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

( ¯∂Y_hⁿ, v_h) +a(Y_hⁿ, v_h) = (fⁿ+U_h, v_h) ∀v_h ∈V_hⁿ, (2.20)

Y_h⁰ = yh,0, (2.21)

−( ¯∂P_hⁿ, v_h) +a(P_hⁿ⁻¹, v_h) = (Y_hⁿ−yⁿ_ds, v_h) ∀v_h ∈V_hⁿ, (2.22)

P_h^N = 0, (2.23)

(U_h+P_hⁿ⁻¹, w_hⁿ−U_h) ≥ 0 ∀w_hⁿ∈U_adⁿ. (2.24) Analogous to the continuous case, we reformulate the discrete optimal control problem (2.13)−(2.14) as

Umin_h∈U_adⁿ j_hⁿ(U_h) := J(U_h, Y_h(U_h)).

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

rⁿ_y(v) := (Lⁿ_h( ¯∂y_hⁿ), v) +a(y_hⁿ, v)−(fⁿ+U_h, v), rⁿ_p(v) := (Lⁿ_h( ¯∂pⁿ_h), v) +a(pⁿ⁻¹_h , v)−(yⁿ_h −y_dsⁿ, v).

We now define the elliptic reconstructions at t = 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−yⁿ_h, v) = −rⁿ_y(v) ∀v ∈H₀¹(Ω), (2.31) a(˜pⁿ⁻¹_h −pⁿ⁻¹_h , v) = −rⁿ_p(v) + (˜y_hⁿ−y_hⁿ, v) ∀v ∈H₀¹(Ω). (2.32) Since r_yⁿ(v_h) = 0 and rⁿ_p(v_h) = 0, ∀v_h ∈ V_hⁿ, we observe that yⁿ_h denotes the elliptic projection of ˜y_hⁿ at time level t =t_n. 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) := (t_n−t)

k_n y˜_hⁿ⁻¹+ (t−tn−1)

k_n y˜_hⁿ, tn−1 ≤t≤t_n, n= 1, . . . , N.

Similarly, we define ˜p(t),t ∈[0, 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.

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

a(˜y−Y_h, v) = −r_y(v) ∀v ∈H₀¹(Ω),

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

where r_y and r_p are piecewise linear interpolant of {r_yⁿ}^N_n=1 and {rⁿ_p}^N_n=1, respectively.

Using elliptic reconstruction, we split the errors as follows:

e_y = (˜y−y(U_h))−(˜y−Y_h) =: ξ_y −η_y, e_p = (˜p−p(U_h))−(˜p−P_h) =: ξ_p−η_p. Using (2.31)−(2.32) in (2.29)−(2.30), we find that

(ξy,t, v) +a(ξy, v) = (ηy,t, v) +a(˜y−y˜_hⁿ, v) + (k⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹, v)

+ (fⁿ−f, v) ∀v ∈H₀¹(Ω), (2.33)

−(ξ_p,t, v) +a(ξ_p, v) =−(η_p,t, v) +a(˜p−p˜ⁿ⁻¹_h , v)−(k_n⁻¹(Lⁿ_h−I)pⁿ⁻¹_h , v)

−(y_dsⁿ −y_ds, v) + (˜yⁿ_h −y(U_h), v) ∀v ∈H₀¹(Ω). (2.34)

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

˜

y−y˜ⁿ_h := − (tn−t)

k_n (˜yⁿ_h −y˜_hⁿ⁻¹) and p˜−p˜ⁿ⁻¹_h := (t−tn−1)

k_n (˜pⁿ_h −p˜ⁿ⁻¹_h ).

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

k_n a(˜yⁿ_h −y˜_hⁿ⁻¹, v) + (k⁻¹_n (Lⁿ_h−I)yⁿ⁻¹_h , v)

+ (fⁿ−f, v) ∀v ∈H₀¹(Ω), (2.35)

−(ξ_p,t, v) +a(ξ_p, v) =−(η_p,t, v) + (t−tn−1)

k_n a(˜pⁿ_h−p˜ⁿ⁻¹_h , v)−(k_n⁻¹(Lⁿ_h−I)pⁿ⁻¹_h , v)

−(y_dsⁿ −y_ds, v) + (˜yⁿ_h −y(U_h), v) ∀v ∈H₀¹(Ω). (2.36) We need the following two propositions for future use (cf. [86]). 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 T_hⁿ and T_hⁿ⁻¹.

Proposition 2.3.1. Let Π_n :H₀¹(Ω)→ V_hⁿ 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^−j_n (ψ−Π_nψ)k ≤ C¯_1,j|ψ|_j, (2.37) and

kh^1/2−j_n (ψ−Π_nψ)k_Σn ≤ C¯_2,j|ψ|_j. (2.38) Proposition 2.3.2. LetΠˆ_n:X →V_hⁿ∩V_hⁿ⁻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 V_hⁿ∩V_hⁿ⁻¹ with mesh size ˆh_n:= max{h_n, hn−1}.

Then the following inequality holds:

kˆh^1/2−j_n (ψ−Πˆ_nψ)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 T_hⁿ⁻¹ to T_hⁿ. Further, the approximation properties (2.37) and (2.38) hold true in the finite element space V_hⁿ∩V_hⁿ⁻¹ with ˆh_n replacing h_n.

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 (˜y_hⁿ,p˜ⁿ⁻¹_h )∈H₀¹(Ω)×H₀¹(Ω) satisfy (2.31)−(2.32). Then, 0≤n≤N, the following estimate holds:

k˜y_hⁿ−yⁿ_hk ≤ C¯_1,2h²_nk(Aⁿ_h− A_el)y_hⁿk+ ¯C_2,2h^3/2_n kJ₁[yⁿ_h]k_Σn. (2.39) For n∈[1 :N], we have

kp˜ⁿ⁻¹_h −pⁿ⁻¹_h k ≤ C¯_1,2h²_nk(Aⁿ_h− A_el)pⁿ⁻¹_h k+ ¯C_2,2h^3/2_n kJ₁[pⁿ⁻¹_h ]k_Σn+k˜y_hⁿ−yⁿ_hk, (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)) = (˜yⁿ_h −y_hⁿ, v) ∀v ∈H₀¹(Ω), a.e. t∈[0, T], (2.41) Further, the solution w of (2.41) satisfies the following regularity result

kwk₂ ≤ C_Rk˜yⁿ_h −y_hⁿk, (2.42)

where C_R denotes to the regularity constant. Setting v = ˜yⁿ_h −y_hⁿ ∈ V_hⁿ in (2.41) and use of (2.31), (2.11) and (2.12) yields

k˜yⁿ_h −y_hⁿk² = a(˜y_hⁿ−y_hⁿ, w)

= a(˜y_hⁿ−y_hⁿ, w−Π_nw)

= ((Aⁿ_h− A_el)y_hⁿ, w−Π_nw)−(J₁[yⁿ_h], w−Π_nw)_Σn.

An application of the Cauchy-Schwarz inequality, Proposition 2.3.1 and (2.42) implies k˜yⁿ_h −y_hⁿk ≤ C¯_1,2h²_hk(Aⁿ_h− A_el)y_hⁿk+ ¯C_2,2h

3

n2kJ₁[yⁿ_h]k, which proves (2.39).

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

we obtain

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

= a(˜pⁿ⁻¹_h −pⁿ⁻¹_h , w−Π_nw) +a(˜pⁿ⁻¹_h −pⁿ⁻¹_h ,Π_nw),

= (Aⁿ_hpⁿ⁻¹_h − A_elpⁿ⁻¹_h , w−Π_nw) + (˜y_hⁿ−yⁿ_h,Π_nw).

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⁻¹_h − pⁿ⁻¹_h yields

kp˜ⁿ⁻¹_h −pⁿ⁻¹_h k ≤ C¯_1,2h²_nk(Aⁿ_h− A_el)pⁿ⁻¹_h k+ ¯C_2,2h^3/2_n kJ₁[pⁿ⁻¹_h ]k_Σn+k˜y_hⁿ−yⁿ_hk, and this completes the proof.

In the following lemma, we derive the bound forξ_y.

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

max

t∈[0,tm]kξ_y(t)k²+ 2α₁ Z tm

0

kξ_yk²₁dt¹₂

≤ kξ_y(0)k+ 4

E1,m² +E2,m²

¹₂ , where

E1,m :=

m

X

n=1

kn β1,n+β2,n+β4,n

, and

E2,m :=

m

X

n=1

β_3,n² k_n.

Here, for n∈[1 :N], β_1,n is the space error estimator and is defined by β_1,n :=

C¯_1,2kˆh²_n∂R¯ ⁿ_yk+ ¯C_2,2khˆ^3/2_n ∂J¯ _yⁿkΣˆn+ ¯C_3,2kˆh^3/2_n ∂J¯ _yⁿkΣˇn\Σˆn

, β_2,n is the temporal error estimator defined by

β_2,n := kk_n∂¯r˜_yⁿk,

and the data approximation error estimators are defined by β3,n := ¯C3,1khnk⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹k, β_4,n := 1

k_n 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 =ξ_y in (2.35) and using the coercive property of the bilinear form, we obtain

1 2

d

dtkξ_yk²+α₁kξ_yk²₁ ≤ (η_y,t, ξ_y)− (t_n−t) kn

a(˜y_hⁿ−y˜_hⁿ⁻¹, ξ_y)

+ (k_n⁻¹(Lⁿ_h −I)y_hⁿ⁻¹, ξ_y) + (fⁿ−f, ξ_y). (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

2kξ_y(t_m)k² −1

2kξ_y(0)k²+α₁ Z tm

0

kξ_yk²₁dt

≤

m

X

n=1

Z tn

tn−1

|(η_y,t, ξ_y)|+|a(˜y_hⁿ−y˜_hⁿ⁻¹, ξ_y)|+|(k⁻¹_n (Lⁿ_h−I)y_hⁿ⁻¹, ξ_y)|

+|(fⁿ−f, ξ_y)| dt =:

m

X

n=1

ζ_n¹+ζ_n²+ζ_n³+ζ_n⁴

=ζ_m. (2.44)

Since ξ_y(t) is continuous in [0, t_m], there exists t_m^∗ ∈[0, t_m] for which

t∈[0,tmaxm]kξy(t)k = kξy(tm^∗)k =: kξ_y^m∗k. (2.45) Integrate (2.43) with respect to time from 0 to t_m^∗ to have

1

2kξ_y(t_m^∗)k²− 1

2kξ_y(0)k²+α₁ Z t_m∗

0

kξ_yk²₁dt

≤

Z tm∗

0

{|(η_y,t, ξ_y)|+|a(˜y_hⁿ−y˜_hⁿ⁻¹, ξ_y)|

+|(k⁻¹_n (Lⁿ_h−I)y_hⁿ⁻¹, ξ_y)|+|(fⁿ−f, ξ_y)|}dt

≤

m

X

n=1

Z tn

tn−1

{|(η_y,t, ξ_y)|+|a(˜yⁿ_h −y˜_hⁿ⁻¹, ξ_y)|

+|(k⁻¹_n (Lⁿ_h−I)y_hⁿ⁻¹, ξ_y)|+|(fⁿ−f, ξ_y)|}dt

=:

m

X

n=1

ζ_n¹+ζ_n²+ζ_n³+ζ_n⁴

=: ζm. In view of (2.45), we have

1

2kξ_y^m∗k²− 1

2kξy(0)k²+α1

Z t_m∗

0

kξyk²₁dt ≤ ζm. (2.46) Combining (2.44) and (2.46), it now leads to

1

2kξ^m_y ^∗k²+α₁ Z tm

0

kξ_yk²₁dt ≤ kξ_y(0)k²+ 2ζ_m. (2.47) Now, we estimate ζ_nⁱ, i = 1, . . . ,4. To estimate the term ζ_n¹, 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

ζ_n¹ = Z tn

tn−1

|(η_y,t, ξ_y)|dt = k_n⁻¹ Z tn

tn−1

|(˜y_hⁿ−y˜ⁿ⁻¹_h −yⁿ_h +y_hⁿ⁻¹, ξ_y)|dt.

CHAPTER 2. L^∞(L²)−A posteriori error estimates for POCP 37 Since ˜yⁿ_h−yⁿ_h is orthogonal toV_hⁿwith respect toa(·,·), the first term inside the brackets is orthogonal to V_hⁿ∩V_hⁿ⁻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 (˜yⁿ_h −y˜_hⁿ⁻¹−y_hⁿ+y_hⁿ⁻¹, ξ_y) = a(˜y_hⁿ−y˜_hⁿ⁻¹−yⁿ_h +y_hⁿ⁻¹, ψ(t))

= a(˜y_hⁿ−y˜_hⁿ⁻¹−yⁿ_h +y_hⁿ⁻¹, ψ(t)−Πˆ_nψ(t)).

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

(˜y_hⁿ−y˜ⁿ⁻¹_h −y_hⁿ+yⁿ⁻¹_h , ξ_y) = (Rⁿ_y −R_yⁿ⁻¹, ψ(t)−Πˆ_nψ(t))

+(J_yⁿ−J_yⁿ⁻¹, ψ(t)−Πˆ_nψ(t))Σˇn, (2.48) where

Rⁿ_y := Lⁿ_h( ¯∂yⁿ_h)−div(∇y_hⁿ)−fⁿ−Uh and J_yⁿ =J1[yⁿ_h].

Equation (2.48) can be expressed as

(˜y_hⁿ−y˜ⁿ⁻¹_h −y_hⁿ+yⁿ⁻¹_h , ξ_y) = k_n( ¯∂Rⁿ_y, ψ(t)−Πˆ_nψ(t)) +k_n( ¯∂J_yⁿ, ψ(t)−Πˆ_nψ(t))Σˇn. By Proposition 2.3.2, it follows that

ζ_n¹ ≤ Z tn

tn−1

|ψ(t)|₂dt

×

C¯_1,2kˆh²_n∂R¯ ⁿ_yk+ ¯C_2,2kˆh^3/2_n ∂J¯ _yⁿk_Σˆ

n+ ¯C_3,2kˆh^3/2_n ∂J¯ _yⁿk_Σˇ

n\Σˆn

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

ζ_n¹ ≤ max

t∈I_n kξ_y(t)k

C¯_1,2khˆ²_n∂R¯ ⁿ_yk+ ¯C_2,2khˆ^3/2_n ∂J¯ _yⁿkΣˆn + ¯C_3,2kˆh^3/2_n ∂J¯ _yⁿkΣˇn\Σˆn

k_n, and hence,

m

X

n=1

ζ_n¹ ≤ kξ_y^m∗k

m

X

n=1

k_nβ_1,n.

In order to estimate ζ_n², which accounts for the time discretization error, we use the definition of elliptic reconstruction (2.31) to obtain

ζ_n² = Z tn

tn−1

|a(˜yⁿ_h −y˜_hⁿ⁻¹, ξ_y)|dt. (2.49)

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

a(˜y_hⁿ, v) = −r˜_yⁿ(v), (2.50)

where ˜rⁿ_y(v) = (Lⁿ_h( ¯∂y_hⁿ), v)−(fⁿ+Uh, v). Altogether (2.50) with (2.49) yields ζ_n² =

Z tn

tn−1

|(k_n∂¯˜r_yⁿ, ξ_y)|dt ≤ Z tn

tn−1

kk_n∂¯r˜_yⁿkkξ_ykdt, and hence,

m

X

n=1

ζ_n² ≤ kξ^m_y ^∗k

m

X

n=1

knβ2,n.

We now bound the term ζ_n³. Since V_hⁿ⊂ker(Lⁿ_h−I), we have ζ_n³ =

Z tn

tn−1

|(k⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹, ξ_y)|dt

= Z tn

tn−1

|(k⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹, ξ_y −Π_nξ_y)|dt

= Z tn

tn−1

|(h_nk_n⁻¹(Lⁿ_h−I)y_hⁿ⁻¹, h⁻¹_n (ξ_y−Π_nξ_y))|dt

≤ Z tn

tn−1

kh_nk⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹kkh⁻¹_n (ξ_y −Π_nξ_y)kdt

≤ C_1,1k^1/2_n kh_nk⁻¹_n (Lⁿ_h −I)y_hⁿ⁻¹kZ tn

tn−1

kξ_yk²₁dt¹₂ . Thus, we have

m

X

n=1

ζ_n³ ≤

m

X

n=1

β_3,nk_n^1/2Z tn

tn−1

kξ_yk²₁dt¹₂ . To estimate ζ_n⁴, we first note that

ζ_n⁴ ≤ maxt∈In

kξ_y(t)kZ tn

tn−1

kfⁿ−fkdt, and hence,

m

X

n=1

ζ_n⁴ ≤ kξ_y^m∗k

m

X

n=1

Z tn

tn−1

kfⁿ−fkdt.

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

2kξ^m_y ^∗k²+α₁ Z tm

0

kξ_yk²₁dt ≤ 1

2kξ_y(0)k²+ 2kξ_y^m∗k

m

X

n=1

k_n β_1,n+β_2,n+β_4,n + 2

m

X

n=1

β_3,nk_n^1/2Z tn

tn−1

kξ_yk²₁ dt1/2

.

CHAPTER 2. L^∞(L²)−A posteriori error estimates for POCP 39 Now, we use a standard inequality due to [48]. 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

√2kξ_y^m∗k, a_n := Z tn

tn−1

kξ_yk²₁dt1/2

, ˆc := 1

√2kξ_y(0)k, b₀ := 2√

2

m

X

n=1

k_n β_1,n+β_2,n+β_4,n

, b_n := 2β_3,nk_n^1/2, 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 ξ_p satisfy (2.36). Then the following estimate holds, 0≤m≤N:

t∈[tmaxm,T]kξ_p(t)k² + 2α₁ Z T

tm

kξ_pk²₁dt1/2

≤4

E3,m² +E4,m²

¹₂ +

Z T tm

kξ_ykdt, where

E3,m :=

N

X

n=m+1

kn δ1,n+δ2,n+δ4,n

, and

E4,m :=

N

X

n=m+1

δ²_3,nkn.

In the above, for n ∈[0 :N], δ1,n represents the space error and is defined by δ_1,n :=

C¯_1,2khˆ²_n∂R¯ ⁿ⁺¹_p k+ ¯C_2,2khˆ^3/2_n ∂J¯ _pⁿkΣˆn+ ¯C_3,2kˆh^3/2_n ∂J¯ _pⁿkΣˇn\Σˆn

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

δ_2,n := kk_n( ¯∂r˜ⁿ_p + ¯∂η_yⁿ⁺¹)k,

and the data approximation error estimators are defined by δ_3,n := ¯C_3,1kh_nk_n⁻¹(Lⁿ_h−I)pⁿ⁻¹_h k, δ_4,n := 1

k_n Z tn

tn−1

kyⁿ_ds−y_dskdt.

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

dtkξ_pk²+α₁kξ_pk²₁

≤ −(η_p,t, ξ_p) + (t−tn−1)

k_n a(˜pⁿ_h−p˜ⁿ⁻¹_h , ξ_p)−(k_n⁻¹(Lⁿ_h−I)pⁿ⁻¹_h , ξ_p)

−(yⁿ_ds−yds, ξp) + (˜y_hⁿ−y(Uh), ξp). (2.51) Integrating (2.51) with respect to time fromt_m toT, we find that

1

2kξ_p(t_m)k²+α₁ Z T

tm

kξ_pk²₁dt

≤ 1

2kξ_p(T)k²+ Z T

tm

n−(η_p,t, ξ_p) + (t−tn−1)

k_n a(˜pⁿ_h −p˜ⁿ⁻¹_h , ξ_p)

−(k⁻¹_n (Lⁿ_h −I)pⁿ⁻¹_h , ξ_p)−(y_dsⁿ −y_ds, ξ_p) + (˜y_hⁿ−y(U_h), ξ_p) o

dt

≤

N

X

n=m+1

Z tn

tn−1

n

|(η_p,t, ξ_p)|+|a(˜pⁿ_h−p˜ⁿ⁻¹_h ), ξ_p)|

+|(k_n⁻¹(Lⁿ_h−I)pⁿ⁻¹_h , ξ_p)|+|(y_dsⁿ −y_ds, ξ_p)|+|(˜yⁿ_h −y(U_h), ξ_p)|o dt

=:

N

X

n=m+1

n

ϑ¹_n+ϑ²_n+ϑ³_n+ϑ⁴_n+ϑ⁵_no

=: Θ_m. (2.52)

Since ξ_p is a continuous in [t_m, T], there exists t_m^∗ ∈[t_m, T] such that max

t∈[tm,T]kξ_pk = kξ_p(t_m^∗)k =: kξ_p^m∗k.

As Lemma 2.3.2, we deduce that 1

2kξ^m_p ^∗k²+α₁ Z T

tm∗

kξ_pk²₁dt ≤ Θ_m. (2.53)

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

2kξ_p^m∗k²+α₁ Z T

t_m^∗

kξ_pk²₁dt ≤ 2Θ_m. (2.54)

Now we estimate each of the summands ϑⁱ_n|_i=1,...,5 appearing on the right hand side of (2.54). To estimate the term ϑ¹_n, we write

ϑ¹_n = Z tn

tn−1

|(η_p,t, ξ_p)|dt = k_n⁻¹ Z tn

tn−1

|(˜pⁿ_h−p˜ⁿ⁻¹_h −pⁿ_h +pⁿ⁻¹_h , ξ_p)|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

(˜pⁿ_h −p˜ⁿ⁻¹_h −pⁿ_h+pⁿ⁻¹_h , ξ_p) = a(˜pⁿ_h −p˜ⁿ⁻¹_h −pⁿ_h+pⁿ⁻¹_h ,Ψ)

= a(˜pⁿ_h −p˜ⁿ⁻¹_h −pⁿ_h+pⁿ⁻¹_h ,Ψ−Πˆ_nΨ) +a(˜pⁿ_h−p˜ⁿ⁻¹_h −pⁿ_h+pⁿ⁻¹_h ,Πˆ_nΨ).

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

(˜pⁿ_h −p˜ⁿ⁻¹_h −pⁿ_h+pⁿ⁻¹_h , ξ_p) = −k_n( ¯∂Rⁿ⁺¹_p ,Ψ−Πˆ_nΨ)−k_n( ¯∂J_pⁿ,Ψ−Πˆ_nΨ)Σˇn

+k_n( ¯∂η_yⁿ⁺¹,Ψ), where

Rⁿ_p :=−Lⁿ_h( ¯∂pⁿ_h)−div(∇pⁿ⁻¹_h )−(yⁿ_h −y_dsⁿ), J_pⁿ := J₁[pⁿ_h].

An application of Proposition 2.3.2 yields ϑ¹_n ≤ Z tn

tn−1

|Ψ|₂dt

×

C¯_1,2kˆh²_n∂R¯ ⁿ⁺¹_p k+ ¯C_2,2kˆh^3/2_n ∂J¯ _pⁿk_Σˆ

n + ¯C_3,2kˆh^3/2_n ∂J¯ _pⁿk_Σˇ

n\Σˆn

+k_nk∂η¯ _yⁿ⁺¹kkΨk.

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

N

X

n=m+1

ϑ¹_n ≤ kξ^m_p ^∗k

N

X

n=m+1

k_nδ_1,n.

To estimate ϑ²_n, use (2.32) together with the definition of rⁿ_p to have

N

X

n=m+1

ϑ²_n ≤ kξ^m_p ^∗k

N

X

n=m+1

k_nkk_n( ¯∂r˜_pⁿ+ ¯∂ηⁿ⁺¹_y )k = kξ^m_p ^∗k

N

X

n=m+1

k_nδ_2,n, where

˜

r_pⁿ :=−Lⁿ_h( ¯∂pⁿ_h)−(y_hⁿ−yⁿ_ds).

Now, we estimate ϑ³_n and ϑ⁴_n. Arguing as in Lemma 2.3.2, we arrive at

N

X

n=m+1

ϑ³_n ≤

N

X

n=m+1

δ_3,nk^1/2_n Z tn

tn−1

kξ_pk²₁dt¹₂ .

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

ϑ⁴_n ≤ maxt∈In

kξ_p(t)kZ tn

tn−1

ky_dsⁿ −y_dskdt, and hence

N

X

n=m+1

ϑ⁴_n ≤ kξ_p^m∗k X^N

n=m+1

Z tn

tn−1

ky_dsⁿ −y_dskdt . Finally, for ϑ⁵_n, we find that

N

X

n=m+1

ϑ⁵_n ≤ kξ_p^m∗k X^N

n=m+1

Z tn

tn−1

kξ_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 (Y_h, P_h, U_h) and (y(U_h), p(U_h)), respectively, be the solutions of (2.20)−(2.24) and (2.25)−(2.28) with uˆ=U_h. 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]kP_h(t)−p(U_h)(t)k ≤ 2 max

n∈[m:N]Fp,n+ 4

E3,n² +E4,n²

1/2

+ Z T

tm

kξ_ykdt, (2.56) with

Fy,n := ¯C_1,2h²_nk(Aⁿ_h−A_el)y_hⁿk+ ¯C_2,2h^3/2_n kJ₁[y_hⁿ]k_Σn,

Fp,n := ¯C_1,2h²_nk(Aⁿ_h −A_el)pⁿ_hk+ ¯C_2,2h^3/2_n kJ₁[pⁿ⁻¹_h ]k_Σn+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

ke_y(t)k ≤ kξ_y(t)k+kη_y(t)k, t ∈I_n

≤ max

t∈[0,t_m]kξ_y(t)k+ max

t∈[0,t_m]kη_y(t)k. (2.57)

Fort ∈In, we note that kη_y(t)k =

(t_n−t)

k_n η_yⁿ⁻¹+ (t−tn−1) k_n η_yⁿ

≤ kη_yⁿ⁻¹k+kη_yⁿk ≤2 max

n∈[1,m]

kηⁿ⁻¹_y k, kη_yⁿk .

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

kη_y(t)k ≤ 2 max

n∈[0,m]kηⁿ_yk ≤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 (Y_h, P_h, U_h) 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:

Y_h−y = (Y_h−y(U_h)) + (y(U_h)−y) =: e_y−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:

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

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

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

k˜e_y(t_m)k²+ Z tm

0

k˜e_yk²₁dt ≤ C_2,1 Z tm

0

ku−U_hk²_L2(Ω)dt, (2.61) and

k˜e_p(t_m)k²+ Z T

tm

k˜e_pk²₁dt ≤ C_2,2 Z T

tm

ku−U_hk²_L2(Ω)dt, (2.62) where C_2,1 andC_2,2 are the positive constants depend on the Poincar´e inequality constant C and the coercivity constant α₁.

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

2 d

dtk˜eyk²+α1k˜eyk²₁ ≤ |(u−Uh,e˜y)|.

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

2 d

dtk˜eyk²+α1k˜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˜e_y(t_m)k²+ 2α₁ Z tm

0

k˜e_yk²₁dt ≤ Z tm

0

ku−U_hk²dt+ Z tm

0

k˜e_yk²dt.

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

To prove (2.62), choosev = ˜e_pin (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˜e_pk²+α₁k˜e_pk²₁ ≤ 1

2k˜e_yk² +1

2k˜e_pk². (2.63) Integrating (2.63) from tm to T and using the fact that ˜ep|t=T = 0, it follows that

k˜e_p(t_m)k²+ 2α₁ Z T

tm

k˜e_pk²₁dt ≤ Z T

tm

k˜e_yk²dt+ Z T

tm

k˜e_pk²dt.

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 thatU_adⁿ ⊂Uad, (Uh+P_hⁿ⁻¹)|K ∈ H¹(K) and w_h ∈U_adⁿ, and there exists a positive constant C_2,3 such that

Z T 0

(U_h+P_hⁿ⁻¹, w_h−u)dt

≤ C_2,3 Z T

0

X

K∈Thⁿ

h_K|U_h+P_hⁿ⁻¹|_H¹_(K)ku−U_hk_L²_(K)dt.

(2.64) Then, we have

ku−U_hk²_L2(0,T;L²(Ω)) ≤ C_2,4Z T 0

X

K∈Thⁿ

h²_K|U_h+P_hⁿ⁻¹|²_H1(K)dt+kP_hⁿ⁻¹−p(U_h)k²_L2(0,T;L²(Ω))

,

where C_2,4 = ³₂max{C_2,3² ,1}, and (y(U_h), p(U_h)) is the solution of (2.25)−(2.28) with ˆ

u=U_h.

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

(u, u−U_h) ≤ −(p, u−U_h),

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

ku−U_hk²_L2(0,T;L²(Ω)) = Z T

0

(u−U_h, u−U_h)dt

≤ Z T

0

{−(p, u−U_h)−(U_h, u−U_h)}dt

= −

Z T 0

(P_hⁿ⁻¹+U_h, u−w_h)dt− Z T

0

(U_h+P_hⁿ⁻¹, w_h−U_h)dt +

Z T 0

(P_hⁿ⁻¹−p(U_h), u−U_h)dt+ Z T

0

(p(U_h)−p, u−U_h)dt.

With an aid of (2.24), we get ku−U_hk²_L2(0,T;L²(Ω)) ≤

Z T 0

(U_h+P_hⁿ⁻¹, w_h−u)dt+ Z T

0

(P_hⁿ⁻¹−p(U_h), u−U_h)dt +

Z T 0

(p(U_h)−p, u−U_h)dt

=: E1+E2+E3. (2.65)

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

|E₁| =

Z T 0

(U_h+P_hⁿ⁻¹, w_h−u)dt

≤ C_2,3 Z T

0

X

K∈T_hⁿ

h_K|U_h +P_hⁿ⁻¹|_H¹_(K)ku−U_hk_L²_(K)dt

≤ 3C_2,3² 4

Z T 0

X

K∈Thⁿ

h²_K|U_h+P_hⁿ⁻¹|²_H1(K)dt +1

4ku−U_hk²_L2(0,T;L²(Ω)). (2.66) The term E₂ is bounded as

|E₂| =

Z T 0

(P_hⁿ⁻¹−p(U_h), u−U_h)dt

≤ 3

4kP_hⁿ⁻¹−p(Uh)k²_L2(0,T;L²(Ω))+1

4ku−Uhk²_L2(0,T;L²(Ω)). (2.67) It now remains to estimate the term E₃. For this, we first note that

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

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

Z T 0

(u−U_h, p(U_h)−p)dt

= Z T

0

−(y−y(U_h), p_t(U_h)−p_t) +a(y−y(U_h), p(U_h)−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(U_h)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., [57]).

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]kY_h(t)−y(t)k ≤ k˜y⁰_h−y_h,0k+ 2 max

n∈[0:m]Fy,n+ 4

E_1,n² +E_2,n² 1/2

+Z tm

0

ku−U_hk²dt¹₂ , max

[tm,T]kP_h(t)−p(t)k ≤ 2 max

n∈[1:m]Fp,n+ 4

E3,n² +E4,n²

1/2

+ Z T

tm

kξ_ykdt +Z T

tm

ku−U_hk²dt¹₂ , and

max

[0,tm]ku(t)−Uh(t)k ≤ C2,4

Z T 0

X

K∈Thⁿ

h²_K|Uh+P_hⁿ⁻¹|²_H1(K)dt 1/2

+ 2 max

n∈[1:m]Fp,n+ 4

E3,n² +E4,n²

1/2

+ Z T

tm

kξ_ykdt hold, where

Fy,n := ¯C_1,2h²_nk(Aⁿ_h −A_el)y_hⁿk+ ¯C_2,2h^3/2_n kJ₁[y_hⁿ]k_Σn,

Fp,n := ¯C_1,2h²_nk(Aⁿ_h −A_el)pⁿ_hk+ ¯C_2,2h^3/2_n kJ₁[pⁿ⁻¹_h ]k_Σn+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. [96]). Similarly, we have

U_h = Π_[ua,ub] −P_hⁿ⁻¹

. (2.70)

Using (2.69) and (2.70), we obtain

ku−U_hk_L^∞_(0,T;L²_(Ω)) ≤ kp−P_hⁿ⁻¹k_L^∞_(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.

In document A POSTERIORI ERROR ESTIMATES FOR FINITE ELEMENT DISCRETIZATIONS OF PARABOLIC OPTIMAL CONTROL PROBLEMS (Page 57-73)