# A posteriori error analysis

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 31 Then the optimality conditions (2.15)−(2.19) can be rewritten as

( ¯∂Yhn, vh) +a(Yhn, vh) = (fn+Uh, vh) ∀vh ∈Vhn, (2.20)

Yh0 = yh,0, (2.21)

−( ¯∂Phn, vh) +a(Phn−1, vh) = (Yhn−ynds, vh) ∀vh ∈Vhn, (2.22)

PhN = 0, (2.23)

(Uh+Phn−1, whn−Uh) ≥ 0 ∀whn∈Uadn. (2.24) Analogous to the continuous case, we reformulate the discrete optimal control problem (2.13)−(2.14) as

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 32 where

rny(v) := (Lnh( ¯∂yhn), v) +a(yhn, v)−(fn+Uh, v), rnp(v) := (Lnh( ¯∂pnh), v) +a(pn−1h , v)−(ynh −ydsn, v).

We now define the elliptic reconstructions at t = tn, n ∈ [1 : N] as follows: For given ynh, pn−1h , seek ˜yhn, p˜n−1h ∈H01(Ω) satisfying

a(˜yhn−ynh, v) = −rny(v) ∀v ∈H01(Ω), (2.31) a(˜pn−1h −pn−1h , v) = −rnp(v) + (˜yhn−yhn, v) ∀v ∈H01(Ω). (2.32) Since ryn(vh) = 0 and rnp(vh) = 0, ∀vh ∈ Vhn, we observe that ynh denotes the elliptic projection of ˜yhn at time level t =tn. Using a sequence of discrete values {y˜hn} for n = 0,1, . . . , N, we set a continuous function of time defined by piecewise linear interpolant

˜

y(t), t∈[0, T] as

˜

y(t) := (tn−t)

knhn−1+ (t−tn−1)

knhn, tn−1 ≤t≤tn, n= 1, . . . , N.

Similarly, we define ˜p(t),t ∈[0, T], from the set of values {˜pnh}, n= 1, . . . , N as

˜

p(t) := (tn−t)

knn−1h + (t−tn−1)

knnh, tn−1 ≤t ≤tn, n = 1, . . . , N.

We note that functions ˜y and ˜p satisfy, for eacht ∈[0, T], the following equations:

a(˜y−Yh, v) = −ry(v) ∀v ∈H01(Ω),

a(˜p−Ph, v) = −rp(v) + (˜y−Yh, v) ∀v ∈H01(Ω),

where ry and rp are piecewise linear interpolant of {ryn}Nn=1 and {rnp}Nn=1, respectively.

Using elliptic reconstruction, we split the errors as follows:

ey = (˜y−y(Uh))−(˜y−Yh) =: ξy −ηy, ep = (˜p−p(Uh))−(˜p−Ph) =: ξp−ηp. Using (2.31)−(2.32) in (2.29)−(2.30), we find that

y,t, v) +a(ξy, v) = (ηy,t, v) +a(˜y−y˜hn, v) + (k−1n (Lnh −I)yhn−1, v)

+ (fn−f, v) ∀v ∈H01(Ω), (2.33)

−(ξp,t, v) +a(ξp, v) =−(ηp,t, v) +a(˜p−p˜n−1h , v)−(kn−1(Lnh−I)pn−1h , v)

−(ydsn −yds, v) + (˜ynh −y(Uh), v) ∀v ∈H01(Ω). (2.34)

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 33 Note that

˜

y−y˜nh := − (tn−t)

kn (˜ynh −y˜hn−1) and p˜−p˜n−1h := (t−tn−1)

kn (˜pnh −p˜n−1h ).

Moreover, the equations (2.33)−(2.34) can be written as (ξy,t, v) +a(ξy, v) = (ηy,t, v)−(tn−t)

kn a(˜ynh −y˜hn−1, v) + (k−1n (Lnh−I)yn−1h , v)

+ (fn−f, v) ∀v ∈H01(Ω), (2.35)

−(ξp,t, v) +a(ξp, v) =−(ηp,t, v) + (t−tn−1)

kn a(˜pnh−p˜n−1h , v)−(kn−1(Lnh−I)pn−1h , v)

−(ydsn −yds, v) + (˜ynh −y(Uh), v) ∀v ∈H01(Ω). (2.36) We need the following two propositions for future use (cf. ). The first one is about the interpolation error estimate for the Cl´ement-type interpolation operator. The second proposition is related to the approximation property which reflects the change of behaviour for the finest common coarsening of Thn and Thn−1.

Proposition 2.3.1. Let Πn :H01(Ω)→ Vhn be the Cl´ement-type interpolation operator.

Then, for sufficiently smooth ψ and finite element polynomial space of degree l, there exist constants C¯1,j and C¯2,j depending only upon the shape-regularity of the family of triangulations such that for j ≤l+ 1

kh−jn (ψ−Πnψ)k ≤ C¯1,j|ψ|j, (2.37) and

kh1/2−jn (ψ−Πnψ)kΣn ≤ C¯2,j|ψ|j. (2.38) Proposition 2.3.2. LetΠˆn:X →Vhn∩Vhn−1 be the Cl´ement-type interpolation operator with respect to the finest common coarsening of Thn and Thn−1, i.e., Tˆ :=Thn∧Thn−1

corresponding to the finite element space Vhn∩Vhn−1 with mesh size ˆhn:= max{hn, hn−1}.

Then the following inequality holds:

kˆh1/2−jn (ψ−Πˆnψ)kΣˇn\Σˆn ≤ C¯3,j|ψ|j,

where the constantC¯3,j depends on the shape regularity of the family of triangulations and on the number of steps required to move from Thn−1 to Thn. Further, the approximation properties (2.37) and (2.38) hold true in the finite element space Vhn∩Vhn−1 with ˆhn replacing hn.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 34 In the following lemma, we provide the bounds for the elliptic reconstruction errors associated with the state and co-state variables.

Lemma 2.3.1 (Elliptic reconstruction errors). Let (˜yhn,p˜n−1h )∈H01(Ω)×H01(Ω) satisfy (2.31)−(2.32). Then, 0≤n≤N, the following estimate holds:

k˜yhn−ynhk ≤ C¯1,2h2nk(Anh− Ael)yhnk+ ¯C2,2h3/2n kJ1[ynh]kΣn. (2.39) For n∈[1 :N], we have

kp˜n−1h −pn−1h k ≤ C¯1,2h2nk(Anh− Ael)pn−1h k+ ¯C2,2h3/2n kJ1[pn−1h ]kΣn+k˜yhn−ynhk, (2.40) where the constants C¯1,2 and C¯2,2 are defined in Proposition 2.3.1.

Proof. The proof of the lemma will proceed by the duality argument. Let w : [0, T] → H01(Ω) be the solution of the elliptic problem in the weak form as

a(v, w(t)) = (˜ynh −yhn, v) ∀v ∈H01(Ω), a.e. t∈[0, T], (2.41) Further, the solution w of (2.41) satisfies the following regularity result

kwk2 ≤ CRk˜ynh −yhnk, (2.42)

where CR denotes to the regularity constant. Setting v = ˜ynh −yhn ∈ Vhn in (2.41) and use of (2.31), (2.11) and (2.12) yields

k˜ynh −yhnk2 = a(˜yhn−yhn, w)

= a(˜yhn−yhn, w−Πnw)

= ((Anh− Ael)yhn, w−Πnw)−(J1[ynh], w−Πnw)Σn.

An application of the Cauchy-Schwarz inequality, Proposition 2.3.1 and (2.42) implies k˜ynh −yhnk ≤ C¯1,2h2hk(Anh− Ael)yhnk+ ¯C2,2h

3

n2kJ1[ynh]k, which proves (2.39).

Next, we shall prove the inequality (2.40). Setting v = ˜pn−1h −pn−1h ∈Vhn in a(v, w) = (˜pn−1h −pn−1h , v),

we obtain

kp˜n−1h −pn−1h k2 = a(˜pn−1h −pn−1h , w),

= a(˜pn−1h −pn−1h , w−Πnw) +a(˜pn−1h −pn−1hnw),

= (Anhpn−1h − Aelpn−1h , w−Πnw) + (˜yhn−ynhnw).

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 35 An application of Proposition 2.3.1 and the Cauchy-Schwarz inequality withw= ˜pn−1h − pn−1h yields

kp˜n−1h −pn−1h k ≤ C¯1,2h2nk(Anh− Ael)pn−1h k+ ¯C2,2h3/2n kJ1[pn−1h ]kΣn+k˜yhn−ynhk, and this completes the proof.

In the following lemma, we derive the bound forξy.

Lemma 2.3.2 (Parabolic error for the state variable). Let ξy satisfy (2.35). Then the following estimate holds, 1≤m≤N:

max

t∈[0,tm]y(t)k2+ 2α1 Z tm

0

yk21dt12

≤ kξy(0)k+ 4

E1,m2 +E2,m2

12 , where

E1,m :=

m

X

n=1

kn β1,n2,n4,n

, and

E2,m :=

m

X

n=1

β3,n2 kn.

Here, for n∈[1 :N], β1,n is the space error estimator and is defined by β1,n :=

1,2kˆh2n∂R¯ nyk+ ¯C2,2khˆ3/2n ∂J¯ ynkΣˆn+ ¯C3,2kˆh3/2n ∂J¯ ynkΣˇn\Σˆn

, β2,n is the temporal error estimator defined by

β2,n := kkn∂¯r˜ynk,

and the data approximation error estimators are defined by β3,n := ¯C3,1khnk−1n (Lnh −I)yhn−1k, β4,n := 1

kn Z tn

tn−1

kfn−fkdt.

The constants C¯i,2|i=1,2,3 and C¯3,1 are positive and independent of the discretization parameters.

Proof. By setting v =ξy in (2.35) and using the coercive property of the bilinear form, we obtain

1 2

d

dtkξyk21yk21 ≤ (ηy,t, ξy)− (tn−t) kn

a(˜yhn−y˜hn−1, ξy)

+ (kn−1(Lnh −I)yhn−1, ξy) + (fn−f, ξy). (2.43)

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 36 Integrating (2.43) with respect to time from 0 to tm, m ∈ [1 : N], and using the fact

(tn−t) kn

≤1 in In, it follows that 1

2kξy(tm)k2 −1

2kξy(0)k21 Z tm

0

yk21dt

m

X

n=1

Z tn

tn−1

|(ηy,t, ξy)|+|a(˜yhn−y˜hn−1, ξy)|+|(k−1n (Lnh−I)yhn−1, ξy)|

+|(fn−f, ξy)| dt =:

m

X

n=1

ζn1n2n3n4

m. (2.44)

Since ξy(t) is continuous in [0, tm], there exists tm ∈[0, tm] for which

t∈[0,tmaxm]y(t)k = kξy(tm)k =: kξymk. (2.45) Integrate (2.43) with respect to time from 0 to tm to have

1

2kξy(tm)k2− 1

2kξy(0)k21 Z tm

0

yk21dt

Z tm

0

{|(ηy,t, ξy)|+|a(˜yhn−y˜hn−1, ξy)|

+|(k−1n (Lnh−I)yhn−1, ξy)|+|(fn−f, ξy)|}dt

m

X

n=1

Z tn

tn−1

{|(ηy,t, ξy)|+|a(˜ynh −y˜hn−1, ξy)|

+|(k−1n (Lnh−I)yhn−1, ξy)|+|(fn−f, ξy)|}dt

=:

m

X

n=1

ζn1n2n3n4

=: ζm. In view of (2.45), we have

1

2kξymk2− 1

2kξy(0)k21

Z tm

0

yk21dt ≤ ζm. (2.46) Combining (2.44) and (2.46), it now leads to

1

2kξmy k21 Z tm

0

yk21dt ≤ kξy(0)k2+ 2ζm. (2.47) Now, we estimate ζni, i = 1, . . . ,4. To estimate the term ζn1, which measures the space error and mesh change, we will exploit the orthogonality property of the elliptic reconstruction definition (2.31). Observe that for each n∈[1 :N], we have

ζn1 = Z tn

tn−1

|(ηy,t, ξy)|dt = kn−1 Z tn

tn−1

|(˜yhn−y˜n−1h −ynh +yhn−1, ξy)|dt.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 37 Since ˜ynh−ynh is orthogonal toVhnwith respect toa(·,·), the first term inside the brackets is orthogonal to Vhn∩Vhn−1.

Letψ : [0, T]→H01(Ω) be such that

a(χ, ψ(t)) = (ξy(t), χ) ∀χ∈H01(Ω), t ∈[0, T].

Using the interpolation operator ˆΠn defined in Proposition 2.3.2, we write (˜ynh −y˜hn−1−yhn+yhn−1, ξy) = a(˜yhn−y˜hn−1−ynh +yhn−1, ψ(t))

= a(˜yhn−y˜hn−1−ynh +yhn−1, ψ(t)−Πˆnψ(t)).

With an aid of (2.31) and Green’s formula, we have

(˜yhn−y˜n−1h −yhn+yn−1h , ξy) = (Rny −Ryn−1, ψ(t)−Πˆnψ(t))

+(Jyn−Jyn−1, ψ(t)−Πˆnψ(t))Σˇn, (2.48) where

Rny := Lnh( ¯∂ynh)−div(∇yhn)−fn−Uh and Jyn =J1[ynh].

Equation (2.48) can be expressed as

(˜yhn−y˜n−1h −yhn+yn−1h , ξy) = kn( ¯∂Rny, ψ(t)−Πˆnψ(t)) +kn( ¯∂Jyn, ψ(t)−Πˆnψ(t))Σˇn. By Proposition 2.3.2, it follows that

ζn1 ≤ Z tn

tn−1

|ψ(t)|2dt

×

1,2kˆh2n∂R¯ nyk+ ¯C2,2kˆh3/2n ∂J¯ ynkΣˆ

n+ ¯C3,2kˆh3/2n ∂J¯ ynkΣˇ

n\Σˆn

. Use of elliptic regularity result, |ψ|2 ≤ kξyk, yields

ζn1 ≤ max

t∈Iny(t)k

1,2khˆ2n∂R¯ nyk+ ¯C2,2khˆ3/2n ∂J¯ ynkΣˆn + ¯C3,2kˆh3/2n ∂J¯ ynkΣˇn\Σˆn

kn, and hence,

m

X

n=1

ζn1 ≤ kξymk

m

X

n=1

knβ1,n.

In order to estimate ζn2, which accounts for the time discretization error, we use the definition of elliptic reconstruction (2.31) to obtain

ζn2 = Z tn

tn−1

|a(˜ynh −y˜hn−1, ξy)|dt. (2.49)

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 38 From (2.31), we have

a(˜yhn, v) = −r˜yn(v), (2.50)

where ˜rny(v) = (Lnh( ¯∂yhn), v)−(fn+Uh, v). Altogether (2.50) with (2.49) yields ζn2 =

Z tn

tn−1

|(kn∂¯˜ryn, ξy)|dt ≤ Z tn

tn−1

kkn∂¯r˜ynkkξykdt, and hence,

m

X

n=1

ζn2 ≤ kξmy k

m

X

n=1

knβ2,n.

We now bound the term ζn3. Since Vhn⊂ker(Lnh−I), we have ζn3 =

Z tn

tn−1

|(k−1n (Lnh −I)yhn−1, ξy)|dt

= Z tn

tn−1

|(k−1n (Lnh −I)yhn−1, ξy −Πnξy)|dt

= Z tn

tn−1

|(hnkn−1(Lnh−I)yhn−1, h−1ny−Πnξy))|dt

≤ Z tn

tn−1

khnk−1n (Lnh −I)yhn−1kkh−1ny −Πnξy)kdt

≤ C1,1k1/2n khnk−1n (Lnh −I)yhn−1kZ tn

tn−1

yk21dt12 . Thus, we have

m

X

n=1

ζn3

m

X

n=1

β3,nkn1/2Z tn

tn−1

yk21dt12 . To estimate ζn4, we first note that

ζn4 ≤ maxt∈In

y(t)kZ tn

tn−1

kfn−fkdt, and hence,

m

X

n=1

ζn4 ≤ kξymk

m

X

n=1

Z tn

tn−1

kfn−fkdt.

Combining the estimates of ζni, i= 1, . . . , 4, together with (2.47), we obtain 1

2kξmy k21 Z tm

0

yk21dt ≤ 1

2kξy(0)k2+ 2kξymk

m

X

n=1

kn β1,n2,n4,n + 2

m

X

n=1

β3,nkn1/2Z tn

tn−1

yk21 dt1/2

.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 39 Now, we use a standard inequality due to . For ˆa = (a0, a1, . . . , an) ∈ Rn+1, ˆb = (b0, b1, . . . , bn)∈Rn+1 and ˆc∈R, if

|ˆa|2 ≤ cˆ2+ ˆa·ˆb, then, we have

|ˆa| ≤ |ˆc|+|ˆb|.

Finally, for 1≤n≤m, we take a0 := 1

√2kξymk, an := Z tn

tn−1

yk21dt1/2

, ˆc := 1

√2kξy(0)k, b0 := 2√

2

m

X

n=1

kn β1,n2,n4,n

, bn := 2β3,nkn1/2, to complete the rest of the proof of the lemma.

The following lemma provides a bound forξp.

Lemma 2.3.3 (Parabolic error for the co-state variable). Let ξp satisfy (2.36). Then the following estimate holds, 0≤m≤N:

t∈[tmaxm,T]p(t)k2 + 2α1 Z T

tm

pk21dt1/2

≤4

E3,m2 +E4,m2

12 +

Z T tm

ykdt, where

E3,m :=

N

X

n=m+1

kn δ1,n2,n4,n

, and

E4,m :=

N

X

n=m+1

δ23,nkn.

In the above, for n ∈[0 :N], δ1,n represents the space error and is defined by δ1,n :=

1,2khˆ2n∂R¯ n+1p k+ ¯C2,2khˆ3/2n ∂J¯ pnkΣˆn+ ¯C3,2kˆh3/2n ∂J¯ pnkΣˇn\Σˆn

+k∂η¯ yn+1k, and the temporal error is defined as

δ2,n := kkn( ¯∂r˜np + ¯∂ηyn+1)k,

and the data approximation error estimators are defined by δ3,n := ¯C3,1khnkn−1(Lnh−I)pn−1h k, δ4,n := 1

kn Z tn

tn−1

kynds−ydskdt.

The constants C¯i,2|i=1,2,3 and C¯3,1 are positive and independent of the discretization parameters.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 40 Proof. Choose v = ξp in (2.36) and use the coercive property of the bilinear form to obtain

−1 2

d

dtkξpk21pk21

≤ −(ηp,t, ξp) + (t−tn−1)

kn a(˜pnh−p˜n−1h , ξp)−(kn−1(Lnh−I)pn−1h , ξp)

−(ynds−yds, ξp) + (˜yhn−y(Uh), ξp). (2.51) Integrating (2.51) with respect to time fromtm toT, we find that

1

2kξp(tm)k21 Z T

tm

pk21dt

≤ 1

2kξp(T)k2+ Z T

tm

n−(ηp,t, ξp) + (t−tn−1)

kn a(˜pnh −p˜n−1h , ξp)

−(k−1n (Lnh −I)pn−1h , ξp)−(ydsn −yds, ξp) + (˜yhn−y(Uh), ξp) o

dt

N

X

n=m+1

Z tn

tn−1

n

|(ηp,t, ξp)|+|a(˜pnh−p˜n−1h ), ξp)|

+|(kn−1(Lnh−I)pn−1h , ξp)|+|(ydsn −yds, ξp)|+|(˜ynh −y(Uh), ξp)|o dt

=:

N

X

n=m+1

n

ϑ1n2n3n4n5no

=: Θm. (2.52)

Since ξp is a continuous in [tm, T], there exists tm ∈[tm, T] such that max

t∈[tm,T]pk = kξp(tm)k =: kξpmk.

As Lemma 2.3.2, we deduce that 1

2kξmp k21 Z T

tm

pk21dt ≤ Θm. (2.53)

Consequently, from (2.52) and (2.53), it follows that 1

2kξpmk21 Z T

tm

pk21dt ≤ 2Θm. (2.54)

Now we estimate each of the summands ϑin|i=1,...,5 appearing on the right hand side of (2.54). To estimate the term ϑ1n, we write

ϑ1n = Z tn

tn−1

|(ηp,t, ξp)|dt = kn−1 Z tn

tn−1

|(˜pnh−p˜n−1h −pnh +pn−1h , ξp)|dt.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 41 Let Ψ : [0, T]→H01(Ω) such that a(χ,Ψ(t)) = (ξp, χ), ∀χ∈H01(Ω). A similar argument as in the case of the state variable and use of interpolation operator ˆΠn defined as in Proposition 2.3.2 leads to

(˜pnh −p˜n−1h −pnh+pn−1h , ξp) = a(˜pnh −p˜n−1h −pnh+pn−1h ,Ψ)

= a(˜pnh −p˜n−1h −pnh+pn−1h ,Ψ−ΠˆnΨ) +a(˜pnh−p˜n−1h −pnh+pn−1h ,ΠˆnΨ).

By the definition of elliptic reconstruction (2.32) and use of integration by parts formula leads to

(˜pnh −p˜n−1h −pnh+pn−1h , ξp) = −kn( ¯∂Rn+1p ,Ψ−ΠˆnΨ)−kn( ¯∂Jpn,Ψ−ΠˆnΨ)Σˇn

+kn( ¯∂ηyn+1,Ψ), where

Rnp :=−Lnh( ¯∂pnh)−div(∇pn−1h )−(ynh −ydsn), Jpn := J1[pnh].

An application of Proposition 2.3.2 yields ϑ1n ≤ Z tn

tn−1

|Ψ|2dt

×

1,2kˆh2n∂R¯ n+1p k+ ¯C2,2kˆh3/2n ∂J¯ pnkΣˆ

n + ¯C3,2kˆh3/2n ∂J¯ pnkΣˇ

n\Σˆn

+knk∂η¯ yn+1kkΨk.

Using elliptic regularity and summing n=m+ 1 to N, we obtain

N

X

n=m+1

ϑ1n ≤ kξmp k

N

X

n=m+1

knδ1,n.

To estimate ϑ2n, use (2.32) together with the definition of rnp to have

N

X

n=m+1

ϑ2n ≤ kξmp k

N

X

n=m+1

knkkn( ¯∂r˜pn+ ¯∂ηn+1y )k = kξmp k

N

X

n=m+1

knδ2,n, where

˜

rpn :=−Lnh( ¯∂pnh)−(yhn−ynds).

Now, we estimate ϑ3n and ϑ4n. Arguing as in Lemma 2.3.2, we arrive at

N

X

n=m+1

ϑ3n

N

X

n=m+1

δ3,nk1/2n Z tn

tn−1

pk21dt12 .

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 42 Note that

ϑ4n ≤ maxt∈In

p(t)kZ tn

tn−1

kydsn −ydskdt, and hence

N

X

n=m+1

ϑ4n ≤ kξpmk XN

n=m+1

Z tn

tn−1

kydsn −ydskdt . Finally, for ϑ5n, we find that

N

X

n=m+1

ϑ5n ≤ kξpmk XN

n=m+1

Z tn

tn−1

ykdt . Proceed as in Lemma 2.3.2 to complete the rest of the proof.

We are now in a position to present the main intermediate results for the state and co-state variables which will be used to derive the main result.

Theorem 2.3.1 (Intermediate error estimates). Let (Yh, Ph, Uh) and (y(Uh), p(Uh)), respectively, be the solutions of (2.20)−(2.24) and (2.25)−(2.28) with uˆ=Uh. Then, for each 1≤m ≤N, the following a posteriori error estimate holds:

max

[0,tm]kYh(t)−y(Uh)(t)k ≤ ky˜h0−y0k+ 2 max

n∈[0:m]Fy,n+ 4

E1,n2 +E2,n2

1/2

, (2.55) and

max

[tm,T]kPh(t)−p(Uh)(t)k ≤ 2 max

n∈[m:N]Fp,n+ 4

E3,n2 +E4,n2

1/2

+ Z T

tm

ykdt, (2.56) with

Fy,n := ¯C1,2h2nk(Anh−Ael)yhnk+ ¯C2,2h3/2n kJ1[yhn]kΣn,

Fp,n := ¯C1,2h2nk(Anh −Ael)pnhk+ ¯C2,2h3/2n kJ1[pn−1h ]kΣn+Fy,n, and Ei,n|i=1,2, Ei,n|i=3,4 are defined in Lemma 2.3.2 and Lemma 2.3.3, respectively.

Proof. By the triangle inequality, we have

key(t)k ≤ kξy(t)k+kηy(t)k, t ∈In

≤ max

t∈[0,tm]y(t)k+ max

t∈[0,tm]y(t)k. (2.57)

Fort ∈In, we note that kηy(t)k =

(tn−t)

kn ηyn−1+ (t−tn−1) kn ηyn

≤ kηyn−1k+kηynk ≤2 max

n∈[1,m]

n−1y k, kηynk .

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 43 Again, for t ∈[0, tm], use of Lemma 2.3.1 yields

y(t)k ≤ 2 max

n∈[0,m]nyk ≤2 max

n∈[0,m]Fy,n. (2.58)

Altogether (2.57), (2.58) and Lemma 2.3.2 yield the first inequality (2.55). The second inequality (2.56) is proved in a similar way by using Lemma 2.3.3. This completes the rest of the proof.

Let (y, p, u) and (Yh, Ph, Uh) be the solutions of (2.6)−(2.10) and (2.20) −(2.24), respectively. In order to derive a posteriori error bounds for the state, co-state and control variables, we decompose the errors as follows:

Yh−y = (Yh−y(Uh)) + (y(Uh)−y) =: ey−e˜y, Ph−p = (Ph−p(Uh)) + (p(Uh)−p) =: ep−˜ep.

From (2.6), (2.8) and (2.25), (2.27), we derive the following error equations:

(˜ey,t, v) +a(˜ey, v) = (u−Uh, v) ∀v ∈H01(Ω), (2.59)

−(˜ep,t, v) +a(˜ep, v) = (˜ey, v) ∀v ∈H01(Ω). (2.60) We now provide the bounds for ˜ey and ˜ep.

Lemma 2.3.4. Let (y, p, u) and (y(Uh), p(Uh)), respectively, be the solutions of (2.6)− (2.10) and (2.25)−(2.28) with uˆ=Uh. Then, for m∈[1 :N], we have

k˜ey(tm)k2+ Z tm

0

k˜eyk21dt ≤ C2,1 Z tm

0

ku−Uhk2L2(Ω)dt, (2.61) and

k˜ep(tm)k2+ Z T

tm

k˜epk21dt ≤ C2,2 Z T

tm

ku−Uhk2L2(Ω)dt, (2.62) where C2,1 andC2,2 are the positive constants depend on the Poincar´e inequality constant C and the coercivity constant α1.

Proof. Set v = ˜ey in (2.59) and use the coercive property of the bilinear form to obtain 1

2 d

dtk˜eyk21k˜eyk21 ≤ |(u−Uh,e˜y)|.

Apply the Cauchy-Schwarz and the Young’s inequality to have 1

2 d

dtk˜eyk21k˜eyk21 ≤ 1

2ku−Uhk2+ 1 2k˜eyk2.

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 44 Integrating the above with respect to time from 0 totm, and use the fact that ˜ey|t=0 = 0 to have

k˜ey(tm)k2+ 2α1 Z tm

0

k˜eyk21dt ≤ Z tm

0

ku−Uhk2dt+ Z tm

0

k˜eyk2dt.

An application of the Poincar´e inequality yields the first inequality (2.61).

To prove (2.62), choosev = ˜epin (2.60). Then, using the coercive property of bilinear form, the Cauchy-Schwarz inequality and the Young’s inequality to obtain

−1 2

d

dtk˜epk21k˜epk21 ≤ 1

2k˜eyk2 +1

2k˜epk2. (2.63) Integrating (2.63) from tm to T and using the fact that ˜ep|t=T = 0, it follows that

k˜ep(tm)k2+ 2α1 Z T

tm

k˜epk21dt ≤ Z T

tm

k˜eyk2dt+ Z T

tm

k˜epk2dt.

An application of the Poincar´e inequality together with (2.61) completes the rest of the proof.

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

Lemma 2.3.5 (Control error estimate). Let (y, p, u) and(Yh, Ph, Uh)be the solutions of (2.6)−(2.10)and(2.20)−(2.24), respectively. Assume thatUadn ⊂Uad, (Uh+Phn−1)|K ∈ H1(K) and wh ∈Uadn, and there exists a positive constant C2,3 such that

Z T 0

(Uh+Phn−1, wh−u)dt

≤ C2,3 Z T

0

X

K∈Thn

hK|Uh+Phn−1|H1(K)ku−UhkL2(K)dt.

(2.64) Then, we have

ku−Uhk2L2(0,T;L2(Ω)) ≤ C2,4Z T 0

X

K∈Thn

h2K|Uh+Phn−1|2H1(K)dt+kPhn−1−p(Uh)k2L2(0,T;L2(Ω))

,

where C2,4 = 32max{C2,32 ,1}, and (y(Uh), p(Uh)) is the solution of (2.25)−(2.28) with ˆ

u=Uh.

Proof. From (2.10) with w=Uh, we have

(u, u−Uh) ≤ −(p, u−Uh),

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 45 using the above inequality, we find that

ku−Uhk2L2(0,T;L2(Ω)) = Z T

0

(u−Uh, u−Uh)dt

≤ Z T

0

{−(p, u−Uh)−(Uh, u−Uh)}dt

= −

Z T 0

(Phn−1+Uh, u−wh)dt− Z T

0

(Uh+Phn−1, wh−Uh)dt +

Z T 0

(Phn−1−p(Uh), u−Uh)dt+ Z T

0

(p(Uh)−p, u−Uh)dt.

With an aid of (2.24), we get ku−Uhk2L2(0,T;L2(Ω))

Z T 0

(Uh+Phn−1, wh−u)dt+ Z T

0

(Phn−1−p(Uh), u−Uh)dt +

Z T 0

(p(Uh)−p, u−Uh)dt

=: E1+E2+E3. (2.65)

In view of the assumption (2.64), it follows that

|E1| =

Z T 0

(Uh+Phn−1, wh−u)dt

≤ C2,3 Z T

0

X

K∈Thn

hK|Uh +Phn−1|H1(K)ku−UhkL2(K)dt

≤ 3C2,32 4

Z T 0

X

K∈Thn

h2K|Uh+Phn−1|2H1(K)dt +1

4ku−Uhk2L2(0,T;L2(Ω)). (2.66) The term E2 is bounded as

|E2| =

Z T 0

(Phn−1−p(Uh), u−Uh)dt

≤ 3

4kPhn−1−p(Uh)k2L2(0,T;L2(Ω))+1

4ku−Uhk2L2(0,T;L2(Ω)). (2.67) It now remains to estimate the term E3. For this, we first note that

y(x,0)−y(Uh)(x,0) = 0 and p(x, T)−p(Uh)(x, T) = 0.

Utilizing (2.6) and (2.25), we write E3 as E3 =

Z T 0

(u−Uh, p(Uh)−p)dt

= Z T

0

−(y−y(Uh), pt(Uh)−pt) +a(y−y(Uh), p(Uh)−p) dt,

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 46 which combine with (2.8) and (2.27) yields

E3 = − Z T

0

ky−y(Uh)k2dt ≤ 0. (2.68)

Altogether (2.65)−(2.68) yields the desired estimate. This completes the proof.

Remark 2.3.1. The assumption (2.64) refers to the auxiliary finite element solution for the control variable, which help us in simplifying the proof of Lemma 2.3.5 for the control variable. It is easy to verify this condition for many applications (cf., ).

By collecting Theorem 2.3.1 and Lemmas 2.3.4− 2.3.5, the main results of this chapter are presented in the following theorem.

Theorem 2.3.2(L(L2)−error estimates). Let(y, p, u)and(Yh, Ph, Uh)be the solutions of (2.6)−(2.10) and (2.20) −(2.24), respectively. Then, for each m ∈ [1 : N], the following a posteriori error estimates

max

[0,tm]kYh(t)−y(t)k ≤ k˜y0h−yh,0k+ 2 max

n∈[0:m]Fy,n+ 4

E1,n2 +E2,n2 1/2

+Z tm

0

ku−Uhk2dt12 , max

[tm,T]kPh(t)−p(t)k ≤ 2 max

n∈[1:m]Fp,n+ 4

E3,n2 +E4,n2

1/2

+ Z T

tm

ykdt +Z T

tm

ku−Uhk2dt12 , and

max

[0,tm]ku(t)−Uh(t)k ≤ C2,4

Z T 0

X

K∈Thn

h2K|Uh+Phn−1|2H1(K)dt 1/2

+ 2 max

n∈[1:m]Fp,n+ 4

E3,n2 +E4,n2

1/2

+ Z T

tm

ykdt hold, where

Fy,n := ¯C1,2h2nk(Anh −Ael)yhnk+ ¯C2,2h3/2n kJ1[yhn]kΣn,

Fp,n := ¯C1,2h2nk(Anh −Ael)pnhk+ ¯C2,2h3/2n kJ1[pn−1h ]kΣn+Fy,n,

and Ei,n, i= 1,2, and Ei,n, i= 3,4, are defined in Lemma 2.3.2 and 2.3.3, respectively.

Proof. From (2.10) and (2.3), we obtain

u(x, t) = Π[ua,ub] −p(x, t)

, (2.69)

CHAPTER 2. L(L2)−A posteriori error estimates for POCP 47 where Π[ua,ub](u(x, t)) := max(ua,min(ub,−p(x, t)) (cf. ). Similarly, we have

Uh = Π[ua,ub] −Phn−1

. (2.70)

Using (2.69) and (2.70), we obtain

ku−UhkL(0,T;L2(Ω)) ≤ kp−Phn−1kL(0,T;L2(Ω)).

Inviting Theorem 2.3.1 together with Lemmas 2.3.4−2.3.5 leads to the desired estimates.

This completes the proof of the theorem.

Outline

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