2],t∈J.


## 4. Superconvergence inW1,p(Ω)

In this Section, we derive the two order global superconvergence (2≤p<∞) and the almost two order global superconvergence (p=∞) estimates on ξ in W1,p(Ω).Theorem 4.1.
Under the assumptions thatut∈Wk+1,p(Ω),utt∈Hk+1(Ω), and uttt∈L2(0,t;Hk+1(Ω)), we have for d≤4 and 2≤p<∞,
(4.1)∥ξ∥1,p≤Chk+2,k>1.Proof.
We first introduce an auxiliary problem.
Forψ∈Lp′(Ω), let ψx be an arbitrary component of ∇ψ, and let Ψ∈W01,p′(Ω) be the solution of
(4.2)A(v,Ψ)=-(v,ψx),∀v∈H01(Ω).
The following priori estimate holds:
(4.3)∥Ψ∥1,p′≤C∥ψ∥0,p′.
Letv=ξ in (4.2), it follows from the integration by parts argument, (2.8) and (3.5) that
(4.4)(ξx,ψ)=A(ξ,Ψ)=A(ξ,RhΨ)=-(ηt+ξt,RhΨ).
From (2.10), the stability of Rh and (4.3), we obtain
(4.5)-(ηt,RhΨ)≤Chk+2∥ut∥k+1,p∥RhΨ∥1,p′≤Chk+2∥ut∥k+1,p∥Ψ∥1,p′≤Chk+2∥ut∥k+1,p∥ψ∥0,p′.
On the other hand, fors=s′=2 and d=1 or 2, or s=2d/(d-2), s′=s/(s-1), and d=3 or 4, Sobolev embedding inequalities (cf. [11]), Lemma 3.2, the stability of Rh, and (4.3) imply that
(4.6)-(ξt,RhΨ)≤C∥ξt∥0,s∥RhΨ∥0,s′≤C∥ξt∥1∥RhΨ∥1,p′≤Chk+2∥ψ∥0,p′.
Combining (4.4), (4.5), and (4.6), we have
(4.7)∥ξx∥0,p=supψ∈Lp′(Ω)(ξx,ψ)∥ψ∥0,p′≤Chk+2.
Therefore, (4.1) follows from summing up all components ξx of ∇ξ.The following theorem can then be obtained immediately by using Lemma3.2 and Theorem 4.1.Theorem 4.2.
Under the assumptions of Lemma3.2 and Theorem 4.1, we have that for d≤4,
(4.8)∥ξ∥1,Q≤Chk+2,k>1.We now turn to the case ofp=∞.Theorem 4.3.
Assume thatut∈L∞(J;Wk+1,p(Ω)),utt∈L∞(J;Hk+1(Ω)), and uttt∈L2(J;Hk+1(Ω)). Then for d=1 and 2,
(4.9)∥ξ∥L∞(J;W1,∞(Ω))≤Chk+2-ε,k>1,
where p is large enough and ε>d/p.Proof.
We first define the Green functions associated with the bilinear formA(·,·).
LetGz*∈H01(Ω) be the pre-Green function, and let ∂zGz* be the directional derivative of Gz* along some direction with respect to z. Let Gzh,∂zGzh∈Sh be the finite element approximations of Gz* and ∂zGz*, respectively. Then we know that (cf. [1, 12])
(4.10)∥Gzh∥+∥Gzh∥1,q≤C,q<2,(4.11)∥∂zGzh∥2+∥∂zGzh∥1,1≤Clog1h.
Now by definitions of Green functions, (3.5), Hölder's inequalities, (2.10), and (3.11), it is true that for all (z,t)∈Q,
(4.12)ξ(z,t)=A(ξ,Gzh)=-(ηt,Gzh)-(ξt,Gzh)≤Chk+2∥ut∥k+1,p∥Gzh∥1,p′+∥ξt∥∥Gzh∥≤Chk+2(∥Gzh∥1,p′+∥Gzh∥),
which together with (4.10) yields
(4.13)∥ξ∥0,∞≤Chk+2.
Similarly, by the inverse property (2.6) and (4.11), we have
(4.14)∂zξ(z,t)=A(ξ,∂zGzh)≤Chk+2(∥∂zGzh∥1,p′+∥∂zGzh∥)≤Chk+2(h-d/p∥∂zGzh∥1,1+∥∂zGzh∥)≤Chk+2-d/plog1h,
which implies that, for p large enough and h sufficiently small,
(4.15)∥∇ξ∥0,∞≤Chk+2-ε,ε>dp.
Inequality (4.9) then follows from (4.13) and (4.15).By the similar arguments used in the proof of Theorems4.1–4.3 and Lemma 3.3, we obtain the following results.Theorem 4.4.
Under the assumptions of Theorems4.1–4.3 with k=1, we have, for d≤4,
(4.16)∥ξ∥1,p≤Ch2,(4.17)∥ξ∥1,Q≤Ch2.
Moreover, for d=1,2,
(4.18)∥ξ∥L∞(J;W1,∞(Ω))≤Ch2.


## 5. Superconvergence inLp(Q)

In this section, we establish the strong superconvergence forξ in Lp(Q) with 2≤p≤∞.We start with the following two order global superconvergence for2≤p<∞.Theorem 5.1.
Assume thatut(0)∈Wk+1,p(Ω),ut∈Lp(J;Wk+1,p(Ω)),utt∈Lp(J;Hk+1(Ω)), and uttt∈Lp(J;L2(0,t;Hk+1(Ω))). Then, for d≤4 and 2≤p<∞, it holds that
(5.1)∥ξ∥0,p,Q≤Chk+3,k>2.Proof.
First, we construct an adjoint problem of (1.1).
LetW∈H01(Ω) satisfy
(5.2)(v,Wt)-A(v,W)=(v,w),v∈H01(Ω),(5.3)W(T)=0inΩ.
By takings=T-t, (5.2) and (5.3) can then be reduced to the weak form of (1.1) and thus we have the regularity estimate (cf. [2])
(5.4)∥W∥Wp′2,1(Q)≤C∥w∥0,p′,Q.
Letv=ξ in (5.2), it follows from (2.8) and (3.12) that
(5.5)(ξ,w)=(ξ,Wt)-A(ξ,W)=ddt(ξ,W)-[(ξt,W)+A(ξ,W)]=ddt(ξ,W)-[(ξt,W)+A(ξ,RhW)]=ddt(ξ,W)-[(ξt,W)-(ηt,RhW)-(ξt,RhW)]=ddt(ξ,W)+(ηt,RhW)+(ξt,RhW-W).
After integrating int, we have
(5.6)(ξ,w)Q=-(ξ(0),W(0))+∫0T(ηt,RhW)dt+∫0T(ξt,RhW-W)dt.
Here the fact that W(T)=0 was used.
Now we estimate the right-hand side of (5.6) term by term.
First of all, by Hölder's inequalities, (3.3), and the Sobolev embedding inequality, we obtain that
(5.7)-(ξ(0),W(0))≤∥ξ(0)∥0,p∥W(0)∥0,p′≤Chk+3∥ut(0)∥k+1,p∥W∥L∞(J;Lp′(Ω))≤Chk+3∥ut(0)∥k+1,p∥W∥W1,1(J;Lp′(Ω))≤Chk+3∥ut(0)∥k+1,p∥W∥Wp′2,1(Q)≤Chk+3∥W∥Wp′2,1(Q),
where
(5.8)∥W∥W1,1(J;Lp′(Ω))=∫0T(∥W∥0,p′+∥Wt∥0,p′)dt.
Secondly, it follows from (2.9), (2.10) and Hölder's inequalities that
(5.9)∫0T(ηt,RhW)dt=∫0T(ηt,W)dt+∫0T(ηt,RhW-W)dt≤Chk+3∫0T∥ut∥k+1,p∥W∥2,p′dt+Chk+2∫0T∥ut∥k+1,p∥RhW-W∥1,p′dt≤Chk+3∫0T∥ut∥k+1,p∥W∥2,p′dt≤Chk+3(∫0T∥ut∥k+1,ppdt)1/p∥W∥Wp′2,1(Q)≤Chk+3∥W∥Wp′2,1(Q).
Finally, by Hölder's inequalities, Sobolev embedding inequalities and Lemma3.2, we have
(5.10)∫0T(ξt,RhW-W)dt≤∫0T∥ξt∥0,4∥RhW-W∥0,4/3dt≤Ch∫0T∥ξt∥1∥W∥1,4/3dt≤Ch∫0T∥ξt∥1∥W∥2,p′dt≤Ch(∫0T∥ξt∥1pdt)1/p∥W∥Wp′2,1(Q)≤Chk+3∥W∥Wp′2,1(Q).
Therefore, (5.1) holds by combining all estimates together with (5.4) and (5.6).We finally establish the almost two order global superconvergence inL∞(Q). We define a function g(t)∈H01(Ω), and its finite element approximation gh(t)∈Sh satisfy that
(5.11)(v,gt)-A(v,g)=0,v∈H01(Ω),(5.12)g(T)=δhinΩ,(5.13)(χ,ght)-A(χ,gh)=0,χ∈Sh,(5.14)gh(T)=δhinΩ,
where δh=δhz(x) is the discrete Delta function which satisfies
(5.15)(δh,χ)=χ(z),∀χ∈Sh.
Then the following estimate holds (cf. [2]):
(5.16)∥gh∥0,1,Q+∥ght∥0,1,Q+∥ϕ2Dx2g∥0,2,Q2≤Clog1h,
where ϕ is the weight function defined by
(5.17)ϕ(t)=(|x-z|2+t+β2)1/2,β=γh,γ≫1.Furthermore, we have the following estimate.Lemma 5.2.
For1<q<2 and its conjugate index q′, we have
(5.18)∥g∥Lq(J;W2,q(Ω))≤Ch-4/q′(log1h)1/2.Proof.
Using the Hölder inequality, it is easy to see that(5.19)∥Dx2g∥0,q,Q≤(∫Qϕ-4q/(2-q)dxdt)(2-q)/2q∥ϕ2Dx2g∥0,2,Q.
Note that (cf. [2])
(5.20)∫Qϕ-λdxdt≤Cβ4-λλ-4,λ>4,
the proof is then completed by the norm equivalence in H01(Ω)∩W2,q(Ω).The following theorem gives the superconvergence ofξ in L∞(Q).Theorem 5.3.
Assume thatut(0)∈Wk+1,p(Ω) and ut∈Lp(J;Wk+1,p(Ω)). Then, for d=1,2,
(5.21)∥ξ∥0,∞,Q≤Chk+3-ε,k>2,
with ε>4/p and p large enough.Proof.
(5.13) and (3.5) yield that
(5.22)Dt(ξ,gh)=(ξt,gh)+(ξ,ght)=(ξt,gh)+A(ξ,gh)=-(ηt,gh).
Then by integrating in t, it follows from (5.15) and (5.14) that
(5.23)ξ(z,T)=(ξ(T),δh)=(ξ(T),gh(T))-(ξ(0),gh(0))+(ξ(0),gh(0))=-∫0T(ηt,gh)dt+(ξ(0),gh(0)).
On the one hand, (2.10) and Lemma 5.2 imply that, for p>2,
(5.24)-∫0T(ηt,gh)dt=∫0T(ηt,g-gh)dt-∫0T(ηt,g)dt≤Chk+2∫0T∥ut∥k+1,p∥g-gh∥1,p′dt+Chk+3∫0T∥ut∥k+1,p∥g∥2,p′dt≤Chk+3∫0T∥ut∥k+1,p∥g∥2,p′dt≤Chk+3-4/p(∫0T∥ut∥k+1,ppdt)1/p∥g∥Lp′(J;W2,p′(Ω))≤Chk+3-4/p(log1h)1/2.
On the other hand, it follows from (2.7), the Sobolev embedding inequality, (3.3), and (5.16) that
(5.25)(ξ(0),gh(0))≤∥ξ(0)∥0,∞∥gh(0)∥0,1≤Ch-d/p∥ξ(0)∥0,p∥gh∥L∞(J;L1(Ω))≤Ch-d/p∥ξ(0)∥0,p∥gh∥W1,1(J;L1(Ω))≤Chk+3-d/plog1h.
Therefore, (5.21) follows from (5.23), (5.24), and (5.25).


## 6. An Application

In this section, we apply the interpolation postprocessing technique to improve the accuracy of the approximate solutionU. Let 𝒯h be a quasi-uniform rectangular partition of Ω⊂ℛ2, and let Sh be the space of continuous piecewise polynomial:
(6.1)Sh={v∈H01(Ω):v∈Qk(T),T∈𝒯h},
where
(6.2)Qk=span{x1ix2j:0≤i,j≤k},k≥1.We introduce the higher interpolation operatorI2hk+2, which satisfies the following properties (cf. [7, 8]), for k≥1,2≤p≤∞,andl=0,1,
(6.3)∥u-I2hk+2u∥l,p≤Chk+3-l∥u∥k+3,p,(6.4)I2hk+2ihk=I2hk+2,(6.5)∥I2hk+2χ∥l,p≤C∥χ∥l,p,χ∈Sh,
where ihk is the finite element interpolation operator.In addition, we assume thatAu=-Δu in (1.1). By replacing the approximate solution U by its interpolation I2hk+2U, we can then establish the two order and the almost two order global superconvergence of u-I2hk+2U in W1,p(Ω) and Lp(Q) for 2≤p≤∞.Theorem 6.1.
Under the assumptions of Theorems4.1–4.4, 5.1 and 5.3, we have for u∈Wk+3,p(Ω)∩Lp(J;Wk+3,p(Ω)) with k>1 and u∈W3,p(Ω)∩Lp(J;W3,p(Ω)) with k=1,
(6.6)∥u-I2hk+2U∥1,p≤Chk+2,2≤p<∞,k>1,∥u-I2hk+2U∥1,Q≤Chk+2,k>1,∥u-I2hk+2U∥L∞(J;W1,∞(Ω))≤Chk+2-ε,k>1,∥u-I2h2U∥1,p≤Ch2,k=1,∥u-I2h2U∥1,Q≤Ch2,k=1,∥u-I2h2U∥L∞(J;W1,∞(Ω))≤Ch2,k=1,∥u-I2hk+2U∥0,p,Q≤Chk+3,2≤p<∞,k>2,∥u-I2hk+2U∥0,∞,Q≤Chk+3-ε,k>2,
with ε>2/p and p large enough.Proof.
From (6.4), we have
(6.7)u-I2hk+2U=u-I2hk+2u+I2hk+2(ihku-Rhu)+I2hk+2(Rhu-U),
which together with the triangular inequality and (6.5) yields that
(6.8)∥u-I2hk+2U∥l,p≤∥u-I2hk+2u∥l,p+C(∥ihku-Rhu∥l,p+∥Rhu-U∥l,p).
Moreover, for2≤p≤∞, the following estimates hold (cf. [7, 8]):
(6.9)∥ihku-Rhu∥0,p≤Chk+3∥u∥k+3,p(log1h)p¯,k>2,∥ihku-Rhu∥1,p≤Chk+2∥u∥k+3,p,k>1,∥ih1u-Rhu∥1,p≤Ch2∥u∥3,p,
where
(6.10)p¯={0,whenp<∞,1,whenp=∞.
Hence the proof is completed by (6.3) and the estimates for ξ in Theorems 4.1–4.4, 5.1, and 5.3.



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*Source: 345196-2009-08-16.xml*