A Priori Error Analysis of the Petrov Galerkin Crank Nicolson ...

A Priori Error Analysis of the 

Petrov Galerkin Crank Nicolson Scheme 

for Parabolic Optimal Control Problems 

Dominik Meidner and Boris Vexler 

Fakultät für Mathematik 

Technische Universität München 

October 10 - 14, 2011 

Boris Vexler Crank Nicolson for parabolic optimal control problems October 10 - 14, 2011 1

Optimal control problem 

Cost functional 

Minimize J(q, u) = 1 2 

State equation 

∫ T 

0 

∫ 

Ω(u(t, x) − û(t, x)) 2 dx dt + α 2 

∂ t u − ∆u = f + Gq in (0, T ) × Ω, 

u = 0 

u = u 0 

on (0, T ) × ∂Ω, 

in {0} × Ω 

with G : Q = L 2 (0, T ; R D ) → L 2 (0, T ; H 1 (Ω)) and 

(Gq)(t, x) = ∑ D 

i=1 q i(t)g i (x), g i ∈ V = H 1 0 (Ω). 

Control constraints 

q a ≤ q(t) ≤ q b a. e. in (0, T ). 

∫ T 

0 

|q(t)| 2 dt 






∫ T 

0 

∫ 

Ω(u(t, x) − û(t, x)) 2 dx dt + α 2 

∂ t u − ∆u = f + Gq in (0, T ) × Ω, 

u = 0 

u = u 0 

on (0, T ) × ∂Ω, 

in {0} × Ω 


(Gq)(t, x) = ∑ D 

i=1 q i(t)g i (x), g i ∈ V = H 1 0 (Ω). 


q a ≤ q(t) ≤ q b a. e. in (0, T ). 

∫ T 

0 

|q(t)| 2 dt 






∫ T 

0 

∫ 

Ω(u(t, x) − û(t, x)) 2 dx dt + α 2 

∂ t u − ∆u = f + Gq in (0, T ) × Ω, 

u = 0 

u = u 0 

on (0, T ) × ∂Ω, 

in {0} × Ω 


(Gq)(t, x) = ∑ D 

i=1 q i(t)g i (x), g i ∈ V = H 1 0 (Ω). 


q a ≤ q(t) ≤ q b a. e. in (0, T ). 

∫ T 

0 

|q(t)| 2 dt 


A priori error analysis 

Temporal discretization of the state 

→ discretization parameter k 

Spatial discretization of the state 

→ discretization parameter h 

Treatment of the control 

→ Goal: 

Error estimate 

‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 

→ optimal control ¯q is not smooth (due to control constraints) 

→ Crank Nicolson scheme is of second order in k 

→ Adjoint scheme 








→ Goal: 


‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 











→ Goal: 


‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 











→ Goal: 


‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 











→ Goal: 


‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 





Existing literature 

Error estimates for parabolic problems without constraints: 

Gunzburger, Hou, Hinze, Deckelnick, Chrysafinos, Winther, . . . 

Apel and Flaig 2011 (Crank-Nicolson) 

Error estimates for parabolic problems with control constraints 

Lasiecka and Malanowski 1977/78, Malanowski 1981 

Rösch 2004 

Meidner and Vexler 2008 

Neitzel and Vexler 2011 (semilinear) 

Meidner and Vexler 2011 (Petrov-Galerkin-Crank-Nicolson) → O(k 2 + h 2 ) 

Error estimates for parabolic problems with state constraints 

Deckelnick and Hinze 2010 

Meidner, Rannacher, and Vexler 2010 








Rösch 2004 














Rösch 2004 














Rösch 2004 








Optimality system and regularity 

Optimality system 

∂ t ū − ∆ū = f + G ¯q, u(0) = u 0 

−∂ t ¯z − ∆¯z = ū − û, z(T ) = 0 

(α¯q + G ∗¯z, δq − ¯q) ≥ 0 

∀δq ∈ Q ad 

→ ¯q = P Qad 

( 

− 

1 

α G ∗¯z ) with the pointwise projection P Qad : Q → Q ad 

Assumption 1 

Regularity 

Ω is polygonal and convex 

f , û ∈ H 1 (0, T ; L 2 (Ω)), f (0), û(T ) ∈ H 1 0 (Ω) u 0 , ∆u 0 ∈ H 1 0 (Ω) 

¯q ∈ W 1,∞ (0, T ; R D ), 

ū, ¯z ∈ H 1 (0, T ; H 2 (Ω)) ∩ H 2 (0, T ; L 2 (Ω)) 




∂ t ū − ∆ū = f + G ¯q, u(0) = u 0 

−∂ t ¯z − ∆¯z = ū − û, z(T ) = 0 

(α¯q + G ∗¯z, δq − ¯q) ≥ 0 


→ ¯q = P Qad 

( 

− 

1 


Assumption 1 

Regularity 



¯q ∈ W 1,∞ (0, T ; R D ), 

ū, ¯z ∈ H 1 (0, T ; H 2 (Ω)) ∩ H 2 (0, T ; L 2 (Ω)) 




∂ t ū − ∆ū = f + G ¯q, u(0) = u 0 

−∂ t ¯z − ∆¯z = ū − û, z(T ) = 0 

(α¯q + G ∗¯z, δq − ¯q) ≥ 0 


→ ¯q = P Qad 

( 

− 

1 


Assumption 1 

Regularity 



¯q ∈ W 1,∞ (0, T ; R D ), 

ū, ¯z ∈ H 1 (0, T ; H 2 (Ω)) ∩ H 2 (0, T ; L 2 (Ω)) 


Temporal discretization 

→ cG(1) Petrov-Galerkin method 

Partitioning of the time interval Ī = [0, T ]: 

Ī = {0} ∪ I 1 ∪ I 2 ∪ · · · ∪ I M 

with subintervals I m = (t m−1, t m] of size k m and time points 

Ansatz space (continuous) 

X 1 k = 

0 = t 0 < t 1 < · · · < t M−1 < t M = T 

{ 

v k ∈ C(Ī , V ) ∣ ∣∣ 

vk 

∣ ∣ 

I m 

∈ P 1(I m, V ), m = 1, 2, . . . , M 

Test space (discontinuous) 

{ 

˜X k 0 = v k ∈ L 2 ∣ } 

(I , V ) ∣ v k ∈ P 

I m 0(I m, V ), m = 1, 2, . . . , M, v k (0) ∈ V 

} 

, 





Ī = {0} ∪ I 1 ∪ I 2 ∪ · · · ∪ I M 



X 1 k = 

0 = t 0 < t 1 < · · · < t M−1 < t M = T 

{ 

v k ∈ C(Ī , V ) ∣ ∣∣ 

vk 

∣ ∣ 

I m 

∈ P 1(I m, V ), m = 1, 2, . . . , M 


{ 

˜X k 0 = v k ∈ L 2 ∣ } 

(I , V ) ∣ v k ∈ P 

I m 0(I m, V ), m = 1, 2, . . . , M, v k (0) ∈ V 

} 

, 





Ī = {0} ∪ I 1 ∪ I 2 ∪ · · · ∪ I M 



X 1 k = 

0 = t 0 < t 1 < · · · < t M−1 < t M = T 

{ 

v k ∈ C(Ī , V ) ∣ ∣∣ 

vk 

∣ ∣ 

I m 

∈ P 1(I m, V ), m = 1, 2, . . . , M 


{ 

˜X k 0 = v k ∈ L 2 ∣ } 

(I , V ) ∣ v k ∈ P 

I m 0(I m, V ), m = 1, 2, . . . , M, v k (0) ∈ V 

} 

, 



Bilinear form: 

B(u k , φ) := (∂ t u k , φ) I ×Ω + (∇u k , ∇φ) I ×Ω + (u k,0 , φ − 0 ). 

Temporal discretization of the state: 

u k ∈ Xk 1 : B(u k , φ) = (f + Gq, φ) I ×Ω + (u 0 , φ − 0 ) ∀φ ∈ ˜X k 0 . 

Temporal discretization of the adjoint state: 

z k ∈ ˜X k 0 : B(φ, z k ) = (u k − û, φ) I ×Ω ∀φ ∈ Xk 1 . 

→ Discrete adjoint state is piecewise constant. 






























cG(1) discretization of the state is a variant of the Crank Nicolson scheme 

→ second order convergence 

Consistent discretization of the adjoint state → piecewise constants 

→ only first order convergence expected 

→ Variational discretization (no control discretization) 

→ only first order convergence 

‖¯q − ¯q k ‖ Q = O(k). 

→ How to achieve O(k 2 ) 









‖¯q − ¯q k ‖ Q = O(k). 










‖¯q − ¯q k ‖ Q = O(k). 










‖¯q − ¯q k ‖ Q = O(k). 




Adjoint discretization 

z k 

Piecewise linear interpolation at the midpoints 

→ Superconvergence result (Meidner & Vexler 2011) 

‖z − π k z k ‖ L2 (I ×Ω) = O(k 2 ) 




z k 


z k 

π k z k 


‖z − π k z k ‖ L2 (I ×Ω) = O(k 2 ) 




z k 


z k 

π k z k 


‖z − π k z k ‖ L2 (I ×Ω) = O(k 2 ) 


Superconvergence result 

Theorem (Meidner & Vexler 2011) 

Let z be solution of 

−∂ t z − ∆z = g, z(T ) = 0 

with g ∈ H 1 (0, T ; L 2 (Ω)), g(T ) ∈ H 1 0 (Ω) and z k ∈ ˜X 0 k 

B(φ, z k ) = (g, φ) I ×Ω ∀φ ∈ X 1 k . 

be defined by 

Then there holds 

‖z − π k z k ‖ L2 (I ×Ω) ≤ c k ( ) 

2 ‖∂t 2 z‖ L2 (I ×Ω) + ‖∂ t ∆z‖ L2 (I ×Ω) 


Superconvergence result: Proof 

Step 1: Semidiscrete stability estimates 

→ diagonal testing is not possible! 

‖∂ t v k ‖ I ×Ω + ‖P k ∆v k ‖ I ×Ω + ‖∇v k ‖ I ×Ω ≤ c‖f ‖ I ×Ω 

with L 2 -Projection P k : L 2 (0, T ; L 2 (Ω)) → ˜X 

k 0. 

Step 2: Supercloseness for the midpoint interpolation 

→ duality arguments & stability estimates (step 1) 

‖Π k z − z k ‖ L2 (I ×Ω) ≤ c k ( ) 

2 ‖∂t 2 z‖ L2 (I ×Ω) + ‖∂ t ∆z‖ L2 (I ×Ω) 

Step 3: Stability of π k in L 2 (I × Ω) for semidiscrete functions 

→ π k is in general not stable w.r.t. L 2 (I × Ω), but 

‖π k y k ‖ I ×Ω ≤ c‖y k ‖ I ×Ω 

∀y k ∈ ˜X 0 k 







k 0. 



‖Π k z − z k ‖ L2 (I ×Ω) ≤ c k ( ) 

2 ‖∂t 2 z‖ L2 (I ×Ω) + ‖∂ t ∆z‖ L2 (I ×Ω) 



‖π k y k ‖ I ×Ω ≤ c‖y k ‖ I ×Ω 

∀y k ∈ ˜X 0 k 







k 0. 



‖Π k z − z k ‖ L2 (I ×Ω) ≤ c k ( ) 

2 ‖∂t 2 z‖ L2 (I ×Ω) + ‖∂ t ∆z‖ L2 (I ×Ω) 



‖π k y k ‖ I ×Ω ≤ c‖y k ‖ I ×Ω 

∀y k ∈ ˜X 0 k 


Postprocessing strategy 

State discretization 

u k 




u k 

Adjoint and control discretization 

z k 




u k 

Adjoint discretization and interpolation 

z k 

π k z k 



Intermediate step: − 1 α π kG ∗ z k 

−α −1 π k G ∗ z k 

Postprocessed ˜q k = P Qad (− 1 α π kG ∗ z k ) 

→ cf. C. Meyer & A. Rösch 2004. 



Intermediate step: − 1 α π kG ∗ z k 

−α −1 π k G ∗ z k 

Postprocessed ˜q k = P Qad (− 1 α π kG ∗ z k ) 

˜q k 

−α −1 π k G ∗ z k 

→ cf. C. Meyer & A. Rösch 2004. 



Assumption 2 

The boundaries of the active sets 

A i = { t ∈ [0, T ] | q i (t) = q a,i or q i (t) = q b,i } , i = 1, 2, . . . , D 

consist of a finite number of points. 

Theorem 

Let Assumptions 1 and 2 be fulfilled. Let ˜q k be defined as 

( 

˜q k = P Qad − 1 ) 

α π kG ∗¯z k . 

Then there holds: 

‖¯q − ˜q k ‖ Q = O(k 2 ). 

◮ D. Meidner and B. Vexler. 

A priori error analysis of the Petrov Galerkin Crank Nicolson scheme for parabolic optimal control 

problems. 

SIAM J. Control Optim., accepted (2011) 



Assumption 2 

The boundaries of the active sets 

A i = { t ∈ [0, T ] | q i (t) = q a,i or q i (t) = q b,i } , i = 1, 2, . . . , D 

consist of a finite number of points. 

Theorem 

Let Assumptions 1 and 2 be fulfilled. Let ˜q k be defined as 

( 

˜q k = P Qad − 1 ) 

α π kG ∗¯z k . 


‖¯q − ˜q k ‖ Q = O(k 2 ). 



problems. 



Spatial discretization 

→ Linear or bilinear finite elements for both state and the adjoint state 

Ansatz space (continuous in time) 

{ 

Xkh 1 = v kh ∈ C(Ī , V h) 

∣ v k 

∣ 

∣Im 

∈ P 1 (I m , V h ), m = 1, 2, . . . , M 

Test space (discontinuous in time) 

{ 

˜X kh 0 = v kh ∈ L 2 ∣ } 

(I , V h ) ∣ v ∣Im kh ∈ P 0 (I m , V h ), m = 1, 2, . . . , M, v kh (0) ∈ V h 

Space-time discretization of the state: 

u kh ∈ X 1 kh : B(u kh , φ) = (f + Gq, φ) I ×Ω + (u 0 , φ − 0 ) ∀φ ∈ ˜X 0 kh. 

Space-time discretization of the adjoint state: 

z kh ∈ ˜X 0 kh : B(φ, z kh ) = (u kh − û, φ) I ×Ω ∀φ ∈ X 1 kh. 

} 

, 





{ 

Xkh 1 = v kh ∈ C(Ī , V h) 

∣ v k 

∣ 

∣Im 

∈ P 1 (I m , V h ), m = 1, 2, . . . , M 


{ 

˜X kh 0 = v kh ∈ L 2 ∣ } 






} 

, 





{ 

Xkh 1 = v kh ∈ C(Ī , V h) 

∣ v k 

∣ 

∣Im 

∈ P 1 (I m , V h ), m = 1, 2, . . . , M 


{ 

˜X kh 0 = v kh ∈ L 2 ∣ } 






} 

, 





{ 

Xkh 1 = v kh ∈ C(Ī , V h) 

∣ v k 

∣ 

∣Im 

∈ P 1 (I m , V h ), m = 1, 2, . . . , M 


{ 

˜X kh 0 = v kh ∈ L 2 ∣ } 






} 

, 





{ 

Xkh 1 = v kh ∈ C(Ī , V h) 

∣ v k 

∣ 

∣Im 

∈ P 1 (I m , V h ), m = 1, 2, . . . , M 


{ 

˜X kh 0 = v kh ∈ L 2 ∣ } 






} 

, 



Theorem 

Let Assumptions 1 and 2 be fulfilled. Let ˜q kh be defined as 

( 

˜q kh = P Qad − 1 ) 

α π kG ∗¯z kh . 


‖¯q − ˜q kh ‖ Q = O(k 2 + h 2 ). 



problems. 



Numerical example: temporal error 

Problem with known exact solution 

( 

¯q(t) := P Qad − π4 { 

exp(aπ 2 t) − exp(aπ 2 T ) }) , 

4 

ū(t, x 1 , x 2 ) := 

−1 

2 + a π2 w a (t, x 1 , x 2 ), 

¯z(t, x 1 , x 2 ) := w a (t, x 1 , x 2 ) − w a (T , x 1 , x 2 ). 

with 

w a (t, x 1 , x 2 ) := exp(aπ 2 t) sin(πx 1 ) sin(πx 2 ), 

q a = −70, q b = −1, a = − √ 5, T = 0.1 


Numerical example: temporal error 

|¯q − ¯q kh | I 

|¯q − ˜q kh | I 

O(k) 

O(k 2 ) 

10 0 10 0 10 1 10 2 10 3 

10 −1 

10 −2 

M 


Numerical example: spatial error 

|¯q − ¯q kh | I 

|¯q − ˜q kh | I 

O(h 2 ) 

10 0 

10 −1 

10 −2 

10 −3 

10 1 10 2 10 3 10 4 10 −4 

N 


Conclusions 

→ cG(1) Petrov-Galerkin discretization → Crank Nicolson scheme 

→ consistent discretization of the adjoint state 

→ second order convergence for the postprocessed control 



problems.

A Priori Error Analysis of the Petrov Galerkin Crank Nicolson ...

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