# Analysis for semidiscrete control problem

CHAPTER 3. L(L)−A posteriori error estimates for POCP 61 The chapter is organized as follows. In Section 3.2, we discuss variational discretiza- tion approximation for the control problem (3.1)−(3.3) and derive a posteriori error estimates for the semidiscrete problem. Section 3.3 is devoted to the fully discrete approximations of the control problem (3.1)−(3.3) and related a posteriori error esti- mates are established. Numerical results are provided to illustrate the performance of the derived estimators in Section 3.4. Some concluding remarks are presented in the last section.

CHAPTER 3. L(L)−A posteriori error estimates for POCP 62 Definition 3.2.1 (Discrete elliptic operator). The discrete elliptic operator associated with the bilinear form a(·,·) and the finite element space Vh0 is the operator −Ah : H01(Ω)→Vh0 +Lhf such that for w∈H01(Ω) and t∈(0, T],

(−Ahw, vh) = a(w, vh), ∀vh ∈Vh0, where Lh be the L2-projection onto the finite element space Vh.

Therefore, we have the following pointwise form of (3.20) and (3.22)

−Ahyh = Lhf +uh −∂yh

∂t ,

−Ahph = yh− Lhyds +∂ph

∂t , respectively.

To begin with, we first establish some intermediate error estimates for the state and co-state variables in the L(0, T;L(Ω))-norm which will enable us to prove the main results of this section. This is accomplished by introducing elliptic reconstructions for the state and co-state variables. For this, we now introduce some auxiliary problems.

For ˆu∈Uad, let the pair (y(ˆu), p(ˆu))∈V ×V be the solutions of the following equations:

∂y(ˆu)

∂t , v

+a(y(ˆu), v) = (f+ ˆu, v) ∀v ∈H01(Ω), (3.26) y(ˆu)(·,0) = y0(x) x∈Ω, (3.27)

−∂p(ˆu)

∂t , v

+a(p(ˆu), v) = (y(ˆu)−yds, v) ∀v ∈H01(Ω), (3.28)

p(ˆu)(·, T) = 0 x∈Ω. (3.29)

Define the errors for the state and co-state variables as follows:

ˆ

ey :=yh−y(uh) and ˆep :=ph−p(uh). (3.30) From (3.20), (3.22), (3.26) and (3.28) with ˆu =uh, we obtain the following error equa- tions for v ∈H01(Ω):

∂eˆy

∂t , v

+a(ˆey, v) = −(Gy, v) + (∇yh,∇v), (3.31)

−∂eˆp

∂t , v

+a(ˆep, v) = −(Gp, v) + (∇ph,∇v) + (ˆey, v), (3.32) where Gy =f +uh−∂yh

∂t and Gp =yh −yds+ ∂ph

∂t .

CHAPTER 3. L(L)−A posteriori error estimates for POCP 63 For t ∈ (0, T], we now define the elliptic reconstructions for the state and co-state variables as follows: For given yh, ph, seek ˜y, p˜∈H01(Ω) such that

a(˜y, v) = (Gy, v) ∀v ∈H01(Ω), (3.33) a(˜p, v) = (Gp, v) + (˜y−yh, v) ∀v ∈H01(Ω). (3.34) With the help of elliptic reconstructions ˜yand ˜p as an intermediate object, we split the errors:

ˆ

ey = (˜y−y(uh))−(˜y−yh) =: ˆξy−ηˆy, ˆ

ep = (˜p−p(uh))−(˜p−ph) =: ˆξp−ηˆp. Using (3.31)−(3.34), for all v ∈H01(Ω), we obtain

∂ξˆy

∂t, v

+a( ˆξy, v) = ∂ηˆy

∂t , v

, (3.35)

−∂ξˆp

∂t , v

+a( ˆξp, v) = −∂ηˆp

∂t , v

+ ( ˆξy, v). (3.36) As a consequence of elliptic error estimate in Lemma 3.1.1, we obtain the following bounds for the elliptic reconstruction errors.

Lemma 3.2.1 (Elliptic reconstruction errors). Let (˜y,p)˜ ∈ H01(Ω) × H01(Ω) satisfy (3.33)−(3.34) and let Lemma 3.1.1 be valid. Then, for each t ∈ [0, T], the following estimates hold:

kˆηy(t)kL(Ω) ≤ C3,3(ln ¯h)2R∞,0(yh(t),Gy(t)), and

kˆηp(t)kL(Ω) ≤ C3,4(ln ¯h)2R∞,0(ph(t),Gp(t)) +kˆηy(t)kL(Ω), where the constants C3,3 and C3,4 depend on the constant C3,1.

We next turn our attention to derive the bounds for ˆξy and ˆξp.

Lemma 3.2.2 (Parabolic errors for the state and co-state variables). Let ξˆy and ξˆp

satisfy (3.35) and (3.36), respectively. Then, for any t ∈[0, T], the following estimates hold:

kξˆy(t)kL(Ω) ≤ kξˆy(0)kL(Ω)+C3,5(ln ¯h)2kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,T], and

kξˆp(t)kL(Ω) ≤ C3,6(ln ¯h)2kR∞,0(∂ph

∂t ,∂Gp

∂t )kL1[0,T]+kξˆy(t)kL(Ω),

where R∞,0 is the L-type residual estimator defined in (3.12). The constants C3,5 and C3,6 are positive and depend on the domain Ω.

CHAPTER 3. L(L)−A posteriori error estimates for POCP 64 Proof. We know that ˆξy satisfies (3.35). For any (x, t)∈Ω×(0, T], use of (3.16) leads to

ξˆy(x, t) = Z

F(x, t;w,0) ˆξy(w,0)dw+ Z t

0

Z

F(x, t;w, s)∂ηˆy

∂t (w, s)dw ds.

An application of the H¨older’s inequality yields

kξˆy(t)kL(Ω) ≤ kF(x, t;w,0)kL1(Ω)kξˆy(0)kL(Ω)+kF(x, t;w, s)kL1(Ω)k∂ηˆy

∂t kL1(0,t;L(Ω)). With an aid of (3.17), we have

kξˆy(t)kL(Ω) ≤ kξˆy(0)kL(Ω)+k∂ηˆy

∂t kL1(0,t;L(Ω)), which combine with Lemma 3.1.1 to obtain

kξˆy(t)kL(Ω) ≤ kξˆy(0)kL(Ω)+C3,5(ln ¯h)2kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,t],

where the constantC3,5 depends on Ω, and this proves the first inequality. The proof of the second inequality can be treated in a similar manner using the fact that ˆξp(T) = 0.

This completes the proof of the lemma.

Let (y, p, u) and (yh, ph, uh) be the solutions of (3.6)−(3.10) and (3.20)−(3.24), respectively. In order to derive a posteriori error bounds for the state and the co-state variables, we decompose the errors as follows:

y−yh = (y−y(uh)) + (y(uh)−yh) := ˆry−eˆy, and

p−ph = (p−p(uh)) + (p(uh)−ph) := ˆrp −eˆp,

where ˆry =y−y(uh), ˆrp =p−p(uh) and ˆey, ˆep are defined in (3.30). With the help of (3.6), (3.8), (3.26) and (3.28), we derive the following error equations for eacht ∈(0, T]:

∂ˆry

∂t , v

+a(ˆry, v) = (u−uh, v) ∀v ∈H01(Ω), (3.37) and

−∂rˆp

∂t , v

+a(ˆrp, v) = (ˆry, v) ∀v ∈H01(Ω). (3.38) In the following lemma, we derive the bounds for ˆry and ˆrp.

CHAPTER 3. L(L)−A posteriori error estimates for POCP 65 Lemma 3.2.3. Let (y, p, u) be the solution of (3.6)−(3.10), and let (y(uh), p(uh)) be the solution of (3.26)−(3.29) with uˆ=uh. Then the following estimates hold:

kˆrykL(0,T;L(Ω)) ≤ C3,7ku−uhkL(0,T;L2(Ω)), and

kˆrpkL(0,T;L(Ω)) ≤ C3,8ku−uhkL(0,T;L2(Ω)), where the constants C3,7 and C3,8 depend on the regularity constant CR. Proof. Note that, for any t∈[0, T]

kˆry(t)kL(Ω) ≤ kˆry(t)kC(Ω).

Using the embedding result H2(Ω),→ C(Ω) and Lemma 3.1.3 we obtain kˆrykL(0,T;L(Ω)) ≤ kˆrykL(0,T;H2(Ω)) ≤ C3,7ku−uhkL(0,T;L2(Ω)), where we have used the fact ˆry(0) = 0, and this proves the first inequality.

Similarly, the second inequality can easily be proved for ˆrp by using the fact ˆrp(T) = 0. This completes the rest of the proof.

The following lemma presents thea posteriorierror estimate for the control variable in the L(0, T;L2(Ω))-norm.

Lemma 3.2.4. Let(y, p, u)and(yh, ph, uh)be the solutions of(3.6)−(3.10)and(3.20)− (3.24), respectively. Assume that (uh +ph)|K ∈ H1(K) and there exists a positive constant C3,9, and wh ∈Uad such that

(uh+ph, wh−u)

≤ C3,9 X

K∈Th

hK|uh+ph|H1(K)ku−uhkL2(K). (3.39)

Then, we have

ku−uhkL(0,T;L2(Ω)) ≤ C3,10h max

t∈[0,T]

X

K∈Th

h2K|uh+ph|2H1(K)

1/2

+kph−p(uh)kL(0,T;L2(Ω))i , whereC3,10=

q3

2 max

1, C3,9 , and(y(ˆu), p(ˆu))is solution of the system(3.26)−(3.29) with uˆ=uh.

CHAPTER 3. L(L)−A posteriori error estimates for POCP 66 Proof. Note that

ku−uhk2L2(Ω) = (u−uh, u−uh)

= (u, u−uh)−(uh, u−uh).

An application of (3.10) and a simple calculation with an aid of (3.24) yields ku−uhk2L2(Ω) ≤ −(p, u−uh)−(uh, u−uh)

= −(uh+ph, u−wh)−(uh+ph, wh−uh) +(ph−p(uh), u−uh) + (p(uh)−p, u−uh)

≤ (ph−p(uh), u−uh) + (p(uh)−p, u−uh) +(uh+ph, wh−u)

=: E1+E2+E3. (3.40)

To boundE1, we use the Cauchy-Schwarz inequality and the Young’s inequality to have E1 ≤ kph−p(uh)kL2(Ω)ku−uhkL2(Ω)

≤ 3

4kph−p(uh)k2L2(Ω)+1

4ku−uhk2L2(Ω). (3.41) Settingv =p(uh)−pin (3.37), and integrate the resulting equation from 0 toT. Then, an integration by parts formula with ˆry(0) = ˆrp(T) = 0 leads to

Z T 0

ˆ ry,∂ˆrp

∂t

dt− Z T

0

a(ˆry,ˆrp)dt = Z T

0

(u−uh, p(uh)−p)dt. (3.42) Again, choosev =y(uh)−y in (3.38) and integrate with respect to time from 0 to T to obtain

Z T 0

∂ˆrp

∂t,rˆy dt−

Z T 0

a(ˆrp,ˆry)dt = Z T

0

(y−y(uh), y(uh)−y)dt. (3.43) Use of (3.42) and (3.43) leads to

E2 = (u−uh, p(uh)−p)

= (y−y(uh), y(uh)−y)

= −ky−y(uh)k2L2(Ω) ≤ 0. (3.44)

Finally to bound of E3, we use (3.39) and the Young’s inequality to have E3 ≤ C3,9 X

K∈Th

hK|uh+ph|H1(K)ku−uhkL2(K)

≤ 3

4C3,92 X

K∈Th

h2K|uh+ph|2H1(K)+1

4ku−uhk2L2(Ω). (3.45)

CHAPTER 3. L(L)−A posteriori error estimates for POCP 67 Altogether (3.40), (3.41), (3.44) and (3.45) yields

ku−uhk2L2(Ω) ≤ 3

2max{1, C3,92 }h X

K∈Th

h2K|uh+ph|2H1(K)+kph−p(uh)k2L2(Ω)

i .

Taking both side maximum over the time domain [0, T], we achieve the desired estimates.

This completes the proof.

By collecting Lemmas 3.2.1−3.2.4, we finally derive the main results for the state and co-state variables in the L(0, T;L(Ω))-norm.

Theorem 3.2.1 (L(L)−error estimates for the state and co-state variables). Let (y, p, u)and(yh, ph, uh)be the solutions of (3.6)−(3.10)and(3.20)−(3.24), respectively.

Let f ∈ L(0, T;L(Ω))∩W1,1(0, T;L(Ω)). Then the following a posteriori error estimates hold for each t∈(0, T]:

ku−uhkL(0,T;L2(Ω)) ≤ C3,11

h

t∈[0,T]max X

K∈Th

h2K|uh+ph|2H1(K)

1/2

+kξˆy(0)kL(Ω) +kˆηy(t)kL(Ω)+ (ln ¯h)2n

kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,T]

+R∞,0(ph(t),Gp(t)) +kR∞,0(∂ph

∂t ,∂Gp

∂t )kL1[0,T]oi ,

where C3,11 depends on the domain Ωand the constantC3,10 as defined in Lemma 3.2.4, ky−yhkL(0,T;L(Ω)) ≤ ky0−yh,0kL(Ω)+C3,12(ln ¯h)2h

R∞,0(yh(0),Gy(0)) +R∞,0(yh(t),Gy(t)) +kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,T]i +C3,7ku−uhkL(0,T;L2(Ω)),

kp−phkL(0,T;L(Ω)) ≤ ky0−yh,0kL(Ω)+C3,13(ln ¯h)2h

R∞,0(ph(t),Gp(t)) +kR∞,0(∂ph

∂t ,∂Gp

∂t )kL1[0,T]+R∞,0(yh(0),Gy(0)) +R∞,0(yh(t),Gy(t)) +kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,t]i +C3,8ku−uhkL(0,T;L2(Ω)),

where the constants C3,12 and C3,13 depend on the domain Ω, and the constants C3,7, C3,8 are defined in Lemma 3.2.3.

Proof. The first inequality follows from Lemmas 3.2.1, 3.2.2 and 3.2.4. To prove the second inequality, we decompose the error in the state variable as

y−yh = (y−y(uh)) + (y(uh)−y) + (˜˜ y−yh) = ˆry−( ˆξy−ηˆy).

CHAPTER 3. L(L)−A posteriori error estimates for POCP 68 For any t∈(0, T], we have

k(y−yh)(t)kL(Ω) ≤ kˆry(t)kL(Ω)+kξˆy(t)kL(Ω)+kˆηy(t)kL(Ω). An application of Lemma 3.2.1 yields

kˆηy(t)kL(Ω) ≤ C3,3(ln ¯h)2R∞,0(yh(t),Gy(t)).

By using Lemma 3.2.2, it now follows that

kξˆy(t)kL(Ω) ≤ ky0−yh,0kL(Ω)+C3,3(ln ¯h)2R∞,0(yh(0),Gy(0)) +C3,5(ln ¯h)2kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,t].

Altogether these estimates and Lemma 3.2.3 leads to the desired result, where C3,12 = max{C3,3, C3,5}.

Similarly for the co-state variable, we use the triangle inequality to write k(p−ph)(t)kL(Ω) ≤ kˆrp(t)kL(Ω)+kξˆp(t)kL(Ω)+kˆηp(t)kL(Ω).

Again, by using Lemmas 3.2.1−3.2.3 and a similar argument as above, we conclude that

k(p−ph)(t)kL(Ω) ≤ ky0−yh,0kL(Ω)+ (ln ¯h)2h

C3,3R∞,0(yh(0),Gy(0)) +C3,3R∞,0(yh(t),Gy(t)) +C3,4R∞,0(ph(t),Gp(t)) +C3,5kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,t]+C3,6kR∞,0(∂ph

∂t ,∂Gp

∂t )kL1[0,t]i +C3,8ku−uhkL(0,T;L2(Ω)).

Setting C3,13= max{C3,3, C3,4, C3,5, C3,6}, we complete the rest of the proof.

Theorem 3.2.2 (L(L)−error estimate for the control variable). Let (y, p, u) and (yh, ph, uh) be the solutions of (3.6)−(3.10) and (3.20)−(3.24), respectively. Assume that all the conditions in Theorem 3.2.1 are valid. Then, for each t∈(0, T], there exists a positive constant C3,14 such that the following error estimate

ku−uhkL(0,T;L(Ω)) ≤ C3,14n

ky0−yh,0kL(Ω)+ (ln ¯h)2h

R∞,0(yh,0, Gy(0)) +R∞,0(yh(t), Gy(t)) +R∞,0(ph(t),Gp(t))

+kR∞,0(∂yh

∂t ,∂Gy

∂t )kL1[0,T]+kR∞,0(∂ph

∂t ,∂Gp

∂t )kL1[0,T]i +

t∈[0,Tmax]

X

K∈Th

h2K|uh+ph|2H1(K)

1/2o ,

holds, where the constant C3,14 depends on the domain Ω, the regularity constant CR and the constant C3,11 as defined in Theorem 3.2.1.

CHAPTER 3. L(L)−A posteriori error estimates for POCP 69 Proof. From (3.11) and (3.25) we obtain

ku−uhkL(0,T;L(Ω)) = kΠ[ua,ub](−p)−Π[ua,ub](−ph)kL(0,T;L(Ω))

≤ kph−pkL(0,T;L(Ω)),

where we have used the Lipschitz continuity of Π[ua,ub] with Lipschitz constant 1. An application of Lemma 3.2.1 and Theorem 3.2.1 completes the rest of the proof.

Outline

Related documents