International Journal of Mathematics and Mathematical Sciences Volume 2009, Article ID 345196,16pages
doi:10.1155/2009/345196
Research Article
Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems
Kening Wang
1and Shuang Li
21Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA
2Derivative Valuation Center, Ernst & Young LLP., New York, NY 10036, USA
Correspondence should be addressed to Kening Wang,[email protected] Received 30 March 2009; Revised 16 June 2009; Accepted 5 July 2009
Recommended by Thomas Witelski
We study the strong superconvergence of a semidiscrete finite element scheme for linear parabolic problems onQ Ω×0, T, whereΩis a bounded domain inRd d≤4with 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 inW1,pΩ andLpQwith 2≤p < ∞and the almost two order superconvergence inW1,∞ΩandL∞Q.
Results of thep∞case are also included in two space dimensionsd1 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.
Copyrightq2009 K. Wang and S. Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1. Introduction
Consider the following initial boundary value parabolic problem:
utAuf, inQ Ω×J, J 0, T, ux, t 0, on ∂Ω×J,
ux,0 u0, in Ω,
1.1
whereΩis a bounded domain inRd d ≤ 4with piecewise smooth boundary∂Ω, andA is a second-order symmetric positive definite elliptic operator. Coefficients ofA, fx, tand u0xtogether with their derivatives up to certain order are bounded in order to guarantee our analysis. Note that our assumptions on udo 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 of1.1in a neighborhood ofu.
Superconvergence of finite element methods for parabolic problems has been studied in many works. For example, Thom´ee 1, Chen and Huang 2studied superconvergence of the gradient inL2norm. In 1989, Thom´ee et al. 3studied 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 5investigated global strong superconvergence of finite element schemes for a class of Sobolev equations in Rd d ≥ 1, and two order superconvergence results are proved inW1,pΩand LpΩ for 2 ≤ p < ∞. In particular, Kwak et al. 6studied superconvergence of a semi-discrete finite element scheme for parabolic problems inR2, in which superconvergence results inW1,pΩ andLpΩare established for 2≤p≤ ∞.
In this paper, we extend superconvergence results obtained in 6. We derive the two order2 ≤ p < ∞and the almost two order p ∞ global superconvergence estimates ofU−RhuinW1,pΩand in LpQ, whereUis the approximate solution, andRhuis the Ritz projection of the exact solution of 1.1. In addition to the results in 6, we establish two order superconvergence estimates inLp 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 operatorscf. 7,8 to the approximate solution U in the rectangular mesh, we obtain the two order global superconvergence of the error betweenuand the interpolation ofU. For a general domainΩ, 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.Section 2provides some preliminaries.
Several useful lemmas are established in Section 3. In Sections 4 and 5, we derive the superconvergence inW1,pΩandLpQrespectively. Finally, an application is presented in Section 6.
2. Preliminaries
We denoteWm,pΩandHmΩ,m≥0 and 1≤ p≤ ∞, the Sobolev spaces onΩassociated with the norms
·m,p ·Wm,pΩ, ·m·HmΩ, ··L2Ω. 2.1
IfXis a normed space with the norm · Xandφ:J → X, then we define
φp
Lp0,t;X t
0
φτp
X dτ, φp
LpJ;X T
0
φτp
X dτ, φ
L∞J;X ess sup
t∈J
φt
X.
2.2
Moreover, we denote
φ, ψ
Ωφψ dx, φ, ψ
Q
Q
φψ dx dt T
0
φ, ψ dt,
2.3
the inner product in L2Ω or L2Ω2 and L2Q or L2Q2, respectively. We also use Sobolev spacesWp2l,lQwith norm
φ
W2l,lp Q
⎛
⎝
Q
|α|2s≤2l
|DαxDstφx, t|pdxdt
⎞
⎠
1/p
. 2.4
We useCto denote a generic positive constant independent ofhthat can take different values at different occurrences.
LetTbe a family of quasiuniform triangulation ofΩ,and letSh ⊂H01Ωbe thekth k≥1degree finite element space satisfying the following propertiescf. 9,10.
Lemma 2.1. For allk≥1,1≤s≤k1 and 1≤p≤ ∞, we have
χ∈Sinfh
v−χ0,phv−χ1,p
≤Chsvs,p, ∀v∈Ws,pΩ∩H01Ω. 2.5
Lemma 2.2. For allχ∈Sh, we have
χ1,p≤Ch−d/pχ1,1, 2≤p <∞, p p
p−1, 2.6
χ0,∞≤Ch−d/pχ0,p, 2≤p <∞. 2.7
Given a functionu∈Wk1,pΩ∩H01Ω, we define its Ritz projectionRhu∈Shthat satisfies
A
Rhu−u, χ
0, ∀χ∈Sh. 2.8
Then we get the following well-known estimate:
Rhu−u0,phRhu−u1,p≤Chk1uk1,p, 1< p <∞. 2.9
Moreover, by using the duality argument and2.9, it is true that forv∈Wl,pΩ,
|Rhu−u, v| ≤Chk1luk1,pvl,p, 0≤l≤k−1, 2.10
where 1< p <∞and 1/p1/p1.
We now turn to the finite element scheme of1.1.
Find a mapUt:J → Shsuch that Ut, χ
A U, χ
ft, χ
, χ∈Sh, t∈J,
U0 U0 inΩ, 2.11
whereU0 ∈Shis defined by
A U0, χ
f0, χ
−
Rhut0, χ
, χ∈Sh, 2.12
andut0 f0−Au0is given by1.1.
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−RhuRhu−u ξη and estimateξin a superconvergent order.
We start with the superconvergence of initial value errors.
Lemma 3.1. LetuandUbe solutions of 1.1and2.11, respectively. Then the following estimates are true:
Ut0 Rhut0, 3.1
ξ01≤
⎧⎨
⎩
Chk2ut0k1, k >1,
Ch2ut02, k1, 3.2
ξ00,p≤Chk3ut0k1,p, 2≤p <∞, k >2. 3.3
Proof. From2.12and2.11, we have for allχ∈Sh
Rhut0, χ
f0, χ
−A U0, χ
Ut0, χ
, 3.4
and thus3.1holds.
By the definition ofξ,2.11,2.8,1.1, and the definition ofη, we obtain that for all χ∈Sh,
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 .
3.5
Lett0, and note thatξt0 0, we obtain A
ξ0, χ −
ηt0, χ
, χ∈Sh. 3.6
Then by takingχξ0, it follows from2.10that
ξ021≤
⎧⎨
⎩
Chk2ut0k1ξ01, k >1,
Ch2ut02ξ0, k1, 3.7
which implies3.2.
Finally we turn to the proof of3.3. To do so, we construct an auxiliary problem. Let Φ∈W01,pΩsatisfy
Av,Φ v, φ
, v∈H01Ω, 3.8
and hence by the regularity estimate, it holds that
Φ2,p≤Cφ0,p, 3.9 wherepp/p−1.
Therefore, it follows from3.8,2.8,3.6,2.10,2.9, and3.9that forφ∈LpΩ, ξ0, φ
Aξ0,Φ Aξ0, RhΦ
ηt0,Φ−RhΦ
−
ηt0,Φ
≤Chk2ut0k1,pΦ−RhΦ1,pChk3ut0k1,pΦ2,p
≤Chk3ut0k1,pΦ2,p
≤Chk3ut0k1,pφ0,p,
3.10
which implies3.3.
The following lemma gives superconvergence estimates forξttand∇ξ.
Lemma 3.2. LetuandUbe solutions of1.1and2.11, respectively. Then fork >1, t
0
ξtt2dτ 1/2
ξt1≤Chk2
⎡
⎣uttk1 t
0
uttt2k1dτ 1/2⎤
⎦, t∈J. 3.11
Proof. By differentiating3.5in time, we have ξtt, χ
A ξt, χ
− ηtt, χ
, χ∈Sh. 3.12
Choosingχξtt,3.12becomes ξtt21
2 d
dtAξt, ξt − ηtt, ξtt
. 3.13
Integrating both sides of 3.13 with respect to t and applying the integration by parts argument, we obtain that from3.1and2.10
t
0
ξtt2dτ1
2Aξt, ξt − t
0
ηtt, ξtt dτ
− ηtt, ξt
t
0
ηttt, ξt dτ
≤Chk2uttk1ξt1Chk2 t
0
utttk1ξt1dτ
≤εξt21C
h2k4
utt2k1 t
0
uttt2k1dτ
t
0
ξt21dτ
.
3.14
Therefore, the proof is completed by eliminatingεξt21and applying the Gronwall inequality.
Furthermore, the result below fork 1 can then be obtained by replacing2.10by 2.9in the proof ofLemma 3.2.
Lemma 3.3. It holds that t
0
ξtt2dτ 1/2
ξt1≤Ch2
⎡
⎣utt2 t
0
uttt22dτ 1/2⎤
⎦, t∈J. 3.15
4. Superconvergence in W
1,pΩ
In this Section, we derive the two order global superconvergence2≤p <∞and the almost two order global superconvergencep∞estimates onξinW1,pΩ.
Theorem 4.1. Under the assumptions that ut ∈ Wk1,pΩ, utt ∈ Hk1Ω, and uttt ∈ L20, t;Hk1Ω, we have ford≤4 and 2≤p <∞,
ξ1,p ≤Chk2, k >1. 4.1
Proof. We first introduce an auxiliary problem.
Forψ ∈ LpΩ, letψxbe an arbitrary component of∇ψ, and letΨ ∈W01,pΩbe the solution of
Av,Ψ − v, ψx
, ∀v∈H01Ω. 4.2
The following priori estimate holds:
Ψ1,p≤Cψ
0,p. 4.3
Letvξin4.2, it follows from the integration by parts argument,2.8and3.5that ξx, ψ
Aξ,Ψ Aξ, RhΨ −
ηtξt, RhΨ .
4.4
From2.10, the stability ofRhand4.3, we obtain
−
ηt, RhΨ
≤Chk2utk1,pRhΨ1,p
≤Chk2utk1,pΨ1,p
≤Chk2utk1,pψ
0,p.
4.5
On the other hand, forss2 andd1 or 2, ors2d/d−2,ss/s−1,and d3 or 4, Sobolev embedding inequalitiescf. 11,Lemma 3.2, the stability ofRh,and4.3 imply that
−ξt, RhΨ≤Cξt0,sRhΨ0,s
≤Cξt1RhΨ1,p
≤Chk2ψ
0,p.
4.6
Combining4.4,4.5, and4.6, we have
ξx0,p sup
ψ∈LpΩ
ξx, ψ ψ
0,p
≤Chk2. 4.7
Therefore,4.1follows from summing up all componentsξxof∇ξ.
The following theorem can then be obtained immediately by usingLemma 3.2and Theorem 4.1.
Theorem 4.2. Under the assumptions ofLemma 3.2andTheorem 4.1, we have that ford≤4,
ξ1,Q≤Chk2, k >1. 4.8
We now turn to the case ofp∞.
Theorem 4.3. Assume that ut ∈ L∞J;Wk1,pΩ, utt ∈ L∞J;Hk1Ω, and uttt ∈ L2J;
Hk1Ω. Then ford1 and 2,
ξL∞J;W1,∞Ω≤Chk2−ε, k >1, 4.9
wherepis large enough andε > d/p.
Proof. We first define the Green functions associated with the bilinear formA·,·.
LetG∗z∈H01Ωbe the pre-Green function, and let∂zG∗zbe the directional derivative of G∗z along some direction with respect to z. Let Ghz, ∂zGhz ∈ Sh be the finite element approximations ofG∗zand∂zG∗z, respectively. Then we know thatcf. 1,12
GhzGhz
1,q≤C, q <2, 4.10
∂zGhz2∂zGhz
1,1≤Clog1
h. 4.11
Now by definitions of Green functions,3.5, H ¨older’s inequalities,2.10, and3.11, it is true that for allz, t∈Q,
ξz, t A ξ, Ghz −
ηt, Ghz
− ξt, Ghz
≤Chk2utk1,pGhz
1,pξtGhz
≤Chk2 Ghz
1,pGhz
,
4.12
which together with4.10yields
ξ0,∞≤Chk2. 4.13
Similarly, by the inverse property2.6and4.11, we have
∂zξz, t A
ξ, ∂zGhz
≤Chk2
∂zGhz1,p∂zGhz
≤Chk2
h−d/p∂zGhz1,1∂zGhz
≤Chk2−d/plog1 h,
4.14
which implies that, forplarge enough andhsufficiently small,
∇ξ0,∞≤Chk2−ε, ε > d
p. 4.15
Inequality4.9then follows from4.13and4.15.
By the similar arguments used in the proof of Theorems4.1–4.3andLemma 3.3, we obtain the following results.
Theorem 4.4. Under the assumptions of Theorems4.1–4.3withk1, we have, ford≤4,
ξ1,p ≤Ch2, 4.16 ξ1,Q≤Ch2. 4.17
Moreover, ford1,2,
ξL∞J;W1,∞Ω≤Ch2. 4.18
5. Superconvergence in L
pQ
In this section, we establish the strong superconvergence forξinLpQwith 2≤p≤ ∞.
We start with the following two order global superconvergence for 2≤p <∞.
Theorem 5.1. Assume thatut0∈Wk1,pΩ, ut∈LpJ;Wk1,pΩ, utt ∈LpJ;Hk1Ω,and uttt∈LpJ;L20, t;Hk1Ω. Then, ford≤4 and 2≤p <∞, it holds that
ξ0,p,Q≤Chk3, k >2. 5.1
Proof. First, we construct an adjoint problem of1.1.
LetW∈H01Ωsatisfy
v, Wt−Av, W v, w, v∈H01Ω, 5.2
WT 0 inΩ. 5.3
By takingsT −t,5.2and5.3can then be reduced to the weak form of1.1and thus we have the regularity estimatecf. 2
WW2,1
pQ≤Cw0,p,Q. 5.4
Letvξin5.2, it follows from2.8and3.12that ξ, w ξ, Wt−Aξ, W
d
dtξ, W− ξt, W Aξ, W d
dtξ, W− ξt, W Aξ, RhW d
dtξ, W−
ξt, W−
ηt, RhW
−ξt, RhW d
dtξ, W
ηt, RhW
ξt, RhW−W.
5.5
After integrating int, we have
ξ, wQ −ξ0, W0 T
0
ηt, RhW dt
T
0
ξt, RhW−Wdt. 5.6
Here the fact thatWT 0 was used.
Now we estimate the right-hand side of5.6term by term.
First of all, by H ¨older’s inequalities,3.3, and the Sobolev embedding inequality, we obtain that
−ξ0, W0≤ ξ00,pW00,p
≤Chk3ut0k1,pWL∞J;LpΩ
≤Chk3ut0k1,pWW1,1J;LpΩ
≤Chk3ut0k1,pWW2,1
pQ
≤Chk3WW2,1
pQ,
5.7
where
WW1,1J;LpΩ T
0
W0,pWt0,p
dt. 5.8
Secondly, it follows from2.9,2.10and H ¨older’s inequalities that T
0
ηt, RhW dt
T
0
ηt, W dt
T
0
ηt, RhW−W dt
≤Chk3 T
0
utk1,pW2,pdtChk2 T
0
utk1,pRhW−W1,pdt
≤Chk3 T
0
utk1,pW2,pdt
≤Chk3 T
0
utpk1,p dt 1/p
WW2,1
pQ
≤Chk3WW2,1
pQ.
5.9
Finally, by H ¨older’s inequalities, Sobolev embedding inequalities andLemma 3.2, we have
T
0
ξt, RhW−Wdt≤ T
0
ξt0,4RhW−W0,4/3dt
≤Ch T
0
ξt1W1,4/3dt
≤Ch T
0
ξt1W2,pdt
≤Ch T
0
ξtp1 dt 1/p
WW2,1
pQ
≤Chk3WW2,1
pQ.
5.10
Therefore,5.1holds by combining all estimates together with5.4and5.6.
We finally establish the almost two order global superconvergence in L∞Q. We define a functiongt∈H01Ω,and its finite element approximationght∈Shsatisfy that
v, gt
−A v, g
0, v∈H01Ω, 5.11
gT δh inΩ, 5.12
χ, ght
−A χ, gh
0, χ∈Sh, 5.13
ghT δh inΩ, 5.14
whereδhδhzxis the discrete Delta function which satisfies δh, χ
χz, ∀χ∈Sh. 5.15
Then the following estimate holdscf. 2:
gh
0,1,Qght
0,1,Qφ2D2xg2
0,2,Q≤Clog1
h, 5.16
whereφis the weight function defined by
φt
|x−z|2tβ21/2
, βγh, γ 1. 5.17
Furthermore, we have the following estimate.
Lemma 5.2. For 1< q <2 and its conjugate indexq, we have g
LqJ;W2,qΩ≤Ch−4/q
log1 h
1/2
. 5.18
Proof. Using the H ¨older inequality, it is easy to see that D2xg
0,q,Q≤
Q
φ−4q/2−q dx dt
2−q/2q
φ2Dx2g0,2,Q. 5.19
Note thatcf. 2
Q
φ−λdx dt≤Cβ4−λ
λ−4, λ >4, 5.20
the proof is then completed by the norm equivalence inH01Ω∩W2,qΩ.
The following theorem gives the superconvergence ofξinL∞Q.
Theorem 5.3. Assume thatut0∈Wk1,pΩandut∈LpJ;Wk1,pΩ.Then, ford1,2,
ξ0,∞,Q≤Chk3−ε, k >2, 5.21
withε >4/pandplarge enough.
Proof. 5.13and3.5yield that
Dt ξ, gh
ξt, gh
ξ, ght
ξt, gh A
ξ, gh −
ηt, gh
.
5.22
Then by integrating int, it follows from5.15and5.14that
ξz, T ξT, δh
ξT, ghT
−
ξ0, gh0
ξ0, gh0 −
T
0
ηt, gh
dt
ξ0, gh0 .
5.23
On the one hand,2.10andLemma 5.2imply that, forp >2,
− T
0
ηt, gh dt
T
0
ηt, g−gh dt−
T
0
ηt, g dt
≤Chk2 T
0
utk1,pg−gh
1,pdtChk3 T
0
utk1,pg
2,pdt
≤Chk3 T
0
utk1,pg
2,pdt
≤Chk3−4/p T
0
utpk1,pdt 1/p
g
LpJ;W2,pΩ
≤Chk3−4/p
log1 h
1/2 .
5.24
On the other hand, it follows from2.7, the Sobolev embedding inequality,3.3, and5.16 that
ξ0, gh0
≤ ξ00,∞gh0
0,1
≤Ch−d/pξ00,pgh
L∞J;L1Ω
≤Ch−d/pξ00,pgh
W1,1J;L1Ω
≤Chk3−d/plog1 h.
5.25
Therefore,5.21follows from5.23,5.24, and5.25.
6. An Application
In this section, we apply the interpolation postprocessing technique to improve the accuracy of the approximate solutionU. LetTh be a quasi-uniform rectangular partition ofΩ ⊂ R2, and letShbe the space of continuous piecewise polynomial:
Sh
v∈H01Ω:v∈QkT, T ∈ Th
, 6.1
where
Qk span
xi1xj2: 0≤i, j≤k
, k≥1. 6.2
We introduce the higher interpolation operator I2hk2, which satisfies the following propertiescf. 7,8, fork≥1, 2≤p≤ ∞, and l0,1,
u−I2hk2u
l,p ≤Chk3−luk3,p, 6.3
I2hk2ikhI2hk2, 6.4
I2hk2χ
l,p≤Cχ
l,p, χ∈Sh, 6.5
whereikhis the finite element interpolation operator.
In addition, we assume thatAu−Δuin1.1. By replacing the approximate solution Uby its interpolationI2hk2U, we can then establish the two order and the almost two order global superconvergence ofu−I2hk2UinW1,pΩandLpQfor 2≤p≤ ∞.
Theorem 6.1. Under the assumptions of Theorems4.1–4.4,5.1and5.3, we have foru∈Wk3,pΩ∩ LpJ;Wk3,pΩwithk >1 andu∈W3,pΩ∩LpJ;W3,pΩwithk1,
u−I2hk2U
1,p≤Chk2, 2≤p <∞, k >1, u−I2hk2U
1,Q≤Chk2, k >1, u−I2hk2U
L∞J;W1,∞Ω≤Chk2−ε, k >1, u−I2h2 U
1,p≤Ch2, k1, u−I2h2 U
1,Q≤Ch2, k1, u−I2h2 U
L∞J;W1,∞Ω≤Ch2, k1, u−I2hk2U
0,p,Q≤Chk3, 2≤p <∞, k >2, u−I2hk2U
0,∞,Q≤Chk3−ε, k >2,
6.6
withε >2/pandplarge enough.
Proof. From6.4, we have
u−I2hk2Uu−I2hk2uI2hk2
ikhu−Rhu
I2hk2Rhu−U, 6.7
which together with the triangular inequality and6.5yields that u−I2hk2U
l,p≤u−I2hk2u
l,pC
ikhu−Rhu
l,pRhu−Ul,p
. 6.8
Moreover, for 2≤p≤ ∞, the following estimates holdcf. 7,8:
ikhu−Rhu
0,p≤Chk3uk3,p
log1 h
p
, k >2, ikhu−Rhu
1,p≤Chk2uk3,p, k >1, i1hu−Rhu
1,p≤Ch2u3,p,
6.9
where
p
⎧⎨
⎩
0, when p <∞,
1, when p∞. 6.10
Hence the proof is completed by6.3and the estimates forξin Theorems4.1–4.4,5.1, and5.3.
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