Analysis for fully discrete control problem

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

ku−u_hk_L^∞_(0,T;L^∞_(Ω)) = kΠ_[ua,ub](−p)−Π_[ua,ub](−p_h)k_L^∞_(0,T;L^∞_(Ω))

≤ kp_h−pk_L^∞_(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.

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 70 where nE is a unit normal vector to E at the point x.

Let Lⁿ_h and Lⁿ_h,0 be the L²-projections onto Vⁿ and V₀ⁿ such that

(Lⁿ_hφ, ψ_n) = (φ, ψ_n) ∀ψ_n∈Vⁿ, and (Lⁿ_h,0φ, ψ_n) = (φ, ψ_n) ∀ψ_n ∈V₀ⁿ. Discrete elliptic operator: The discrete elliptic operator associated with the bilinear form a(·,·) and the finite element space V₀ⁿ is the operator Aⁿ_h : H₀¹(Ω) → V₀ⁿ+Lⁿ_hfⁿ such that for v ∈H₀¹(Ω) and 0≤n≤N,

(−Aⁿ_hv, w_h) =a(v, w_h), ∀w_h ∈V₀ⁿ.

The fully discrete variational discretization approximations of the problem (3.18)− (3.19) is defined as follows: Find (y_hⁿ, uⁿ_h)∈V₀ⁿ×U_ad, forn ∈[1 :N], such that

uminⁿ_h∈U_ad

1 2

N

X

n=1

Z

In

ky_hⁿ−yⁿ_dsk²_L2(Ω)+kuⁿ_hk²_L2(Ω) dt (3.46) subject to











( ¯∂y_hⁿ, v_h) +a(yⁿ_h, v_h) = (fⁿ+uⁿ_h, v_h) ∀v_h ∈V₀ⁿ, y_h⁰ =y_h,0,

(3.47)

where y_h,0 is the suitable approximation or projection of y₀ inV₀⁰.

The optimal control problem (3.46)−(3.47) admits a unique solution (y_hⁿ, uⁿ_h) if and only if there exists a co-state pⁿ⁻¹_h ∈ V₀ⁿ such that the following optimality conditions are satisfied: For each n∈[1 :N],

( ¯∂y_hⁿ, v_h) +a(y_hⁿ, v_h) = (fⁿ+uⁿ_h, v_h) ∀v_h ∈V₀ⁿ, (3.48)

y_h⁰ = y_h,0, (3.49)

−( ¯∂pⁿ_h, v_h) +a(pⁿ⁻¹_h , v_h) = (yⁿ_h −y_dsⁿ, v_h) ∀v_h ∈V₀ⁿ, (3.50)

p^N_h = 0, (3.51)

(uⁿ_h +pⁿ⁻¹_h , w_hⁿ−uⁿ_h) ≥ 0 ∀w_hⁿ∈Uad. (3.52) Given a sequence of discrete values {y_hⁿ}, n = 0,1, . . . , N, we associate a continuous function of time defined by the continuous piecewise linear interpolant Y_h(t), t∈I_n as

Y_h(t) := (t_n−t)

k_n y_hⁿ⁻¹+ (t−tn−1) k_n y_hⁿ.

Similarly, we define Ph(t), t∈In, from the set of values {pⁿ_h}, n= 0,1, . . . , N as P_h(t) := (t_n−t)

k_n pⁿ⁻¹_h +(t−tn−1) k_n pⁿ_h,

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 71 and

Uh(t)|t∈In :=uⁿ_h.

Finally, we define Y_h,tⁿ = ^∂Y_∂t^h|_In and P_h,tⁿ = ^∂P_∂t^h|_In. Further, we note that the values of Y_h(t) andP_h(t) at the nodal point t=t_n, n= 1, 2, . . . , N are coincided with yⁿ_h and pⁿ_h, respectively.

The weak-form of fully discrete schemes (3.48) and (3.50) can be easily transformed into the pointwise form as

yⁿ_h − Lⁿ_h,0yⁿ⁻¹_h

k_n − Aⁿ_hy_hⁿ = Lⁿ_hfⁿ+uⁿ_h,

−pⁿ_h− Lⁿ_h,0pⁿ⁻¹_h

k_n − Aⁿ_hpⁿ⁻¹_h = yⁿ_h − Lⁿ_hy_dsⁿ. This implies

Y_h,tⁿ − Aⁿ_hy_hⁿ=Lⁿ_hfⁿ+uⁿ_h +Lⁿ_h,0y_hⁿ⁻¹−y_hⁿ⁻¹

k_n , n≥1, (3.53)

−P_h,tⁿ − Aⁿ_hpⁿ⁻¹_h =yⁿ_h − Lⁿ_hy_dsⁿ − Lⁿ_h,0pⁿ⁻¹_h −pⁿ⁻¹_h

k_n , n≥1. (3.54)

Then the optimality conditions (3.48)−(3.52) can be stated as follows:

( ¯∂Y_hⁿ, v_h) +a(Y_hⁿ, v_h) = (fⁿ+U_h, v_h) ∀v_h ∈V₀ⁿ, (3.55)

Y_h⁰ = y_h,0, (3.56)

−( ¯∂P_hⁿ, v_h) +a(P_hⁿ⁻¹, v_h) = (Y_hⁿ−yⁿ_ds, v_h) ∀v_h ∈V₀ⁿ, (3.57)

P_h^N = 0, (3.58)

(U_h+P_hⁿ⁻¹, w_hⁿ−U_h) ≥ 0 ∀w_hⁿ∈U_ad. (3.59) Analogous to the continuous case, we reformulate the discrete optimal control problem (3.46)−(3.47) as

min

Uh∈U_adj_hⁿ(U_h) :=J(U_h, Y_h(U_h)).

As in the case of semidiscrete error analysis, we first derive some intermediate error estimates for the state and co-state variables in the L^∞(0, T;L^∞(Ω))-norm. Here, the fully discrete analogue of elliptic reconstructions for the state and co-state variables are treated as intermediate objects in the error analysis.

For the purpose of error analysis, we shall define the errors for the state and co-state variables as follows:

e_y :=Y_h−y(U_h) and e_p :=P_h−p(U_h).

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 72 From (3.26), (3.28), (3.55) and (3.57) with ˆu = Uh, we have the following error equations forv ∈H₀¹(Ω):

∂e_y

∂t , v

+a(e_y, v) = −ωⁿ_y(v) +a(Y_h−y_hⁿ, v) + (fⁿ−f, v), (3.60)

−∂e_p

∂t , v

+a(e_p, v) = ωⁿ_p(v) +a(P_h−pⁿ⁻¹_h , v) + (y_hⁿ−y(U_h), v)

+(y_ds−y_dsⁿ, v), (3.61)

where

ω_yⁿ(v) := (fⁿ− Lⁿ_hfⁿ, v) +

y_hⁿ⁻¹− Lⁿ_h,0y_hⁿ⁻¹ k_n , v

, ω_pⁿ(v) := (y_dsⁿ − Lⁿ_hy_dsⁿ, v) +pⁿ⁻¹_h − Lⁿ_h,0pⁿ⁻¹_h

k_n , v .

We now define the elliptic reconstructions att =tn,n ∈[1 :N] as follows: For given yⁿ_h, pⁿ⁻¹_h , seek ˜y_hⁿ, p˜ⁿ⁻¹_h ∈H₀¹(Ω) satisfying

a(˜y_hⁿ, v) = (Gyⁿ, v) ∀v ∈H₀¹(Ω), (3.62) and

a(˜pⁿ⁻¹_h , v) = (Gpⁿ, v) + (˜yⁿ_h −y_hⁿ, v) ∀v ∈H₀¹(Ω), (3.63) where

G_yⁿ=











−A⁰_hy⁰_h+f⁰− L⁰_hf⁰ n = 0, fⁿ+uⁿ_h −Y_h,tⁿ n ≥1, and

Gpⁿ =











−A⁰_hp⁰_h+y⁰_ds− L⁰_hy⁰_ds n= 0, yⁿ_h − Lⁿ_hy_dsⁿ +P_h,tⁿ n≥1.

Using a sequence of discrete values{˜y_hⁿ}forn = 0,1, . . . , N, we set a continuous function of time defined by piecewise linear interpolant ˜y(t) as

˜

y(t) := (t_n−t) kn

˜

y_hⁿ⁻¹+ (t−tn−1) kn

˜

y_hⁿ tn−1 ≤t≤t_n, n= 1, . . . , N.

Similarly, we define ˜p(t) from the set of values {˜pⁿ_h}, n= 1, . . . , N as

˜

p(t) := (t_n−t)

k_n p˜ⁿ⁻¹_h +(t−tn−1)

k_n p˜ⁿ_h tn−1 ≤t≤t_n, n= 1, . . . , N.

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 73 We note that functions ˜y and ˜psatisfy, for each t∈[0, T], the following equations:

a(˜y−Yh, v) = ωy(v) ∀v ∈H₀¹(Ω),

a(˜p−P_h, v) = −ω_p(v) + (˜y−Y_h, v) ∀v ∈H₀¹(Ω).

From (3.53) and (3.54), we obtain

Gyⁿ = fⁿ+uⁿ_h −Y_h,tⁿ =−Aⁿ_hyⁿ_h +fⁿ− Lⁿ_hfⁿ− Lⁿ_h,0yⁿ⁻¹_h −yⁿ⁻¹_h

k_n , n≥1, Gpⁿ = y_hⁿ−yⁿ_ds+P_h,tⁿ =−Aⁿ_hpⁿ⁻¹_h +Lⁿ_hyⁿ_ds−y_dsⁿ + Lⁿ_h,0pⁿ⁻¹_h −pⁿ⁻¹_h

k_n , n≥1.

Using elliptic reconstruction, we decompose the errors as

e_y = (˜y−y(U_h))−(˜y−Y_h) =: ξ_y−η_y, and e_p = (˜p−p(U_h))−(˜p−P_h) =: ξ_p−η_p. Note that

˜

y−y˜_hⁿ := −(1−l(t))(˜yⁿ_h −y˜_hⁿ⁻¹) and p˜−p˜ⁿ⁻¹_h := l(t)(˜pⁿ_h −p˜ⁿ⁻¹_h ), wherel(t) = ^t−t_kⁿ⁻¹

n . Using (3.62)−(3.63) in (3.60)−(3.61), for allv ∈H₀¹(Ω), we obtain ∂ξy

∂t, v

+a(ξ_y, v) = ∂ηy

∂t , v

+ (fⁿ−f, v) + (1−l(t))(Gyⁿ⁻¹−Gyⁿ, v),(3.64)

−∂ξp

∂t , v

+a(ξp, v) = −∂ηp

∂t , v

+ (yds−y_dsⁿ, v) + (˜y_hⁿ−y(Uh), v)

+l(t)(G_pⁿ−G_pⁿ⁺¹, v). (3.65)

The fully discrete analogue of Lemma 3.2.1 is stated in the following lemma.

Lemma 3.3.1 (Elliptic reconstruction errors). Let (˜y_hⁿ,p˜ⁿ⁻¹_h )∈H₀¹(Ω)×H₀¹(Ω) satisfy (3.62)−(3.63). Then, 0≤n ≤N, we have

k˜y_hⁿ−y_hⁿk_L^∞_(Ω) ≤ C3,15(ln ˆhn)²R∞,0(y_hⁿ,Gyⁿ).

Moreover, for n∈[1 :N], we have

kp˜ⁿ⁻¹_h −pⁿ⁻¹_h k_L^∞_(Ω) ≤ C_3,16(ln ˆh_n)²R∞,0(pⁿ⁻¹_h ,Gpⁿ) +ky˜_hⁿ−y_hⁿk_L^∞_(Ω), where the constants C3,15 and C3,16 depend on the domain Ω.

In the following lemma, we derive the bounds forξ_y and ξ_p.

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 74 Lemma 3.3.2 (Parabolic error estimates for the state and co-state variables). Let ξy and ξp satisfy (3.64) and (3.65), respectively. Then, for any 1 ≤ m ≤ N with ˆh_m = min

1≤n≤m min

K∈Thⁿ

h_K, the following estimates hold:

kξ_y(t_m)k_L^∞_(Ω) ≤ kξ_y(0)k_L^∞_(Ω)+

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+k_n

2 kGyⁿ⁻¹−Gyⁿk_L^∞_(Ω) +C_3,17(ln ˆh_m)²

m

X

n=1

k_nRˆ∞,0

y_hⁿ−yⁿ⁻¹_h

k_n ,G_yⁿ−G_yⁿ⁻¹;T_hⁿ⁻¹,T_hⁿ ,

(3.66) kξ_p(t_m)k_L^∞_(Ω) ≤ kξ_y(t_m)k_L^∞_(Ω)+

N

X

n=m+1

Z

In

ky_ds−y_dsⁿk_L^∞_(Ω)ds+k_n

2 kGpⁿ−Gpⁿ⁺¹k_L^∞_(Ω) +C_3,18(ln ˆh_m)²

N

X

n=m+1

k_nRˆ∞,0

pⁿ⁻¹_h −pⁿ_h

k_n ,G_pⁿ−G_pⁿ⁺¹;T_hⁿ⁻¹,T_hⁿ . (3.67) In the above, the constants C_3,17 andC_3,18 are positive and depend on the constant C_3,2. Proof. Note that ξy satisfies (3.64). For any tm ∈[0, T] and a fixedxm ∈Ω, an applica- tion of (3.16) leads to

|ξ_y(x_m, t_m)| ≤ Z

Ω

|F(x_m, t_m;w,0)ξ_y(w,0)|dw +

Z tm

0

Z

Ω

|F(xm, tm;w, s)∂η_y

∂t (w, s)|dw ds +

m

X

n=1

Z

In

Z

Ω

|F(x_m, t_m;w, s) (fⁿ−f)|dw ds +

m

X

n=1

Z

In

Z

Ω

|F(x_m, t_m;w, s) (1−l(s))(Gyⁿ⁻¹−Gyⁿ)|dw ds,

using the H¨older’s inequality and (3.17) with |ξ_y(x_m, t_m)| =kξ_y(t_m)k_L^∞_(Ω) (since x_m is fixed), we obtain

kξy(tm)kL^∞(Ω) ≤ kξy(0)kL^∞(Ω)+k∂η_y

∂t k_L¹_([0,tm];L^∞_(Ω))+

m

X

n=1

Z

In

kfⁿ−fkL^∞(Ω))ds +k_n

2 kGyⁿ⁻¹−Gyⁿk_L^∞(Ω).

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 75 Use of Lemma 3.1.2 leads to

kξ_y(t_m)k_L^∞_(Ω) ≤ kξ_y(0)k_L^∞_(Ω)+

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+k_n

2 kGyⁿ⁻¹−Gyⁿk_L^∞_(Ω) +C_3,17(ln ˆh_m)²

m

X

n=1

k_nRˆ∞,0

yⁿ_h −y_hⁿ⁻¹

k_n ,Gyⁿ−Gyⁿ⁻¹;T_hⁿ⁻¹,Thⁿ

,

and this completes the proof of (3.66).

To prove (3.67), we first note that ξ_p satisfies (3.65). For any t_m ∈ [0, T] and fixed x_m ∈Ω, a similar argument as before leads to

|ξ_p(x_m, t_m)| ≤ Z

Ω

|F(x_m, t_m;w, T)ξ_p(w, T)|dw +

Z T tm

Z

Ω

|F(x_m, t_m;w, s)∂ηp

∂t (w, s)|dw ds +

N

X

n=m+1

Z

In

Z

Ω

|F(x_m, t_m;w, s) (y_ds−yⁿ_ds)dw|ds

+

N

X

n=m+1

Z

In

Z

Ω

|F(x_m, t_m;w, s) (˜y_hⁿ−y(U_h))|dw ds

+

N

X

n=m+1

Z

In

Z

Ω

|F(x_m, t_m;w, s)l(s)(Gpⁿ−Gpⁿ⁺¹)|dw ds.

An application of the H¨older’s inequality and (3.17) with |ξ_p(x_m, t_m)| = kξ_p(t_m)k_L^∞_(Ω) yields

kξ_p(t_m)k_L^∞_(Ω) ≤ kξ_p(T)k_L^∞_(Ω)+k∂η_p

∂t k_L¹_([tm,T];L^∞_(Ω))+

N

X

n=m+1

Z

In

ky_ds−y_dsⁿk_L^∞_(Ω)ds

+

N

X

n=m+1

Z

In

k˜y_hⁿ−y(U_h)k_L^∞_(Ω)ds+ k_n

2 kG_pⁿ−G_pⁿ⁺¹k_L^∞_(Ω). Utilization of Lemma 3.1.2 and ξ_p(T) = 0 imply

kξ_p(t_m)k_L^∞_(Ω) ≤ C_3,18(ln ˆh_m)²

m

X

n=1

k_nRˆ∞,0

pⁿ⁻¹_h −pⁿ_h

k_n ,Gpⁿ−Gpⁿ⁺¹;T_hⁿ⁻¹,Thⁿ

+

N

X

n=m+1

Z

In

kyds−yⁿ_dsk_L^∞_(Ω)ds+ k_n

2 kGpⁿ−Gpⁿ⁺¹k_L^∞_(Ω)+kξy(tm)k_L^∞_(Ω), which completes the rest of the proof.

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 76 Let (y, p, u) and (Yh, Ph, Uh) be the solutions of (3.6)−(3.10) and (3.55)−(3.59), respectively. In order to derive a posteriori error bounds for the state and co-state variables, we decompose the errors as follows:

y−Y_h = (y−y(U_h)) + (y(U_h)−Y_h) =:r_y−e_y, p−P_h = (p−p(U_h)) + (p(U_h)−P_h) =:r_p−e_p.

From (3.6), (3.8), (3.26) and (3.28) with ˆu=U_h, we derive the following error equations:

∂r_y

∂t , v

+a(ry, v) = (u−Uh, v) ∀v ∈H₀¹(Ω), (3.68)

−∂r_p

∂t, v

+a(r_p, v) = (r_y, v) ∀v ∈H₀¹(Ω). (3.69) The following lemma provides the bounds for r_y and r_p.

Lemma 3.3.3. Let(y, p, u)be the solution of the problem(3.6)−(3.10), and let(y(ˆu), p(ˆu)) be the solution of the problem (3.26)−(3.29) with uˆ=U_h. Then, for any 1≤m ≤N, we have

kry(tm)k_L^∞_(Ω) ≤ C3,19ku−Uhk_L^∞_(0,T;L²_(Ω)), (3.70) and

krp(tm)kL^∞(Ω) ≤ C3,20ku−Uhk_L^∞_(0,T;L²_(Ω)), (3.71) where the constants C_3,19 and C_3,20 depend on the regularity constant C_R.

Proof. Following the lines of argument of Lemma 3.2.3, the proof of inequalities (3.70) and (3.71) can easily be obtained. The details are thus omitted.

In the following lemma, we derive the a posteriori error estimate for the control variable in the L²(0, T;L²(Ω))-norm.

Lemma 3.3.4. Let(y, p, u)and(Y_h, P_h, U_h)be the solutions of(3.6)−(3.10)and(3.55)−

(3.59), respectively. Assume that(U_h+P_hⁿ⁻¹)|_K ∈H¹(K)andw_h ∈U_ad, and there exists a positive constant C_3,21 such that

(U_h+P_hⁿ⁻¹, w_h−u)

≤ C_3,21 X

K∈Thⁿ

h_K|U_h+P_hⁿ⁻¹|_H¹_(K)ku−U_hk_L²_(K). Then, we have

ku−U_hk_L^∞_(0,T;L²_(Ω)) ≤ C_3,22h

n∈[1,N]max X

K∈Thⁿ

h²_K|U_h+P_hⁿ⁻¹|²_H1(K)

1/2

+kP_hⁿ⁻¹−p(U_h)k_L^∞_(0,T;L^∞_(Ω))i ,

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 77 whereC3,22=

q3

2 max

1, C3,21 , and(y(ˆu), p(ˆu))is defined by the system(3.26)−(3.29) with uˆ=Uh.

Proof. From (3.10) with w=U_h, we have

(u, u−U_h) ≤ −(p, u−U_h).

Using the above inequality, it follows that ku−Uhk²_L2(Ω) = (u−Uh, u−Uh)

≤ −(p, u−U_h)−(U_h, u−U_h)

= −(P_hⁿ⁻¹+U_h, u−w_h)−(U_h +P_hⁿ⁻¹, w_h−U_h) +(P_hⁿ⁻¹−p(U_h), u−U_h) + (p(U_h)−p, u−U_h).

Inviting (3.59) we obtain

ku−U_hk²_L2(Ω) ≤ (U_h+P_hⁿ⁻¹, w_h−u) + (P_hⁿ⁻¹−p(U_h), u−U_h) +(p(U_h)−p, u−U_h)

=: Eˆ₁+ ˆE₂+ ˆE₃.

Following the idea of Lemma 3.2.4, it is easy to bound the term ˆE_i, i = 1,2,3. So, we omit the details. This completes the proof.

By collecting Lemmas 3.3.1−3.3.4, we finally derive the main results of this section.

Theorem 3.3.1 ( L^∞(L^∞)−error estimates for the state and co-state variables). Let (y, p, u)and(Y_h, P_h, U_h)be the solutions of (3.6)−(3.10)and(3.55)−(3.59), respectively.

Then there exists positive constants C_3,24, C_3,25 (depend on Ω), for each t ∈(0, T], and any 1≤m≤N with ˆh_m = min

1≤n≤m min

K∈Thⁿ

h_K, the following estimates

ku−U_hk_L^∞_(0,T;L²_(Ω)) ≤ C_3,23h

n∈[1,N]max X

K∈Thⁿ

h²_K|U_h +P_hⁿ⁻¹|²_H1(K)

1/2

+kξ_pk_L^∞_(0,T;L^∞_(Ω))+kη_pk_L^∞_(0,T;L^∞_(Ω))i

, (3.72)

where the constant C_3,23 depends on the domain Ω and the constant C_3,22 as defined in

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 78 Lemma 3.3.4,

ky−Y_hk_L^∞_(0,T;L^∞_(Ω)) ≤ ky₀−y_h,0k_L^∞_(Ω)+C_3,24(ln ˆh_m)²

×h

R^∞,0(y_h,0,Gy⁰) +R^∞,0(y_h^m,Gy^m) +

m

X

n=1

k_nRˆ^∞,0 y_hⁿ−yⁿ⁻¹_h

k_n ,Gyⁿ−Gyⁿ⁻¹;Thⁿ⁻¹,Thⁿ

i

+

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+k_n

2 kGyⁿ⁻¹ −Gyⁿk_L^∞_(Ω)

+C_3,19ku−U_hk_L^∞_(0,T;L²_(Ω)), (3.73)

and

kp−P_hk_L^∞_(0,T;L^∞_(Ω)) ≤ ky₀−y_h,0k_L^∞_(Ω)+C_3,25(ln ˆh_m)²h

R∞,0(y^m_h,G_y^m) +R^∞,0(y_h,0,Gy⁰) +R^∞,0(p^m_h,Gp^m)

+

m

X

n=1

k_nRˆ∞,0

y_hⁿ−y_hⁿ⁻¹

k_n ,Gyⁿ−Gyⁿ⁻¹;T_hⁿ⁻¹,Thⁿ

+

N

X

n=m+1

k_nRˆ^∞,0 pⁿ⁻¹_h −pⁿ_h

k_n ,Gpⁿ−Gpⁿ⁺¹;Thⁿ⁻¹,Thⁿ

i

+

N

X

n=m+1

Z

In

ky_ds−yⁿ_dsk_L^∞_(Ω)ds+ k_n

2 kGpⁿ−Gpⁿ⁺¹k_L^∞_(Ω) +

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+ k_n

2 kGyⁿ⁻¹−Gyⁿk_L^∞_(Ω)

+C_3,20ku−U_hk_L^∞_(0,T;L²_(Ω)) (3.74) hold, where the constants C_3,19 and C_3,20 are defined in Lemma 3.3.3.

Proof. The first inequality (3.72) follows from Lemma 3.3.4. Next, to prove error esti- mate for the state variable, we write

y−Y_h = (y−y(U_h)) + (y(U_h)−y) + (˜˜ y−Y_h) =r_y−(ξ_y−η_y).

For a fix xm ∈Ω and tm ∈(0, T], we have

k(y−Yh)(tm)k_L^∞_(Ω) ≤ kry(tm)k_L^∞_(Ω)+kξy(tm)k_L^∞_(Ω)+kηy(tm)k_L^∞_(Ω). Using Lemma 3.3.1, the last term of the right hand side is bounded as

kη_y(t_m)k_L^∞_(Ω) ≤ C_3,15(ln ˆh_m)²R∞,0(y_h^m,Gy^m).

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 79 By Lemma 3.3.2, we have

kξ_y(t_m)k_L^∞_(Ω) ≤ ky₀−y_h,0k_L^∞_(Ω)+C_3,15(ln ˆh_m)²R∞,0(y_h,0,G_y⁰) +C_3,17(ln ˆh_m)²

m

X

n=1

k_nRˆ^∞,0(y_hⁿ−y_hⁿ⁻¹

k_n ,Gyⁿ−Gyⁿ⁻¹;Thⁿ⁻¹,Thⁿ) +

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+k_n

2 kGyⁿ⁻¹−Gyⁿk_L^∞_(Ω). An application of Lemma 3.3.3 yields

kr_y(t_m)k_L^∞_(Ω) ≤ C_3,19ku−U_hk_L^∞_(0,T;L²_(Ω)).

Combining the above estimates and setting C3,24 = max{C3,15, C3,17}, we accomplish (3.73).

Next, we estimate the error for the co-state variable. By the triangle inequality, for any t_m ∈(0, T], we have

k(p−P_h)(t_m)k_L^∞_(Ω) ≤ kr_p(t_m)k_L^∞_(Ω)+kξ_p(t_m)k_L^∞_(Ω)+kη_p(t_m)k_L^∞_(Ω). We apply Lemmas 3.3.1−3.3.3 to arrive at

k(p−Ph)(tm)k_L^∞_(Ω) ≤ (ln ˆhm)² h

C3,15R∞,0(y^m_h,Gy^m) +C3,16R∞,0(p^m_h,Gp^m) +C3,18

N

X

n=m+1

knRˆ∞,0

pⁿ⁻¹_h −pⁿ_h

k_n ,Gpⁿ−Gpⁿ⁺¹;T_hⁿ⁻¹,Thⁿ

i

+

N

X

n=m+1

Z

In

ky_ds−yⁿ_dsk_L^∞_(Ω)ds+ kn

2 kGpⁿ−Gpⁿ⁺¹k_L^∞_(Ω) +kξ_yk_L^∞_(0,T;L^∞_(Ω))+C_3,20ku−U_hk_L^∞_(0,T;L²_(Ω)).

Substituting the bound of ξ_y and settingC_3,25= max{C_3,15, C_3,16, C_3,17, C_3,18}, we com- plete the rest of the proof.

Theorem 3.3.2 (L^∞(L^∞)−error estimate for the control variable). Let (y, p, u) and (Y_h, P_h, U_h) be the solutions of (3.6)−(3.10) and (3.55)−(3.59), respectively. Assume that all the conditions in Theorem 3.3.1 are valid. For each t ∈ (0, T], there exists a

CHAPTER 3. L^∞(L^∞)−A posteriori error estimates for POCP 80 positive constant C3,26 such that the following error estimate

ku−UhkL^∞(0,T;L^∞(Ω)) ≤ C3,26

h

ky0−yh,0kL^∞(Ω)+ (ln ˆhm)²

nR∞,0(yh,0,Gy⁰) +R∞,0(y_h^m,Gy^m) +R∞,0(p^m_h,Gp^m)

+

m

X

n=1

k_nRˆ∞,0

y_hⁿ−yⁿ⁻¹_h

k_n ,G_yⁿ−G_yⁿ⁻¹;T_hⁿ⁻¹,T_hⁿ +

N

X

n=m+1

k_nRˆ∞,0

pⁿ⁻¹_h −pⁿ_h

k_n ,Gpⁿ−Gpⁿ⁺¹;T_hⁿ⁻¹,Thⁿ

o

+

m

X

n=1

Z

In

kfⁿ−fk_L^∞_(Ω)ds+kn

2 kGyⁿ⁻¹−Gyⁿk_L^∞_(Ω) +

N

X

n=m+1

Z

In

kyds−y_dsⁿkL^∞(Ω)ds+k_n

2 kGpⁿ−Gpⁿ⁺¹kL^∞(Ω)

+

n∈[1,N]max X

K∈T_hⁿ

h²_K|U_h+P_hⁿ⁻¹|²_H1(K)dt1/2i

holds, where the constant C_3,26 depends on the domain Ω, the regularity constant C_R, and the constant C_3,23 as defined in Theorem 3.3.1.

Proof. Use of pointwise projection of u and U_h leads to

ku−U_hk_L^∞_(0,T;L^∞_(Ω)) = kΠ_[ua,ub](−p)−Π_[ua,ub](−P_hⁿ⁻¹)k_L^∞_(0,T;L^∞_(Ω))

≤ kp−P_hⁿ⁻¹k_L^∞_(0,T;L^∞_(Ω)).

In the above, we have used the Lipschitz continuity of Π_[ua,ub] with Lipschitz constant 1. Inviting Theorem 3.3.1, we complete the rest of the proof.

In document A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT DISCRETIZATIONS OF PARABOLIC OPTIMAL CONTROL PROBLEMS (Page 95-106)