Strong Superconvergence of Finite Element Methods for Linear Parabolic Problems

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

_{and Shuang Li}

1_{Department of Mathematics and Statistics, University of North Florida, Jacksonville, FL 32224, USA} 2_{Derivative 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,p_Ω

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

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 inW}1,p_{Ωand 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,p_{ΩandLpQrespectively. Finally, an application is presented in}

Section 6.

2. Preliminaries

We denoteWm,p_ΩandHm_{Ω,m≥0 and 1≤ p≤ ∞, the Sobolev spaces onΩassociated}

with the norms

·m,p ·Wm,p_Ω, ·_m·_Hm_Ω, ··_L

2Ω. 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_L

pJ;X

_T

0

φτp_X dτ,

φ_L

∞J;X ess sup

t∈J

φt_X.

Moreover, we denote

φ, ψ

Ωφψ dx,

φ, ψ_Q

Q

φψ dx dt

_T

0

φ, ψdt,

2.3

the inner product in L2Ω or L2Ω2 and L2Q or L2Q2, respectively. We also use

Sobolev spacesWp2l,lQwith norm

φ_W2l,l p Q

⎛ ⎝

Q

|α|2s≤2l |Dα

xDstφx, 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

inf

χ∈Sh

v−χ_0,phv−χ_1,p≤Chsv_s,p, ∀v∈Ws,pΩ∩H₀1Ω. 2.5

Lemma 2.2. For allχ∈Sh, we have

χ_1,p≤Ch−d/pχ_1,1, 2≤p <∞, p

p

p−1, 2.6

χ_0,∞≤Ch−d/pχ_0,p, 2≤p <∞. 2.7

Given a functionu∈Wk1,p_Ω∩H1

0Ω, we define its Ritz projectionRhu∈Shthat

satisfies

ARhu−u, χ

0, ∀χ∈Sh. 2.8

Then we get the following well-known estimate:

Moreover, by using the duality argument and2.9, it is true that forv∈Wl,p_Ω,

|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

AU, χft, χ, χ∈Sh, t∈J,

U0 U0 inΩ,

2.11

whereU0 ∈Shis defined by

AU0, 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≤

⎧ ⎨ ⎩

Chk2_ut0

k1, k >1,

Ch2_ut0

2, k1,

3.2

ξ0_0,p≤Chk3ut0_k1,p, 2≤p <∞, k >2. 3.3

Proof. From2.12and2.11, we have for allχ∈Sh

Rhut0, χ

f0, χ−AU0, χ

Ut0, χ, 3.4

By the definition ofξ,2.11,2.8,1.1, and the definition ofη, we obtain that for all

χ∈Sh,

ξt, χ

Aξ, χUt, χ

AU, χ−Rhut, χ

−ARhu, χ

f, χ−Rhut, χ

−Au, χ ut, χ

−Rhut, χ

−ηt, χ

.

3.5

Lett0, and note thatξt0 0, we obtain

Aξ0, χ−ηt0, χ, χ∈Sh. 3.6

Then by takingχξ0, it follows from2.10that

ξ02₁≤

⎧ ⎨ ⎩

Chk2_ut0

k1ξ01, k >1,

Ch2_ut0

2ξ0, 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≤Cφ_0,p, 3.9

wherepp/p−1.

Therefore, it follows from3.8,2.8,3.6,2.10,2.9, and3.9that forφ∈LpΩ,

ξ0, φAξ0,Φ

Aξ0, RhΦ

ηt0,Φ−RhΦ−ηt0,Φ

≤Chk2ut0k1,pΦ−RhΦ1,pChk3ut0k1,pΦ2,p

≤Chk3ut0k1,pΦ2,p

≤Chk3ut0k1,pφ0,p,

3.10

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

⎡

⎣utt_k1

t

0

uttt2_k1dτ

1/2⎤

⎦, t∈J. 3.11

Proof. By differentiating3.5in time, we have

ξtt, χ

Aξt, 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

≤Chk2utt_k1ξt₁Chk2

_t

0

uttt_k1ξt₁dτ

≤εξt2₁C

h2k4

utt2_k1

t

0

uttt2_k1dτ

t

0

ξt2₁dτ

.

3.14

Therefore, the proof is completed by eliminatingεξt2

1and 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

_Ω

In this Section, we derive the two order global superconvergence2≤p <∞and the almost

Theorem 4.1. Under the assumptions that ut ∈ Wk1,pΩ, utt ∈ Hk1Ω, and uttt ∈ L20, t;Hk1Ω, we have ford≤4 and 2≤p <∞,

ξ_1,p ≤Chk2, k >1. 4.1

Proof. We first introduce an auxiliary problem.

Forψ ∈ LpΩ, letψxbe an arbitrary component of∇ψ, and letΨ ∈W1,p

0 Ωbe the

solution of

Av,Ψ −v, ψx

, ∀v∈H₀1Ω. 4.2

The following priori estimate holds:

Ψ_1,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

≤Chk2ut_k1,pRhΨ_1,p

≤Chk2ut_k1,pΨ_1,p

≤Chk2ut_k1,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 ≤Cξt₁RhΨ_1,p

≤Chk2ψ_0,p.

Combining4.4,4.5, and4.6, we have

ξx_0,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,

ξ_1,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;

Hk1_{Ω. Then ford1 and 2,}

ξ_L∞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 Gh

z, ∂zGhz ∈ Sh be the finite element

approximations ofG∗_zand∂zG∗z, respectively. Then we know thatcf. 1,12

Gh_zGh_z

1,q≤C, q <2, 4.10

∂zGhz

2

∂zGhz_1,1≤Clog

1

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 Aξ, Gh_z

−ηt, Ghz

−ξt, Ghz

≤Chk2_ut

k1,pGhz_1,pξtGhz

≤Chk2Gh_z

1,p

Gh_z

,

which together with4.10yields

ξ_0,∞≤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/p_log1 h,

4.14

which implies that, forplarge enough andhsufficiently small,

∇ξ_0,∞≤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,

ξ_1,p ≤Ch2, 4.16

ξ_1,Q≤Ch2. 4.17

Moreover, ford1,2,

ξ_L∞J;W1,∞_Ω≤Ch2. 4.18

5. Superconvergence in

L

Q

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,p_{Ω, u}

t∈LpJ;Wk1,pΩ, utt ∈LpJ;Hk1Ω,and uttt∈LpJ;L20, t;Hk1Ω. Then, ford≤4 and 2≤p <∞, it holds that

Proof. First, we construct an adjoint problem of1.1.

LetW∈H1

0Ωsatisfy

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

W_W2,1

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, w_Q −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≤ ξ0_0,pW0_0,p

≤Chk3ut0_k1,pW_L∞J;L

pΩ

≤Chk3ut0_k1,pW_W1,1_J;L pΩ

≤Chk3ut0_k1,pW_W2,1 pQ

≤Chk3W_W2,1 pQ,

where

W_W1,1_J;L pΩ

_T

0

W_0,pWt_0,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

ut_k1,pW_2,pdtChk2

T

0

ut_k1,pRhW−W1,pdt

≤Chk3

_T

0

ut_k1,pW_2,pdt

≤Chk3

T

0

utp_k1,p dt

1/p

W_W2,1 pQ

≤Chk3W_W2,1 pQ.

5.9

Finally, by H ¨older’s inequalities, Sobolev embedding inequalities andLemma 3.2, we

have

T

0

ξt, RhW−Wdt≤

T

0

ξt_0,4RhW−W0,4/3dt

≤Ch

_T

0

ξt₁W_1,4/3dt

≤Ch

T

0

ξt₁W_2,pdt

≤Ch

T

0

ξtp₁ dt

1/p

W_W2,1 pQ

≤Chk3_W

W2,1 pQ.

5.10

We finally establish the almost two order global superconvergence in L∞Q. We define a functiongt∈H1

0Ω,and its finite element approximationght∈Shsatisfy that

v, gt

−Av, g0, v∈H1

0Ω, 5.11

gT δh inΩ, 5.12

χ, ght

−Aχ, gh

0, χ∈Sh, 5.13

ghT δh inΩ, 5.14

whereδhδ_hzxis the discrete Delta function which satisfies

δh, χ

χz, ∀χ∈Sh. 5.15

Then the following estimate holdscf. 2:

_gh

0,1,Qght0,1,Q

φ2D2_xg2

0,2,Q≤Clog

1

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_L

qJ;W2,qΩ≤Ch −4/q

log1

h

1/2

. 5.18

Proof. Using the H ¨older inequality, it is easy to see that

D2_xg

0,q,Q≤

Q

φ−4q/2−q dx dt

2−q/2q

φ2D_x2g_0,2,Q. 5.19

Note thatcf. 2

Q

φ−λ_{dx dt≤C}β4−λ

λ−4, λ >4, 5.20

the proof is then completed by the norm equivalence inH1

The following theorem gives the superconvergence ofξinL∞Q.

Theorem 5.3. Assume thatut0∈Wk1,p_Ωandu

t∈LpJ;Wk1,pΩ.Then, ford1,2,

ξ_0,∞,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

Aξ, gh

−ηt, gh

.

5.22

Then by integrating int, it follows from5.15and5.14that

ξz, T ξT, δh

ξT, ghT−ξ0, gh0ξ0, gh0

−

T

0

ηt, gh

dtξ0, 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

ut_k1,pg−gh_1,pdtChk3

_T

0

ut_k1,pg_2,pdt

≤Chk3

T

0

ut_k1,pg_2,pdt

≤Chk3−4/p

T

0

utp_k1,pdt

1/p

_g

LpJ;W2,pΩ

≤Chk3−4/p

log1

h

1/2

.

On the other hand, it follows from2.7, the Sobolev embedding inequality,3.3, and5.16

that

ξ0, gh0≤ ξ0_0,∞gh0_0,1

≤Ch−d/pξ0_0,pgh_L∞J;L1Ω

≤Ch−d/pξ0_0,pgh_W1,1_J;L

1Ω

≤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 spanxi₁xj₂: 0≤i, j≤k, k≥1. 6.2

We introduce the higher interpolation operator I₂k_h2, which satisfies the following

propertiescf. 7,8, fork≥1, 2≤p≤ ∞, and l0,1,

u−I₂k_h2u

l,p ≤Ch k3−l_u

k3,p, 6.3

I₂k_h2ik_hI₂k_h2, 6.4

Ik2

2h χ_l,p≤Cχl,p, χ∈Sh, 6.5

whereik_his the finite element interpolation operator.

In addition, we assume thatAu−Δuin1.1. By replacing the approximate solution

Uby its interpolationIk2

2h U, we can then establish the two order and the almost two order

global superconvergence ofu−Ik2

Theorem 6.1. Under the assumptions of Theorems4.1–4.4,5.1and5.3, we have foru∈Wk3,p_Ω∩ LpJ;Wk3,p_{Ωwithk >1 andu∈W}3,p_Ω∩LpJ;W3,p_Ωwithk1,

u−I₂k_h2U

1,p≤Ch

k2_{, 2≤p <∞, k >1,}

u−I₂k_h2U

1,Q≤Ch

k2_{, k >1,}

u−I₂k_h2U

L∞J;W1,∞_Ω≤Ch

k2−ε_{, k >1,}

u−I₂2_hU

1,p≤Ch

2_{, k1,}

u−I₂2_hU

1,Q≤Ch

2_{, k1,}

u−I₂2_hU

L∞J;W1,∞_Ω≤Ch

2_{, k1,}

u−I₂k_h2U

0,p,Q≤Ch

k3_{, 2≤p <∞, k >2,}

u−I₂k_h2U

0,∞,Q≤Ch

k3−ε_{, k >2,}

6.6

withε >2/pandplarge enough.

Proof. From6.4, we have

u−I₂k_h2Uu−I₂k_h2uI₂k_h2ik_hu−Rhu

I₂k_h2Rhu−U, 6.7

which together with the triangular inequality and6.5yields that

u−I₂k_h2U

l,p≤

u−I₂k_h2u

l,pC

ik_hu−Rhu

l,pRhu−Ul,p

. 6.8

Moreover, for 2≤p≤ ∞, the following estimates holdcf. 7,8:

ik_hu−Rhu

0,p≤Ch k3_u

k3,p

log1

h

p

, k >2,

ik_hu−Rhu

1,p≤Ch k2_u

k3,p, k >1,

i1_hu−Rhu

1,p≤Ch

2_u 3,p,

6.9 where p ⎧ ⎨ ⎩

0, when p <∞,

1, when p∞.

Hence the proof is completed by6.3and the estimates forξin Theorems4.1–4.4,5.1, and5.3.

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