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Adaptive fully-discrete finite element methods for nonlinear quadratic parabolic boundary optimal control

Zuliang Lu

Author Affiliations

School of Mathematics and Statistics, Chongqing Three Gorges University, Chongqing, 404000, P.R. China

College of Civil Engineering and Mechanics, Xiangtan University, Xiangtan, 411105, P.R. China

Boundary Value Problems 2013, 2013:72  doi:10.1186/1687-2770-2013-72

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/72


Received:18 January 2013
Accepted:14 March 2013
Published:4 April 2013

© 2013 Lu; licensee Springer

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

The aim of this work is to study adaptive fully-discrete finite element methods for quadratic boundary optimal control problems governed by nonlinear parabolic equations. We derive a posteriori error estimates for the state and control approximation. Such estimates can be used to construct reliable adaptive finite element approximation for nonlinear quadratic parabolic boundary optimal control problems. Finally, we present a numerical example to show the theoretical results.

1 Introduction

In this paper, we study the fully-discrete finite element approximation for quadratic boundary optimal control problems governed by nonlinear parabolic equations. Optimal control problems are very important models in engineering numerical simulation. They have various physical backgrounds in many practical applications. Finite element approximation of optimal control problems plays a very important role in the numerical methods for these problems. The finite element approximation of a linear elliptic optimal control problem is well investigated by Falk [1] and Geveci [2]. The discretization for semilinear elliptic optimal control problems is discussed by Arada, Casas, and Tröltzsch in [3]. Systematic introductions of the finite element method for optimal control problems can be found in [4-6].

As one of important kinds of optimal control problems, the boundary optimal control is widely used in scientific and engineering computing. The literature in this aspect is huge; see, e.g., [7-10]. For some quadratic boundary optimal control problems, Liu and Yan [11,12] investigated a posteriori error estimates and adaptive finite element methods. Alt and Mackenroth [13] were concerned with error estimates of finite element approximations to state constrained convex parabolic boundary optimal control problems. Arada et al. discussed the numerical approximation of boundary optimal control problems governed by semilinear elliptic equations with pointwise constraints on the control in [14]. Although a priori error estimates and a posteriori error estimates of finite element approximation are widely used in numerical simulations, they have not yet been utilized in nonlinear parabolic boundary optimal control problems.

Adaptive finite element approximation is the most important method to boost accuracy of the finite element discretization. It ensures a higher density of nodes in a certain area of the given domain, where the solution is discontinuous or more difficult to approximate, using a posteriori error indicator. A posteriori error estimates are computable quantities in terms of the discrete solution that measure the actual discrete errors without the knowledge of exact solutions. They are essential in designing algorithms for mesh which equidistribute the computational effort and optimize the computation. Recently, in [15-18], we derived a priori error estimates, a posteriori error estimates and superconvergence for optimal control problems using mixed finite element methods.

In this paper, we adopt the standard notation W m , p ( Ω ) for Sobolev spaces on Ω with a norm m , p given by v m , p p = | α | m D α v L p ( Ω ) p and a semi-norm | | m , p given by | v | m , p p = | α | = m D α v L p ( Ω ) p . We set W 0 m , p ( Ω ) = { v W m , p ( Ω ) : v | Ω = 0 } . For p = 2 , we denote H m ( Ω ) = W m , 2 ( Ω ) , H 0 m ( Ω ) = W 0 m , 2 ( Ω ) , and m = m , 2 , = 0 , 2 . We denote by L s ( 0 , T ; W m , p ( Ω ) ) the Banach space of all L s integrable functions from J into W m , p ( Ω ) with the norm v L s ( J ; W m , p ( Ω ) ) = ( 0 T v W m , p ( Ω ) s d t ) 1 s for s [ 1 , ) , and the standard modification for s = . The details can be found in [19].

In this paper, we derive a posteriori error estimates for a class of boundary optimal control problems governed by a nonlinear parabolic equation. To our best knowledge, in the context of nonlinear parabolic boundary optimal control problems, these estimates are new. The problem that we are interested in is the following nonlinear quadratic parabolic boundary optimal control problem:

min u ( t ) K { 0 T ( 1 2 y y 0 2 + α 2 u 2 ) d t } (1)

subject to the state equations

y t ( x , t ) ( A y ( x , t ) ) + ϕ ( y ( x , t ) ) = f ( x , t ) , x Ω , t J , (2)

( A y ( x , t ) ) n = B u ( x , t ) + z b , x Ω , t J , (3)

y ( x , 0 ) = y 0 ( x ) , x Ω , (4)

where the bounded open set Ω R 2 is 2 regular convex polygon with boundary Ω, J = ( 0 , T ] , f L 2 ( J ; L 2 ( Ω ) ) , y 0 H 1 ( Ω ) , z b L 2 ( Ω ) , and α is a positive constant. For any I > 0 , the function ϕ ( ) W 2 , ( I , I ) , ϕ ( y ) L 2 ( Ω ) for any y L 2 ( J ; H 1 ( Ω ) ) , and ϕ ( y ) 0 . We assume the coefficient matrix A ( x ) = ( a i , j ( x ) ) 2 × 2 ( W 1 , ( Ω ) ) 2 × 2 is a symmetric positive definite matrix, and there is a constant c > 0 satisfying for any vector X R 2 , X t A X c X R 2 2 . Here, K denotes the admissible set of the control variable defined by

K = { u ( x , t ) L 2 ( J ; L 2 ( Ω ) ) : u ( x , t ) 0  a.e.  x Ω , t J } . (5)

The plan of this paper is as follows. In the next section, we present a finite element discretization for nonlinear quadratic parabolic boundary optimal control problems. A posteriori error estimates are established for the finite element approximation solutions in Section 3. In Section 4, we give a numerical example to prove the theoretical results.

2 Finite element methods for parabolic boundary optimal control

We shall now describe a finite element discretization of nonlinear quadratic parabolic boundary optimal control problem (1)-(4). Let V = H 1 ( Ω ) , W = L 2 ( Ω ) , U = L 2 ( Ω ) . Let

a ( y , w ) = Ω ( A y ) w , y , w V , (6)

( f 1 , f 2 ) = Ω f 1 f 2 , ( f 1 , f 2 ) W × W , (7)

( u , v ) U = Ω u v , ( u , v ) U × U . (8)

Then quadratic parabolic boundary optimal control problem (1)-(4) can be restated as

min u ( t ) K { 0 T ( 1 2 y y 0 2 + α 2 u 2 ) d t } (9)

subject to

( y t , w ) + a ( y , w ) + ( ϕ ( y ) , w ) = ( f , w ) + ( B u + z b , w ) U , w V , t J , (10)

y ( x , 0 ) = y 0 ( x ) , x Ω , (11)

where the inner product in L 2 ( Ω ) or L 2 ( Ω ) 2 is indicated by ( , ) , and B is a continuous linear operator from U to L 2 ( Ω ) .

It is well known (see, e.g., [12]) that the optimal control problems have at least a solution ( y , u ) , and that if a pair ( y , u ) is the solution of (9)-(11), then there is a co-state p V such that the triplet ( y , p , u ) satisfies the following optimality conditions:

( y t , w ) + a ( y , w ) + ( ϕ ( y ) , w ) = ( f , w ) + ( B u + z b , w ) U , w V = H 1 ( Ω ) , (12)

y ( x , 0 ) = y 0 ( x ) , x Ω , (13)

( p t , w ) + a ( q , p ) + ( ϕ ( y ) p , q ) = ( y y 0 , q ) , q V = H 1 ( Ω ) , (14)

p ( x , T ) = 0 , x Ω , (15)

0 T ( α u + B p , v u ) U d t 0 , v K U = L 2 ( Ω ) , (16)

where B is the adjoint operator of B. In the rest of the paper, we shall simply write the product as ( , ) whenever no confusion should be caused.

Let us consider the finite element approximation of control problem (9)-(11). Again, here we consider only n-simplex elements and conforming finite elements.

Let T h be a regular partition of Ω. Associated with T h is a finite dimensional subspace V h of C ( Ω ¯ ) such that χ | τ are polynomials of m-order ( m 1 ) χ V h and τ T h . It is easy to see that V h V . Let E h be a partition of Ω into disjoint regular ( n 1 ) -simplices s, so that Ω = s E h s ¯ . Associated with E h is another finite dimensional subspace U h of L 2 ( Ω ) such that χ | τ are polynomials of m-order ( m 0 ) χ U h and s E h . Let h τ ( h s ) denote the maximum diameter of the element τ ( s ) in T h ( E h ) , h = max τ T h { h τ } , and h U = max s E h { h s } . In addition C or c denotes a general positive constant independent of h.

By the definition of a finite element subspace, the finite element discretization of (9)-(11) is as follows: compute ( y h , u h ) V h × K h such that

min u h K h { 0 T ( 1 2 y h y 0 2 + α 2 u h 2 ) } (17)

( y h t , w h ) + a ( y h , w h ) + ( ϕ ( y h ) , w h ) = ( f , w h ) + ( B u h + z b , w h ) U , w h V h , (18)

y h ( x , 0 ) = y 0 h ( x ) , x Ω , (19)

where K h = K U h , y 0 h V h is an approximation of y 0 .

Again, it follows that optimal control problem (17)-(19) has at least a solution ( y h , u h ) , and that if a pair ( y h , u h ) is the solution of (17)-(19), then there is a co-state p h V h such that the triplet ( y h , p h , u h ) satisfies the following optimality conditions:

( y h t , w h ) + a ( y h , w h ) + ( ϕ ( y h ) , w h ) = ( f , w h ) + ( B u h + z b , w h ) U , w h V h , (20)

y h ( x , 0 ) = y 0 h ( x ) , x Ω , (21)

( p h t , w h ) + a ( q h , p h ) + ( ϕ ( y h ) p h , q h ) = ( y h y 0 , q h ) , q h V h , (22)

p h ( x , T ) = 0 , x Ω , (23)

0 T ( α u h + B p h , v h u h ) U d t 0 , v h K h . (24)

We now consider the fully discrete approximation for the semidiscrete problem. Let Δ t > 0 , N = T / Δ t Z , and let t i = i Δ t , i R . Also, let

ψ i = ψ i ( x ) = ψ ( x , t i ) , d t ψ i = ψ i ψ i 1 Δ t .

For i = 1 , 2 , , N , we construct the finite element spaces V i h V with the mesh T h i (similar to V h ). Similarly, we construct the finite element spaces U i h L 2 ( Ω ) with the mesh T h i (similar to U h ). Let h τ i ( h s i ) denote the maximum diameter of the element τ i ( s i ) in T h i ( ( E h ) i ) . Define mesh functions τ ( ) , s ( ) and mesh size functions h τ ( ) , h s ( ) such that τ ( t ) | t ( t i 1 , t i ] = τ i , s ( t ) | t ( t i 1 , t i ] = s i , h τ ( t ) | t ( t i 1 , t i ] = h τ i , h s ( t ) | t ( t i 1 , t i ] = h s i . For ease of exposition, we denote τ ( t ) , s ( t ) , h τ ( t ) , and h s ( t ) by τ, s, h τ , and h s , respectively.

Then the fully discrete finite element approximation of (17)-(19) is as follows. Compute ( y h i , u h i ) V i h × K i h , i = 1 , 2 , , N , such that

min u h i K i h { i = 1 N Δ t ( 1 2 y h i y 0 2 + α 2 u h i 2 ) } (25)

( d t y h i , w h ) + a ( y h i , w h ) + ( ϕ ( y h i ) , w h ) = ( f ( x , t i ) , w h ) + ( B u h i + z b , w h ) U , (26)

w h V i h , i = 1 , 2 , , N , y h 0 ( x ) = y 0 h ( x ) , x Ω , (27)

where K i h = K U i h , y 0 h V h is an approximation of y 0 .

Now, it follows that optimal control problem (25)-(27) has at least a solution ( Y h i , U h i ) , i = 1 , 2 , , N , and that if a pair ( Y h i , U h i ) , i = 1 , 2 , , N , is the solution of (25)-(27), then there is a co-state P h i 1 V i h , i = 1 , 2 , , N , such that the triplet ( Y h i , P h i 1 , U h i ) satisfies the following optimality conditions:

( d t Y h i , w h ) + a ( Y h i , w h ) + ( ϕ ( Y h i ) , w h ) = ( f , w h ) + ( B U h i + z b , w h ) U , w h V i h , (28)

i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω , (29)

( d t P h i , q h ) + a ( q h , P h i 1 ) + ( ϕ ( Y h i 1 ) P h i 1 , q h ) = ( Y h i y 0 , q h ) , q h V i h , (30)

i = N , , 2 , 1 , P h N ( x ) = 0 , x Ω , (31)

( α U h i + B P h i , v h U h i ) 0 , v h K i h , i = 1 , 2 , , N . (32)

For i = 1 , 2 , , N , let

Y h | ( t i 1 , t i ] = ( ( t i t ) Y h i 1 + ( t t i 1 ) Y h i ) / Δ t , (33)

P h | ( t i 1 , t i ] = ( ( t i t ) P h i 1 + ( t t i 1 ) P h i ) / Δ t , (34)

U h | ( t i 1 , t i ] = U h i . (35)

For any function w C ( 0 , T ; L 2 ( Ω ) ) , let w ˆ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i ) , w ˜ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i 1 ) . Then the optimality conditions (28)-(32) can be restated as follows:

( Y h t , w h ) + a ( Y ˆ h , w h ) + ( ϕ ( Y ˆ h ) , w h ) = ( f ˆ , w h ) + ( B U h + z b , w h ) U , w h V i h , (36)

i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω , (37)

( P h t , q h ) + a ( q h , P ˜ h ) + ( ϕ ( Y ˜ h ) P ˜ h , q h ) = ( Y ˆ h y 0 , q h ) , q h V i h , (38)

i = N , , 2 , 1 , P h ( x , T ) = 0 , x Ω , (39)

( α U h + B P ˜ h , v h U h ) 0 , v h K i h , i = 1 , 2 , , N . (40)

In the rest of the paper, we shall use some intermediate variables. For any control function U h K , we define that the state solution ( y ( U h ) , p ( U h ) ) satisfies

( y t ( U h ) , w ) + a ( y ( U h ) , w ) + ( ϕ ( y ( U h ) ) , w ) = ( f , w ) + ( B U h + z b , w ) , w V , (41)

y ( U h ) ( x , 0 ) = y 0 ( x ) , x Ω , (42)

( p t ( U h ) , q ) + a ( q , p ( U h ) ) + ( ϕ ( y ( U h ) ) p ( U h ) , q ) = ( y ( U h ) y 0 , q ) , q V , (43)

p ( U h ) ( x , T ) = 0 , x Ω . (44)

Now we restate the following well-known estimates in [19].

Lemma 2.1Let π ˆ h be the Clément-type interpolation operator defined in[19]. Then for any v H 1 ( Ω ) and all elementτ,

v π ˆ h v L 2 ( τ ) + h τ ( v π ˆ h v ) L 2 ( τ ) C h τ τ ¯ τ ¯ | v | L 2 ( τ ) , (45)

v π ˆ h v L 2 ( l ) C h l 1 / 2 l τ ¯ | v | L 2 ( τ ) , (46)

wherelis the edge of the element.

For φ W h , we write

ϕ ( φ ) ϕ ( ρ ) = ϕ ˜ ( φ ) ( ρ φ ) = ϕ ( ρ ) ( ρ φ ) + ϕ ˜ ( φ ) ( ρ φ ) 2 , (47)

where

ϕ ˜ ( φ ) = 0 1 ϕ ( φ + s ( ρ φ ) ) d s , ϕ ˜ ( φ ) = 0 1 ( 1 s ) ϕ ( ρ + s ( φ ρ ) ) d s

are bounded functions in Ω ¯ [20].

3 A posteriori error estimates

In this section we obtain a posteriori error estimates for nonlinear quadratic parabolic boundary optimal control problems. Firstly, we estimate the error y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) .

Theorem 3.1Let ( y ( U h ) , p ( U h ) ) and ( Y h , P h ) be the solutions of (41)-(44) and (36)-(40), respectively. Then

y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 6 η i 2 , (48)

where

η 1 2 = 0 T τ T h h τ 2 τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) 2 , η 2 2 = 0 T l Ω = ϕ h l l [ A Y ˆ h n ] 2 , η 3 2 = 0 T l Ω h l l ( A Y ˆ h n B U h z b ) 2 , η 4 2 = Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 , η 5 2 = Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 , η 6 2 = f f ˆ L 2 ( J ; L 2 ( Ω ) ) 2 ,

wherelis a face of an elementτ, h l is the size of the facel, [ A y h n ] is theA-normal derivative jump over the interior faceldefined by

[ A Y h n ] l = ( A Y h | τ l 1 A Y h | τ l 2 ) n ,

wherenis the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 outwards τ l 1 .

Proof Let e y = y ( U h ) Y h , and let e I y be the Clément-type interpolator of e y defined in Lemma 2.1. Note that

0 T ( y t ( U h ) Y h t , e y ) d t = 0 T Ω ( y t ( U h ) Y h t ) e y d x d t = 1 2 Ω ( ( y ( U h ) Y h ) ( x , T ) ) 2 d x 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 .

Thus

0 T ( y t ( U h ) Y h t , e y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 0 .

Using equations (36) and (41), we infer that

c e y L 2 ( J ; H 1 ( Ω ) ) 2 0 T ( A ( y ( U h ) Y h ) , e y ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y ) d t = 0 T ( A ( y ( U h ) Y h ) , ( e y e I y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y e I y ) d t + 0 T ( A ( y ( U h ) Y h ) , ( e I y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e I y ) d t 0 T ( A ( y ( U h ) Y h ) , ( e y e I y ) ) + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y e I y ) + 0 T ( y t ( U h ) Y h t , e y e I y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 + 0 T ( A ( y ( U h ) Y h ) , ( e I y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e I y ) d t + 0 T ( y t ( U h ) Y h t , e I y ) d t = 0 T τ T h τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) ( e y e I y ) d t + 0 T τ T h τ ( A Y ˆ h n ) ( e y e I y ) d s d t + 0 T Ω ( A Y ˆ h n B U h z b ) ( e y e I y ) d s d t + 0 T ( A ( y ( U h ) Y h ) , ( e y ) ) d t + 0 T ( ϕ ( y ( U h ) ) ϕ ( Y h ) , e y ) d t + 0 T ( f f ˆ , e y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 = 0 T τ T h τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) ( e y e I y ) d t + 0 T l Ω = ϕ l ( A Y ˆ h n ) ( e y e I y ) d s d t + 0 T l Ω l ( A Y ˆ h n B U h z b ) ( e y e I y ) d s d t + 0 T ( A ( Y h Y ˆ h ) , ( e y ) ) d t + 0 T ( ϕ ( Y h ) ϕ ( Y ˆ h ) , e y ) d t + 0 T ( f f ˆ , e y ) d t + 1 2 Y h ( x , 0 ) y 0 ( x ) L 2 ( Ω ) 2 K 1 + K 2 + K 3 + K 4 + K 5 + K 6 + K 7 . (49)

Let us bound each of the terms on the right-hand side of (49). By Lemma 2.1 we have

K 1 = 0 T τ T h τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) ( e y e I y ) d t C 0 T τ T h h τ 2 τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) 2 d t + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 d t C 0 T τ T h h τ 2 τ ( f ˆ Y h t + div ( A Y ˆ h ) ϕ ( Y ˆ h ) ) 2 d t + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 . (50)

Next, using Lemma 2.1, we get

K 2 = 0 T l Ω = ϕ l ( A Y ˆ h n ) ( e y e I y ) d s d t C 0 T l Ω = ϕ h l l [ A Y ˆ h n ] 2 + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 + C δ 0 T τ T h τ | ( e y e I y ) | 2 C 0 T l Ω = ϕ h l l [ A Y ˆ h n ] 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 , (51)

and

K 3 = 0 T l Ω l ( A Y ˆ h n B U h z b ) ( e y e I y ) C 0 T l Ω h l l ( A Y ˆ h n B U h z b ) 2 + C δ 0 T τ T h h τ 2 τ | e y e I y | 2 + C δ 0 T τ T h τ | ( e y e I y ) | 2 C 0 T l Ω h l l ( A Y ˆ h n B U h z b ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 . (52)

For K 4 - K 6 , the Schwarz inequality implies

K 4 = 0 T ( A ( Y h Y ˆ h ) , ( e y ) ) d t C Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 , (53)

and

K 5 = 0 T ( ϕ ( Y h ) ϕ ( Y ˆ h ) , e y ) d t = 0 T ( ϕ ˜ ( Y h ) ( Y h Y ˆ h ) , e y ) d t C Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 , (54)

and

K 6 = 0 T ( f f ˆ , e y ) d t C f f ˆ L 2 ( J ; L 2 ( Ω ) ) 2 + C δ e y L 2 ( J ; H 1 ( Ω ) ) 2 . (55)

Finally, add inequalities (49)-(55) to obtain

y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 6 η i 2 . (56)

This completes the proof. □

Analogously to Theorem 3.1, we show the following estimates.

Theorem 3.2Let ( y ( U h ) , p ( U h ) ) and ( Y h , P h ) be the solutions of (41)-(44) and (36)-(40), respectively. Then

p ( U h ) P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 11 η i 2 , (57)

where

η 7 2 = τ T h h τ 2 τ ( Y ˆ h y 0 + P h t + div ( A P ˜ h ) ϕ ( Y ˜ h ) P ˜ h ) 2 , η 8 2 = 0 T l Ω = ϕ h l l [ A P ˜ h n ] 2 , η 9 2 = 0 T l Ω h l l ( A P ˜ h n ) 2 , η 10 2 = Y h Y ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 , η 11 2 = P h P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 ,

where η 1 - η 6 are defined in Theorem 3.1, lis a face of an elementτ, [ A P ˜ h n ] is theA-normal derivative jump over the interior faceldefined by

[ A P ˜ h n ] l = ( A P ˜ h | τ l 1 A P ˜ h | τ l 2 ) n ,

wherenis the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 outwards τ l 1 .

Proof Let e p = p ( U h ) P h , and let e I p = π ˆ h e p , where π ˆ h is the Clément-type interpolator defined in Lemma 2.1. Note that ( p ( U h ) P h ) ( x , T ) = 0 , then we obtain

0 T ( p t ( U h ) P h t , e p ) d t 0 .

Using equations (38) and (43), we obtain

c e p L 2 ( J ; H 1 ( Ω ) ) 2 0 T ( e p , A ( p ( U h ) P h ) ) d t + 0 T ( ϕ ( y ( U h ) ) ( p ( U h ) P h ) , e p ) d t = 0 T ( e p , A ( p ( U h ) P ˜ h ) ) d t + 0 T ( ϕ ( y ( U h ) ) p ( U h ) ϕ ( Y ˜ h ) P ˜ h , e p ) d t 0 T ( p t ( U h ) P h t , e p ) d t + 0 T ( ϕ ( Y ˜ h ) P ˜ h ϕ ( y ( U h ) ) P h , e p ) d t + 0 T ( e p , A ( P ˜ h P h ) ) d t = 0 T ( ( e p e I p ) , A ( p ( U h ) P ˜ h ) ) d t 0 T ( p t ( U h ) P h t , e p e I p ) d t + 0 T ( ϕ ( y ( U h ) ) p ( U h ) ϕ ( Y ˜ h ) P ˜ h , e p e I p ) d t + 0 T ( y ( U h ) Y ˆ h , e I p ) d t + 0 T ( e p , A ( P ˜ h P h ) ) d t + 0 T ( ϕ ( Y ˜ h ) P ˜ h ϕ ( y ( U h ) ) P h , e p ) d t = 0 T ( Y ˆ h y 0 + P h t + div ( A P ˜ h ) ϕ ( Y ˜ h ) P ˜ h , e p e I p ) d t + 0 T l Ω = ϕ l ( A P ˜ h n ) ( e p e I p ) d s d t + 0 T l Ω l ( A P ˜ h n ) ( e p e I p ) d s d t + 0 T ( y ( U h ) Y ˆ h , e p ) d t + 0 T ( e p , A ( P ˜ h P h ) ) d t + 0 T ( ϕ ( Y ˜ h ) P ˜ h ϕ ( y ( U h ) ) P h , e p ) d t L 1 + L 2 + L 3 + L 4 + L 5 + L 6 . (58)

Now let us bound each of the terms on the right-hand side of (58). By Lemma 2.1 we have

L 1 = 0 T τ T h τ ( Y ˆ h y 0 + P h t + div ( A P ˜ h ) ϕ ( Y ˜ h ) P ˜ h ) ( e p e I p ) d t C 0 T τ T h h τ 2 τ ( Y ˆ h y 0 + P h t + div ( A P ˜ h ) ϕ ( Y ˜ h ) P ˜ h ) 2 d t + C δ 0 T τ T h h τ 2 τ | e p e I p | 2 d t C 0 T τ T h h τ 2 τ ( Y ˆ h y 0 + P h t + div ( A P ˜ h ) ϕ ( Y ˜ h ) P ˜ h ) 2 d t + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 . (59)

Next, using Lemma 2.1, we get

L 2 = 0 T l Ω = ϕ l ( A P ˜ h n ) ( e p e I p ) d s d t C 0 T l Ω = ϕ h l l [ A P ˜ h n ] 2 + C δ 0 T τ T h h τ 2 τ | e p e I p | 2 + C δ 0 T τ T h τ | ( e p e I p ) | 2 C 0 T l Ω = ϕ h l l [ A P ˜ h n ] 2 + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 (60)

and

L 3 = 0 T l Ω l ( A P ˜ h n ) ( e p e I p ) C 0 T l Ω h l l ( A P ˜ h n ) 2 + C δ 0 T τ T h h τ 2 τ | e p e I p | 2 + C δ 0 T τ T h τ | ( e p e I p ) | 2 C 0 T l Ω h l l ( A P ˜ h n ) 2 + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 . (61)

The Schwarz inequality implies

L 4 = 0 T ( y ( U h ) Y ˆ h , e p ) d t = 0 T ( ( y ( U h ) Y h ) + ( Y h Y ˆ h ) , e p ) d t C y ( U h ) Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + Y h Y ˆ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 (62)

and

L 5 = 0 T ( e p , A ( P ˜ h P h ) ) d t C P ˜ h P h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 . (63)

Next, for L 6 , we obtain

L 6 = 0 T ( ϕ ( Y ˜ h ) P ˜ h ϕ ( y ( U h ) ) P h , e p ) d t = 0 T ( ϕ ( Y ˜ h ) ( P ˜ h P h ) , e p ) d t + 0 T ( ( ϕ ( Y ˜ h ) ϕ ( Y h ) ) P h , e p ) d t + 0 T ( ( ϕ ( Y ˜ h ) ϕ ( y ( U h ) ) ) P h , e p ) d t = 0 T ( ϕ ( Y ˜ h ) ( P ˜ h P h ) , e p ) d t + 0 T ( ( ϕ ( Y ˜ h ) ϕ ( Y h ) ) P h , e p ) d t + 0 T ( ϕ ˜ ( Y ˜ h ) ( Y h y ( U h ) ) P h , e p ) d t C y ( U h ) Y h L 2 ( J ; H 1 ( Ω ) ) 2 + C Y h Y ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 + C P h P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 + C δ e p L 2 ( J ; H 1 ( Ω ) ) 2 . (64)

Finally, add inequalities (58)-(64) and combine Theorem 3.1 to obtain

p ( U h ) P ˜ h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 11 η i 2 . (65)

This completes the proof. □

For given u K , let M be the inverse operator of the state equation (12) such that y ( u ) = M B u is the solution of the state equation (12). Similarly, for given U h K h , Y h ( U h ) = M h B U h is the solution of the discrete state equation (36). Let

S ( u ) = 1 2 M B u y 0 2 + α 2 u 2 , S h ( U h ) = 1 2 M h B U h y 0 2 + α 2 U h 2 .

It is clear that S and S h are well defined and continuous on K and K h . Also, the functional S h can be naturally extended on K. Then (9) and (25) can be represented as

min u K { S ( u ) } , (66)

min U h K h { S h ( U h ) } . (67)

It can be shown that

( S ( u ) , v ) = ( α u + B p , v ) , ( S ( U h ) , v ) = ( α U h + B p ( U h ) , v ) , ( S h ( U h ) , v ) = ( α U h + B P ˜ h , v ) ,

where p ( U h ) is the solution of equations (41)-(43).

In many applications, S ( ) is uniform convex near the solution u (see, e.g., [21]). The convexity of S ( ) is closely related to the second-order sufficient conditions of the control problems, which are assumed in many studies on numerical methods of the problems. If S ( ) is uniformly convex, then there is a c > 0 such that

0 T ( S ( u ) S ( U h ) , u U h ) c u U h L 2 ( J ; L 2 ( Ω ) ) 2 , (68)

where u and U h are the solutions of (66) and (67), respectively. We assume the above inequality throughout this paper.

In order to have sharp a posteriori error estimates, we divide Ω into some subsets:

Ω i = { x Ω : ( B P ˜ h ) ( x , t i ) 0 } , Ω i = { x Ω : ( B P ˜ h ) ( x , t i ) > 0 , U h i = 0 } , Ω i + = { x Ω : ( B P ˜ h ) ( x , t i ) > 0 , U h i > 0 } .

Then it is clear that three subsets do not intersect each other, and Ω = Ω i Ω i Ω i + , i = 1 , 2 , , N .

Let p ( U h ) be the solution of (41)-(44). We establish the following error estimate, which can be proved similarly to the proofs given in [22].

Theorem 3.3Letuand U h be the solutions of (66) and (67), respectively. Then

u U h L 2 ( J ; L 2 ( Ω ) ) 2 C ( η 12 2 + P ˜ h p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 ) , (69)

where

η 12 2 = i = 1 N t i 1 t i Ω i | B P ˜ h + α U h | 2 .

Proof It follows from the inequality (68) that

c u U h L 2 ( J ; L 2 ( Ω ) ) 2 0 T ( S ( u ) , u U h ) ( S ( u h ) , u U h ) d t 0 T ( S ( U h ) , u U h ) d t = 0 T ( S h ( U h ) , U h u ) d t + 0 T ( S h ( U h ) S ( U h ) , u U h ) d t . (70)

Note that

0 T ( S h ( U h ) , U h u ) d t = i = 1 N t i 1 t i Ω i ( B P ˜ h + α U h ) ( U h u ) + i = 1 N t i 1 t i Ω i ( B P ˜ h + α U h ) ( U h u ) + i = 1 N t i 1 t i Ω i + ( B P ˜ h + α U h ) ( u ) . (71)

It is easy to see that

Ω i ( B P ˜ h + α U h ) ( U h u ) Ω i | B P ˜ h + α U h | 2 d x + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 = C η 12 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 . (72)

Since U h is piecewise constant, U h | s > 0 if s Ω i + is not empty. If u h | s > 0 , there exists ε > 0 and β U h such that β 0 , β L ( s ) = 1 and ( u h ε β ) | s 0 . For example, one can always find such a required β from one of the shape functions on s. Hence, u ˆ h K h , where u ˆ h = U h ε β as x s and otherwise u ˆ = U h . Then it follows from (40) that

s ( B P ˜ h + α U h ) β = ε 1 s ( B P ˜ h + α U h ) ( U h ( U h ε β ) ) ε 1 Ω ( B P ˜ h + α U h ) ( U h ( U h ε β ) ) 0 . (73)

Note that on Ω i + , B P ˜ h + α U h B P ˜ h > 0 , and from (72) we have that

s Ω i + | B P ˜ h + α U h | β = s Ω i + ( B P ˜ h + α U h ) β s Ω i ( B P ˜ h + α U h ) β s Ω i | B P ˜ h + α U h | . (74)

Let s ˆ be the reference element of s, s 0 = s Ω i + , and s ˆ 0 s ˆ be a part mapped from s ˆ 0 . Note that ( s | | 2 ) 1 / 2 , s | | β are both norms on L 2 ( s ) . In such a case, for the function β fixed above, it follows from the equivalence of the norm in the finite-dimensional space that

s Ω i + | B P ˜ h + α U h | 2 = s 0 | B P ˜ h + α U h | 2 C h s 2 s ˆ 0 | B P ˜ h + α U h | 2 C h s 2 ( s ˆ 0 | B P ˜ h + α U h | β ) 2 C h s 2 ( s Ω i + | B P ˜ h + α U h | β ) 2 C h s 2 ( s Ω i | B P ˜ h + α U h | ) 2 C s Ω i | B P ˜ h + α U h | 2 , (75)

where the constant C can be made independent of β since it is always possible to find the required β from the shape functions on s so that

Ω i + ( B P ˜ h + α U h ) ( U h u ) C Ω i + | B P ˜ h + α U h | 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C Ω i | B P ˜ h + α U h | 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C η 12 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 . (76)

It follows from the definition of Ω i that B P ˜ h + α U h > 0 on Ω i . Note that u 0 , we have that

Ω i ( B P ˜ h + α U h ) ( u ) 0 . (77)

It is easy to show that

( S h ( U h ) S ( U h ) , u U h ) = ( B P ˜ h + α U h , u U h ) ( B p ( U h ) + α U h , u U h ) = ( B ( P ˜ h p ( U h ) ) , u U h ) C P ˜ h p ( U h ) L 2 ( J ; L 2 ( Ω ) ) 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 C P ˜ h p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 + δ u U h L 2 ( J ; L 2 ( Ω ) ) 2 . (78)

Therefore, (69) follows from (70)-(72) and (76)-(78). □

Hence, we combine Theorems 3.1-3.3 to conclude the following.

Theorem 3.4Let ( y , p , u ) and ( Y h , P h , U h ) be the solutions of (12)-(16) and (36)-(40), respectively. Then

u U h L 2 ( J ; L 2 ( Ω ) ) 2 + y Y h L 2 ( J ; H 1 ( Ω ) ) 2 + p P h L 2 ( J ; H 1 ( Ω ) ) 2 C i = 1 12 η i 2 , (79)

where η 1 , η 2 ,  , and η 12 are defined in Theorems 3.1-3.3, respectively.

Proof From (12)-(15) and (41)-(44), we obtain the error equations

( y t y t ( U h ) , w ) + a ( y y ( U h ) , w ) + ( ϕ ( y ) ϕ ( y ( U h ) ) , w ) = ( B ( u U h ) , w ) , (80)

( p t p t ( U h ) , q ) + a ( q , p p ( U h ) ) + ( ϕ ( y ) p ϕ ( y ( U h ) ) p ( U h ) , q ) = ( y y ( U h ) , q ) , (81)

for all w V and q V . Thus it follows from (80)-(81) that

( y t y t ( U h ) , w ) + a ( y y ( U h ) , w ) + ( ϕ ( y ) ( y y ( U h ) ) , w ) = ( B ( u U h ) , w ) , (82)

( p t p t ( U h ) , q ) + a ( q , p p ( U h ) ) + ( ϕ ( y ( U h ) ) ( p p ( U h ) ) , q ) = ( ϕ ˜ ( y ( U h ) ) ( y ( U h ) y ) p , q ) . (83)

By using the stability results in [23], we can prove that

y y ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 C u U h L 2 ( J ; L 2 ( Ω ) ) 2 (84)

and

p p ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 y y ( U h ) L 2 ( J ; H 1 ( Ω ) ) 2 C u U h L 2 ( J ; L 2 ( Ω ) ) 2 . (85)

Finally, combining Theorems 3.1-3.3 and (84)-(85) leads to (79). □

4 Numerical example

In the section, we use a posteriori error estimates presented in our paper as an indicator for the adaptive finite element approximation. The optimization problem is solved numerically by a preconditioned projection algorithm, with codes developed based on AFEPACK. The optimal control problem is

min u ( t ) K { 0 T ( 1 2 y y 0 2 + 1 2 u 2 ) d t } y t Δ y + y 3 = f , x Ω ; y n = u , x Ω , y ( x , 0 ) = 0 , x Ω .

In the example, we choose the domain Ω = [ 0 , 1 ] × [ 0 , 1 ] and K = { u L 2 ( J ; L 2 ( Ω ) ) : u 0 } . Let Ω be partitioned into T h as described in Section 2. We use η 12 as the control mesh refinement indicator and η 1 - η 11 as the states and co-states.

For the constrained optimization problem min u K S ( u ) , where S ( u ) is a convex functional on U, the iterative scheme reads ( n = 0 , 1 , 2 , )

b ( u n + 1 2 , v ) = b ( u n , v ) ρ n ( S ( u n ) , v ) , u n + 1 = P k b ( u n + 1 2 ) , v K , (86)

where b ( , ) is a symmetric and positive definite bilinear form such that there exist constants c 0 and c 1 satisfying

b ( u , u ) c 0 u U 2 , | b ( u , v ) | c 1 u U v U , u , v U , (87)

and the projection operator P K b U K is defined as follows. For given w U , find P K b w K such that

b ( P K b w w , P K b w w ) = min u K b ( u w , u w ) . (88)

The bilinear form b ( , ) provides suitable preconditioning for the projection algorithm. An application of (86) to the discretized nonlinear parabolic boundary optimal control problem yields the following algorithm:

b ( u n + 1 2 i , v h ) = b ( u n i , v h ) ρ n ( u n i + p n i , v h ) , v h K i h , (89)

( y n i y n i 1 Δ t , w h ) + a ( y n i , w h ) + ( y n i , 3 , w h ) = ( f , w h ) + ( u n i , w h ) U , w h V i h , (90)

( p n i 1 p n i Δ t , q h ) + a ( q h , p n i ) + ( 3 y n i , 2 p n i , q h ) = ( y n i y 0 , q h ) , q h V i h , (91)

u n + 1 i = P k b ( u n + 1 2 i ) , u n + 1 2 i , u n i K i h . (92)

The main computational effort is to solve the state and co-state equations and to compute the projection P K b u n + 1 2 i . In this paper we use a fast algebraic multigrid solver to solve the state and co-state equations. Then it is clear that the key to saving computing time is finding how to compute P K b u n + 1 2 i efficiently. For the piecewise constant elements, K h = { u h K : u h 0 } and b ( u , v ) = ( u , v ) U , then

P K b u n + 1 2 i | T = max ( 0 , avg ( u n + 1 2 i ) | T ) ,

where avg ( u n + 1 2 i ) | T is the average of u n + 1 2 i over T. In solving our discretized optimal control problem, we use the preconditioned projection gradient method (89)-(92) with b ( u , v ) = ( u , v ) K h and a fixed step size ρ = 0.8 . In the numerical simulation, we use a piecewise linear finite element space for the approximation of y and p, and a piecewise constant for u.

It can be clearly seen from Table 1 that on the adaptive meshes one may use less degree of freedom to produce a given control error reduction. Then it is clear that these a posteriori error estimates are very good for the parabolic boundary optimal control, and the adaptive finite element method is more efficient.

Table 1. Comparison of uniform mesh and adaptive mesh

Competing interests

The author declares that he has no competing interests.

Author’s contributions

ZL participated in the design of all the study and drafted the manuscript.

Acknowledgements

This work is supported by National Science Foundation of China (11201510), Mathematics TianYuan Special Funds of the National Natural Science Foundation of China (11126329), China Postdoctoral Science Foundation funded project (2011M500968), Natural Science Foundation Project of CQ CSTC (cstc2012jjA00003), and Natural Science Foundation of Chongqing Municipal Education Commission (KJ121113). The author expresses his thanks to the referees for their helpful suggestions, which led to improvements of the presentation.

References

  1. Falk, FS: Approximation of a class of optimal control problems with order of convergence estimates. J. Math. Anal. Appl.. 44, 28–47 (1973). Publisher Full Text OpenURL

  2. Geveci, T: On the approximation of the solution of an optimal control problem governed by an elliptic equation. RAIRO. Anal. Numér.. 13, 313–328 (1979)

  3. Arada, N, Casas, E, Tröltzsch, F: Error estimates for the numerical approximation of a semilinear elliptic control problem. Comput. Optim. Appl.. 23, 201–229 (2002). Publisher Full Text OpenURL

  4. French, DA, King, JT: Approximation of an elliptic control problem by the finite element method. Numer. Funct. Anal. Optim.. 12, 299–315 (1991). Publisher Full Text OpenURL

  5. Gunzburger, MD, Hou, S: Finite dimensional approximation of a class of constrained nonlinear control problems. SIAM J. Control Optim.. 34, 1001–1043 (1996). Publisher Full Text OpenURL

  6. Neittaanmaki, P, Tiba, D: Optimal Control of Nonlinear Parabolic Systems: Theory, Algorithms and Applications, Dekker, New York (1994)

  7. Gong, W, Yan, N: A posteriori error estimates for boundary control problems governed by the parabolic partial differential equations. J. Comput. Math.. 27, 68–88 (2009)

  8. Liu, H, Yan, N: Superconvergence and a posteriori error estimates for boundary control by Stokes equations. J. Comput. Math.. 24, 343–356 (2006). PubMed Abstract OpenURL

  9. Chen, Y, Lu, Z: A posteriori error estimates for semilinear boundary control problems. Lect. Notes Comput. Sci. Eng.. 78, 455–462 (2011). Publisher Full Text OpenURL

  10. Lu, Z: A posteriori error estimates of finite element methods for nonlinear quadratic boundary optimal control problem. Numer. Anal. Appl.. 4, 210–222 (2011). Publisher Full Text OpenURL

  11. Liu, W, Yan, N: A posteriori error estimates for some model boundary control problems. J. Comput. Appl. Math.. 120, 159–173 (2000). Publisher Full Text OpenURL

  12. Liu, W, Yan, N: A posteriori error estimates for convex boundary control problems. SIAM J. Numer. Anal.. 39, 73–99 (2001). Publisher Full Text OpenURL

  13. Alt, W, Mackenroth, U: Convergence of finite element approximations to state constrained convex parabolic boundary control problems. SIAM J. Control Optim.. 27, 718–736 (1989). Publisher Full Text OpenURL

  14. Arada, N, Casas, E, Tröltzsch, F: Error estimates for the numerical approximation of a boundary semilinear elliptic control problem. Comput. Optim. Appl.. 31, 193–219 (2005). Publisher Full Text OpenURL

  15. Chen, Y, Lu, Z: Error estimates of fully discrete mixed finite element methods for semilinear quadratic parabolic optimal control problems. Comput. Methods Appl. Mech. Eng.. 199, 1415–1423 (2010). Publisher Full Text OpenURL

  16. Chen, Y, Lu, Z: Error estimates for parabolic optimal control problem by fully discrete mixed finite element methods. Finite Elem. Anal. Des.. 46, 957–965 (2010). Publisher Full Text OpenURL

  17. Lu, Z, Chen, Y: L -error estimates of triangular mixed finite element methods for optimal control problem govern by semilinear elliptic equation. Numer. Anal. Appl.. 2, 74–86 (2009). Publisher Full Text OpenURL

  18. Lu, Z, Chen, Y: A posteriori error estimates of triangular mixed finite element methods for semilinear optimal control problems. Adv. Appl. Math. Mech.. 1, 242–256 (2009)

  19. Lions, JL: Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin (1971)

  20. Miliner, FA: Mixed finite element methods for quasilinear second-order elliptic problems. Math. Comput.. 44, 303–320 (1985). Publisher Full Text OpenURL

  21. Liu, W, Yan, N: A posteriori error estimates for control problems governed by nonlinear elliptic equation. Adv. Comput. Math.. 15, 285–309 (2001). Publisher Full Text OpenURL

  22. Chen, Y, Liu, W: A posteriori error estimates for mixed finite element solutions of convex optimal control problems. J. Comput. Appl. Math.. 211, 76–89 (2008). PubMed Abstract | Publisher Full Text OpenURL

  23. Thomée, V: Galerkin Finite Element Methods for Parabolic Problems, Springer, Berlin (1997)