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

Boundary Value Problems20132013:72

DOI: 10.1186/1687-2770-2013-72

Received: 18 January 2013

Accepted: 14 March 2013

Published: 4 April 2013

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 [46].

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., [710]. 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 [1518], 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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq1_HTML.gif for Sobolev spaces on Ω with a norm m , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq2_HTML.gif given by v m , p p = | α | m D α v L p ( Ω ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq3_HTML.gif and a semi-norm | | m , p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq4_HTML.gif given by | v | m , p p = | α | = m D α v L p ( Ω ) p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq5_HTML.gif. We set W 0 m , p ( Ω ) = { v W m , p ( Ω ) : v | Ω = 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq6_HTML.gif. For p = 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq7_HTML.gif, we denote H m ( Ω ) = W m , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq8_HTML.gif, H 0 m ( Ω ) = W 0 m , 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq9_HTML.gif, and m = m , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq10_HTML.gif, = 0 , 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq11_HTML.gif. We denote by L s ( 0 , T ; W m , p ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq12_HTML.gif the Banach space of all L s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq13_HTML.gif integrable functions from J into W m , p ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq1_HTML.gif with the norm v L s ( J ; W m , p ( Ω ) ) = ( 0 T v W m , p ( Ω ) s d t ) 1 s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq14_HTML.gif for s [ 1 , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq15_HTML.gif, and the standard modification for s = http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq16_HTML.gif. 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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ1_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ2_HTML.gif
(2)
( A y ( x , t ) ) n = B u ( x , t ) + z b , x Ω , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ3_HTML.gif
(3)
y ( x , 0 ) = y 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ4_HTML.gif
(4)
where the bounded open set Ω R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq17_HTML.gif is 2 regular convex polygon with boundary Ω, J = ( 0 , T ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq18_HTML.gif, f L 2 ( J ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq19_HTML.gif, y 0 H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq20_HTML.gif, z b L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq21_HTML.gif, and α is a positive constant. For any I > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq22_HTML.gif, the function ϕ ( ) W 2 , ( I , I ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq23_HTML.gif, ϕ ( y ) L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq24_HTML.gif for any y L 2 ( J ; H 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq25_HTML.gif, and ϕ ( y ) 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq26_HTML.gif. We assume the coefficient matrix A ( x ) = ( a i , j ( x ) ) 2 × 2 ( W 1 , ( Ω ) ) 2 × 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq27_HTML.gif is a symmetric positive definite matrix, and there is a constant c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq28_HTML.gif satisfying for any vector X R 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq29_HTML.gif, X t A X c X R 2 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq30_HTML.gif. 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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ5_HTML.gif
(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 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq31_HTML.gif, W = L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq32_HTML.gif, U = L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq33_HTML.gif. Let
a ( y , w ) = Ω ( A y ) w , y , w V , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ6_HTML.gif
(6)
( f 1 , f 2 ) = Ω f 1 f 2 , ( f 1 , f 2 ) W × W , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ7_HTML.gif
(7)
( u , v ) U = Ω u v , ( u , v ) U × U . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ8_HTML.gif
(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 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ9_HTML.gif
(9)
subject to
( y t , w ) + a ( y , w ) + ( ϕ ( y ) , w ) = ( f , w ) + ( B u + z b , w ) U , w V , t J , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ10_HTML.gif
(10)
y ( x , 0 ) = y 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ11_HTML.gif
(11)

where the inner product in L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq34_HTML.gif or L 2 ( Ω ) 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq35_HTML.gif is indicated by ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq36_HTML.gif, and B is a continuous linear operator from U to L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq34_HTML.gif.

It is well known (see, e.g., [12]) that the optimal control problems have at least a solution ( y , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq37_HTML.gif, and that if a pair ( y , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq37_HTML.gif is the solution of (9)-(11), then there is a co-state p V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq38_HTML.gif such that the triplet ( y , p , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq39_HTML.gif 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ12_HTML.gif
(12)
y ( x , 0 ) = y 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ13_HTML.gif
(13)
( p t , w ) + a ( q , p ) + ( ϕ ( y ) p , q ) = ( y y 0 , q ) , q V = H 1 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ14_HTML.gif
(14)
p ( x , T ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ15_HTML.gif
(15)
0 T ( α u + B p , v u ) U d t 0 , v K U = L 2 ( Ω ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ16_HTML.gif
(16)

where B http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq40_HTML.gif is the adjoint operator of B. In the rest of the paper, we shall simply write the product as ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq36_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq41_HTML.gif be a regular partition of Ω. Associated with T h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq41_HTML.gif is a finite dimensional subspace V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq42_HTML.gif of C ( Ω ¯ ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq43_HTML.gif such that χ | τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq44_HTML.gif are polynomials of m-order ( m 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq45_HTML.gif) χ V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq46_HTML.gif and τ T h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq47_HTML.gif. It is easy to see that V h V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq48_HTML.gif. Let E h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq49_HTML.gif be a partition of Ω into disjoint regular ( n 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq50_HTML.gif-simplices s, so that Ω = s E h s ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq51_HTML.gif. Associated with E h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq49_HTML.gif is another finite dimensional subspace U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq52_HTML.gif of L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq53_HTML.gif such that χ | τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq44_HTML.gif are polynomials of m-order ( m 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq54_HTML.gif) χ U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq55_HTML.gif and s E h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq56_HTML.gif. Let h τ ( h s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq57_HTML.gif denote the maximum diameter of the element τ ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq58_HTML.gif in T h ( E h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq59_HTML.gif, h = max τ T h { h τ } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq60_HTML.gif, and h U = max s E h { h s } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq61_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq62_HTML.gif such that
min u h K h { 0 T ( 1 2 y h y 0 2 + α 2 u h 2 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ17_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ18_HTML.gif
(18)
y h ( x , 0 ) = y 0 h ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ19_HTML.gif
(19)

where K h = K U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq63_HTML.gif, y 0 h V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq64_HTML.gif is an approximation of y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq65_HTML.gif.

Again, it follows that optimal control problem (17)-(19) has at least a solution ( y h , u h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq66_HTML.gif, and that if a pair ( y h , u h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq66_HTML.gif is the solution of (17)-(19), then there is a co-state p h V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq67_HTML.gif such that the triplet ( y h , p h , u h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq68_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ20_HTML.gif
(20)
y h ( x , 0 ) = y 0 h ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ21_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ22_HTML.gif
(22)
p h ( x , T ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ23_HTML.gif
(23)
0 T ( α u h + B p h , v h u h ) U d t 0 , v h K h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ24_HTML.gif
(24)
We now consider the fully discrete approximation for the semidiscrete problem. Let Δ t > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq69_HTML.gif, N = T / Δ t Z http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq70_HTML.gif, and let t i = i Δ t http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq71_HTML.gif, i R http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq72_HTML.gif. Also, let
ψ i = ψ i ( x ) = ψ ( x , t i ) , d t ψ i = ψ i ψ i 1 Δ t . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equa_HTML.gif

For i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq73_HTML.gif, we construct the finite element spaces V i h V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq74_HTML.gif with the mesh T h i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq75_HTML.gif (similar to V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq76_HTML.gif). Similarly, we construct the finite element spaces U i h L 2 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq77_HTML.gif with the mesh T h i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq75_HTML.gif (similar to U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq78_HTML.gif). Let h τ i ( h s i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq79_HTML.gif denote the maximum diameter of the element τ i ( s i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq80_HTML.gif in T h i ( ( E h ) i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq81_HTML.gif. Define mesh functions τ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq82_HTML.gif, s ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq83_HTML.gif and mesh size functions h τ ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq84_HTML.gif, h s ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq85_HTML.gif such that τ ( t ) | t ( t i 1 , t i ] = τ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq86_HTML.gif, s ( t ) | t ( t i 1 , t i ] = s i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq87_HTML.gif, h τ ( t ) | t ( t i 1 , t i ] = h τ i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq88_HTML.gif, h s ( t ) | t ( t i 1 , t i ] = h s i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq89_HTML.gif. For ease of exposition, we denote τ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq90_HTML.gif, s ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq91_HTML.gif, h τ ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq92_HTML.gif, and h s ( t ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq93_HTML.gif by τ, s, h τ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq94_HTML.gif, and h s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq95_HTML.gif, 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq96_HTML.gif, i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq97_HTML.gif, 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 ) } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ25_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ26_HTML.gif
(26)
w h V i h , i = 1 , 2 , , N , y h 0 ( x ) = y 0 h ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ27_HTML.gif
(27)

where K i h = K U i h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq98_HTML.gif, y 0 h V h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq64_HTML.gif is an approximation of y 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq65_HTML.gif.

Now, it follows that optimal control problem (25)-(27) has at least a solution ( Y h i , U h i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq99_HTML.gif, i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq100_HTML.gif, and that if a pair ( Y h i , U h i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq99_HTML.gif, i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq100_HTML.gif, is the solution of (25)-(27), then there is a co-state P h i 1 V i h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq101_HTML.gif, i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq100_HTML.gif, such that the triplet ( Y h i , P h i 1 , U h i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq102_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ28_HTML.gif
(28)
i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ29_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ30_HTML.gif
(30)
i = N , , 2 , 1 , P h N ( x ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ31_HTML.gif
(31)
( α U h i + B P h i , v h U h i ) 0 , v h K i h , i = 1 , 2 , , N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ32_HTML.gif
(32)
For i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq100_HTML.gif, let
Y h | ( t i 1 , t i ] = ( ( t i t ) Y h i 1 + ( t t i 1 ) Y h i ) / Δ t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ33_HTML.gif
(33)
P h | ( t i 1 , t i ] = ( ( t i t ) P h i 1 + ( t t i 1 ) P h i ) / Δ t , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ34_HTML.gif
(34)
U h | ( t i 1 , t i ] = U h i . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ35_HTML.gif
(35)
For any function w C ( 0 , T ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq103_HTML.gif, let w ˆ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq104_HTML.gif, w ˜ ( x , t ) | t ( t i 1 , t i ] = w ( x , t i 1 ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq105_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ36_HTML.gif
(36)
i = 1 , 2 , , N , Y h 0 ( x ) = y 0 h ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ37_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ38_HTML.gif
(38)
i = N , , 2 , 1 , P h ( x , T ) = 0 , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ39_HTML.gif
(39)
( α U h + B P ˜ h , v h U h ) 0 , v h K i h , i = 1 , 2 , , N . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ40_HTML.gif
(40)
In the rest of the paper, we shall use some intermediate variables. For any control function U h K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq106_HTML.gif, we define that the state solution ( y ( U h ) , p ( U h ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq107_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ41_HTML.gif
(41)
y ( U h ) ( x , 0 ) = y 0 ( x ) , x Ω , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ42_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ43_HTML.gif
(43)
p ( U h ) ( x , T ) = 0 , x Ω . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ44_HTML.gif
(44)

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

Lemma 2.1 Let π ˆ h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq108_HTML.gif be the Clément-type interpolation operator defined in [19]. Then for any v H 1 ( Ω ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq109_HTML.gif and all element τ,
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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ45_HTML.gif
(45)
v π ˆ h v L 2 ( l ) C h l 1 / 2 l τ ¯ | v | L 2 ( τ ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ46_HTML.gif
(46)

where l is the edge of the element.

For φ W h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq110_HTML.gif, we write
ϕ ( φ ) ϕ ( ρ ) = ϕ ˜ ( φ ) ( ρ φ ) = ϕ ( ρ ) ( ρ φ ) + ϕ ˜ ( φ ) ( ρ φ ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ47_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equb_HTML.gif

are bounded functions in Ω ¯ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq111_HTML.gif [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 ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq112_HTML.gif.

Theorem 3.1 Let ( y ( U h ) , p ( U h ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq107_HTML.gif and ( Y h , P h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq113_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ48_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equc_HTML.gif
where l is a face of an element τ, h l http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq114_HTML.gif is the size of the face l, [ A y h n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq115_HTML.gif is the A-normal derivative jump over the interior face l defined by
[ A Y h n ] l = ( A Y h | τ l 1 A Y h | τ l 2 ) n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equd_HTML.gif

where n is the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq116_HTML.gif outwards τ l 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq117_HTML.gif.

Proof Let e y = y ( U h ) Y h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq118_HTML.gif, and let e I y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq119_HTML.gif be the Clément-type interpolator of e y http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq120_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Eque_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equf_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ49_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ50_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ51_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ52_HTML.gif
(52)
For K 4 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq121_HTML.gif- K 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq122_HTML.gif, 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ53_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ54_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ55_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ56_HTML.gif
(56)

This completes the proof. □

Analogously to Theorem 3.1, we show the following estimates.

Theorem 3.2 Let ( y ( U h ) , p ( U h ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq107_HTML.gif and ( Y h , P h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq113_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ57_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equg_HTML.gif
where η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq123_HTML.gif- η 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq124_HTML.gif are defined in Theorem  3.1, l is a face of an element τ, [ A P ˜ h n ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq125_HTML.gif is the A-normal derivative jump over the interior face l defined by
[ A P ˜ h n ] l = ( A P ˜ h | τ l 1 A P ˜ h | τ l 2 ) n , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equh_HTML.gif

where n is the unit normal vector on l = τ ¯ l 1 τ ¯ l 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq116_HTML.gif outwards τ l 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq117_HTML.gif.

Proof Let e p = p ( U h ) P h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq126_HTML.gif, and let e I p = π ˆ h e p http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq127_HTML.gif, where π ˆ h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq108_HTML.gif is the Clément-type interpolator defined in Lemma 2.1. Note that ( p ( U h ) P h ) ( x , T ) = 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq128_HTML.gif, then we obtain
0 T ( p t ( U h ) P h t , e p ) d t 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equi_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ58_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ59_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ60_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ61_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ62_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ63_HTML.gif
(63)
Next, for L 6 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq129_HTML.gif, 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ64_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ65_HTML.gif
(65)

This completes the proof. □

For given u K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq130_HTML.gif, let M be the inverse operator of the state equation (12) such that y ( u ) = M B u http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq131_HTML.gif is the solution of the state equation (12). Similarly, for given U h K h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq132_HTML.gif, Y h ( U h ) = M h B U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq133_HTML.gif 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equj_HTML.gif
It is clear that S and S h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq134_HTML.gif are well defined and continuous on K and K h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq135_HTML.gif. Also, the functional S h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq134_HTML.gif can be naturally extended on K. Then (9) and (25) can be represented as
min u K { S ( u ) } , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ66_HTML.gif
(66)
min U h K h { S h ( U h ) } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ67_HTML.gif
(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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equk_HTML.gif

where p ( U h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq136_HTML.gif is the solution of equations (41)-(43).

In many applications, S ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq137_HTML.gif is uniform convex near the solution u (see, e.g., [21]). The convexity of S ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq137_HTML.gif 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 ( ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq137_HTML.gif is uniformly convex, then there is a c > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq28_HTML.gif such that
0 T ( S ( u ) S ( U h ) , u U h ) c u U h L 2 ( J ; L 2 ( Ω ) ) 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ68_HTML.gif
(68)

where u and U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq78_HTML.gif 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 } . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equl_HTML.gif

Then it is clear that three subsets do not intersect each other, and Ω = Ω i Ω i Ω i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq138_HTML.gif, i = 1 , 2 , , N http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq97_HTML.gif.

Let p ( U h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq136_HTML.gif be the solution of (41)-(44). We establish the following error estimate, which can be proved similarly to the proofs given in [22].

Theorem 3.3 Let u and U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq78_HTML.gif 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ69_HTML.gif
(69)
where
η 12 2 = i = 1 N t i 1 t i Ω i | B P ˜ h + α U h | 2 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equm_HTML.gif
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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ70_HTML.gif
(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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ71_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ72_HTML.gif
(72)
Since U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq78_HTML.gif is piecewise constant, U h | s > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq139_HTML.gif if s Ω i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq140_HTML.gif is not empty. If u h | s > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq141_HTML.gif, there exists ε > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq142_HTML.gif and β U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq143_HTML.gif such that β 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq144_HTML.gif, β L ( s ) = 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq145_HTML.gif and ( u h ε β ) | s 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq146_HTML.gif. For example, one can always find such a required β from one of the shape functions on s. Hence, u ˆ h K h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq147_HTML.gif, where u ˆ h = U h ε β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq148_HTML.gif as x s http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq149_HTML.gif and otherwise u ˆ = U h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq150_HTML.gif. 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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ73_HTML.gif
(73)
Note that on Ω i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq151_HTML.gif, B P ˜ h + α U h B P ˜ h > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq152_HTML.gif, 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 | . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ74_HTML.gif
(74)
Let s ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq153_HTML.gif be the reference element of s, s 0 = s Ω i + http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq154_HTML.gif, and s ˆ 0 s ˆ http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq155_HTML.gif be a part mapped from s ˆ 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq156_HTML.gif. Note that ( s | | 2 ) 1 / 2 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq157_HTML.gif, s | | β http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq158_HTML.gif are both norms on L 2 ( s ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq159_HTML.gif. 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ75_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ76_HTML.gif
(76)
It follows from the definition of Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq160_HTML.gif that B P ˜ h + α U h > 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq161_HTML.gif on Ω i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq160_HTML.gif. Note that u 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq162_HTML.gif, we have that
Ω i ( B P ˜ h + α U h ) ( u ) 0 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ77_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ78_HTML.gif
(78)

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

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

Theorem 3.4 Let ( y , p , u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq39_HTML.gif and ( Y h , P h , U h ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq163_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ79_HTML.gif
(79)

where η 1 , η 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq164_HTML.gif , and η 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq165_HTML.gif 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ80_HTML.gif
(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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ81_HTML.gif
(81)
for all w V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq166_HTML.gif and q V http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq167_HTML.gif. 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 ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ82_HTML.gif
(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 ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ83_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ84_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ85_HTML.gif
(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 . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equn_HTML.gif

In the example, we choose the domain Ω = [ 0 , 1 ] × [ 0 , 1 ] http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq168_HTML.gif and K = { u L 2 ( J ; L 2 ( Ω ) ) : u 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq169_HTML.gif. Let Ω be partitioned into T h http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq170_HTML.gif as described in Section 2. We use η 12 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq165_HTML.gif as the control mesh refinement indicator and η 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq123_HTML.gif- η 11 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq171_HTML.gif as the states and co-states.

For the constrained optimization problem min u K S ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq172_HTML.gif, where S ( u ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq173_HTML.gif is a convex functional on U, the iterative scheme reads ( n = 0 , 1 , 2 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq174_HTML.gif)
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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ86_HTML.gif
(86)
where b ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq175_HTML.gif is a symmetric and positive definite bilinear form such that there exist constants c 0 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq176_HTML.gif and c 1 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq177_HTML.gif satisfying
b ( u , u ) c 0 u U 2 , | b ( u , v ) | c 1 u U v U , u , v U , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ87_HTML.gif
(87)
and the projection operator P K b U K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq178_HTML.gif is defined as follows. For given w U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq179_HTML.gif, find P K b w K http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq180_HTML.gif such that
b ( P K b w w , P K b w w ) = min u K b ( u w , u w ) . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ88_HTML.gif
(88)
The bilinear form b ( , ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq175_HTML.gif 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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ89_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ90_HTML.gif
(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 , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ91_HTML.gif
(91)
u n + 1 i = P k b ( u n + 1 2 i ) , u n + 1 2 i , u n i K i h . http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equ92_HTML.gif
(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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq181_HTML.gif. 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq181_HTML.gif efficiently. For the piecewise constant elements, K h = { u h K : u h 0 } http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq182_HTML.gif and b ( u , v ) = ( u , v ) U http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq183_HTML.gif, then
P K b u n + 1 2 i | T = max ( 0 , avg ( u n + 1 2 i ) | T ) , http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_Equo_HTML.gif

where avg ( u n + 1 2 i ) | T http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq184_HTML.gif is the average of u n + 1 2 i http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq185_HTML.gif 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 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq186_HTML.gif and a fixed step size ρ = 0.8 http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq187_HTML.gif. 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

Mesh information

Uniform mesh

Adaptive mesh

u-elements

15,872

1,148

u-sides

23,968

1,868

u-nodes

8,097

721

y,p-elements

15,872

1,754

y,p-sides

23,968

2,725

y,p-nodes

8,097

972

u u h L 2 ( J ; L 2 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq188_HTML.gif

4.38920e-02

4.27977e-02

y y h L 2 ( J ; H 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq189_HTML.gif

9.80281e-02

9.62631e-02

p p h L 2 ( J ; H 1 ( Ω ) ) http://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-72/MediaObjects/13661_2013_Article_325_IEq190_HTML.gif

4.39287e-03

4.17962e-03

Author’s contributions

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

Declarations

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.

Authors’ Affiliations

(1)
School of Mathematics and Statistics, Chongqing Three Gorges University
(2)
College of Civil Engineering and Mechanics, Xiangtan University

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