ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC

(1)

Differential Inclusions, Control and Optimization 27 (2007 ) 7–22

ERROR ESTIMATES FOR FINITE ELEMENT APPROXIMATIONS OF ELLIPTIC

CONTROL PROBLEMS

Walter Alt

Friedrich-Schiller-Universit¨ at Jena Institute for Applied Mathematics

D–07740 Jena, Germany Nils Br¨ autigam

Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg Institute for Applied Mathematics

D–91058 Erlangen, Germany and

Arnd R¨ osch

Johann Radon Institute for Computational and Applied Mathematics (RICAM), A–4040 Linz, Austria

Abstract

We investigate finite element approximations of one-dimensional elliptic control problems. For semidiscretizations and full discretizations with piecewise constant controls we derive error estimates in the maximum norm.

Keywords: Linear quadratic optimal control problems, elliptic equations, finite element approximations, error estimates.

2000 Mathematics Subject Classification: 49K20, 49M25, 65N30.

(2)

1. Introduction

The paper discusses the one-dimensional elliptic optimal control problem (CP1) min J(z, u) = 1

2 Z

T

0

|z(t) − z

d

(t)|

²

+ ν |u(t)|

²

dt s.t.

−¨z(t) + Az(t) = Bu(t) + e(t) for a.a. t ∈ [0, T ] , z(0) = z(T ) = 0 ,

a ≤ u(t) ≤ b for a.a. t ∈ [0, T ] ,

where u ∈ L

²

(0, T ; R

^m

), z, z

_d

∈ W

2²

(0, T ; R

ⁿ

), e ∈ L

^∞

(0, T ; R

^m

), A ∈ R

^n×n

is positive semidefinite, B ∈ R

^n×m

and a, b ∈ R

^m

, a < b.

Hinze [6] and Meyer/R¨osch [8] consider multidimensional elliptic control problems and derive error estimates in the L

²

-norm for finite element discretizations. We consider only one-dimensional problems, however, we deal with vector-valued controls and states. Based on the ideas of Sendov/Popov [10] we derive error estimates in the L

^∞

-norm, and we prove a convergence result according to Meyer/R¨osch [8].

The following notations are used. By X(0, T ; R

ⁿ

) we denote a space of functions on [0, T ] with values in R

ⁿ

. By L

²

(0, T ; R

ⁿ

) we denote the Hilbert space of square-integrable functions with the usual scalar product (·, ·) and the corresponding norm k · k

2

. By L

^∞

(0, T ; R

ⁿ

) we denote the space of essentially bounded functions with norm k · k

∞

, by W

_p^k

(0, T ; R

ⁿ

) the space of absolutely continuous functions which are k times differentiable a.e. on [0, T ] and whose k-th derivative belongs to L

^p

(0, T ; R

ⁿ

). The norm of a function z ∈ W

2¹

(0, T ; R

ⁿ

) is defined by kzk

1,2

= kzk

²2

+ k ˙zk

²2

¹₂

. By W

_p,0^k

(0, T ; R

ⁿ

) we denote the space of functions z ∈ W

p^k

(0, T ; R

ⁿ

) satisfying the boundary conditions z(0) = z(T ) = 0. For a matrix A ∈ R

^n×n

we use the matrix norm kAk = sup

|x|=1

|Ax|, where | · | denotes the Euclidian norm of a vector in R

ⁿ

.

2. Optimality conditions

In the case of one-dimensional domains considered here the state equation

(1) −¨z(t) + Az(t) = y(t) for a.a. t ∈ [0, T ] ,

z(0) = 0 , z(T ) = 0 ,

(3)

is equivalent to its variational formulation

(2) ( ˙z, ˙v) + (Az, v) = (y, v) ∀v ∈ V

0

= W

_2,0¹

(0, T ; R

ⁿ

)

(see e.g., Br¨autigam [2]). By S : L

²

(0, T ; R

ⁿ

) → W

2,0¹

(0, T ; R

ⁿ

) we denote the continuous linear operator which assigns to each y ∈ L

²

(0, T ; R

ⁿ

) the unique solution z = S(y) of (2). With the aid of the operator S, Problem (CP1) can be equivalently written in the form

(CP2) min F (u) = J(S(Bu + e), u) s.t. u ∈ U

ad

,

where U

_ad

= {u ∈ L

²

(0, T ; R

^m

) | a ≤ u(t) ≤ b for a.a.t ∈ [0, T ]}.

Problem (CP2) has a unique solution ¯ u which is characterized by the pointwise variational inequality

(3)

B

^T

p(t) + ν ¯ ¯ u(t)

T

(u − ¯u(t)) ≥ 0 ∀u ∈ U

for a.a. t ∈ [0, T ], where U = {u ∈ R

^m

| a ≤ u ≤ b}, ¯p = S

^∗

(¯ z − z

d

) = S(¯ z − z

d

) is the solution of the adjoint equation, and ¯ z = S(B ¯ u + e) is the solution of (2). Equation (3) implies that ¯ u(t) is the projection of −

¹ν

B

^T

p(t) ¯ onto [a, b], i.e.,

(4) u(t) = P r ¯

_[a,b]

− 1

ν B

^T

p(t) ¯

(compare e.g. Malanowski [7]). This further implies ¯ u ∈ W

∞¹

(0, T ; R

^m

).

3. Discretization of the state equation

For discretizations of Problem (CP2) we use a finite element discretization of the state equation. To this end we define a mesh size h = T /N for N ≥ 2 and a uniform grid

(5) G = {t

ⁱ

= ih | i = 0, . . . , N}

on [0, T ]. In the variational equation (2) we replace the space V

₀

by the

finite-dimensional subspace V

_h,0

of continuous, piecewise linear functions v

_h

(4)

on the grid G satisfying the boundary conditions v

_h

(0) = v

_h

(T ) = 0. In this way we obtain the finite element discretization

(6) ( ˙z, ˙v

_h

) + (Az, v

_h

) = (y, v

_h

) ∀v

h

∈ V

h,0

of the state equation.

The functions v

_h^(j)

, j = 1, . . . , N − 1, defined by

(7) v

_h^(j)

(t) =



 



 



t − t

j−1

h , t ∈ [t

j−1

, t

_j

] t

_j+1

− t

h , t ∈ [t

j

, t

_j+1

]

0, else,

form a basis of V

_h,0

.

By S

_h

: L

²

(0, T ; R

ⁿ

) → V

h,0

we denote the continuous linear operator which assigns to each y ∈ L

²

(0, T ; R

ⁿ

) the unique solution z

h

= S

h

(y) of (6).

The operators S

h

are stable in the following sense:

Theorem 1. There are nonnegative constants c

₁

and c

₂

independent of h such that

kS

h

(y)k

2

≤ c

1

kyk

2

∀y ∈ L

²

(0, T ; R

ⁿ

) , and

kS

^h

(y)k

^∞

≤ c

²

kyk

^∞

∀y ∈ L

^∞

(0, T ; R

ⁿ

) . P roof. The inequalities directly follow from (6) with v

_h

= z

_h

.

Estimates for the discretization error S(y) − S

h

(y) play a crucial role in the proof of error estimates for discretizations of Problem (CP2). For estimates in the L

²

-norm we use the Aubin-Nitsche-Lemma (compare Aubin [1], Nitsche [9], Ciarlet [4]).

Lemma 2. For an arbitrary y ∈ L

²

(0, T ; R

ⁿ

) the error estimate kS(y) − S

^h

(y)k

²

≤ c kyk

²

h

²

holds with a constant c ≥ 0 independent of y and h.

For error estimates in the L

^∞

-norm Sendov/Popov [10] state a result for

scalar equations (see [10], Theorem 7.8). The following extension to systems

of equations has been proved in Br¨autigam [2].

(5)

Theorem 3. Let y ∈ L

^∞

(0, T ; R

ⁿ

). Then the error estimate kS(y) − S

h

(y)k

∞

≤ c kyk

∞

h

²

holds with a constant c ≥ 0 independent of y and h.

4. Semi-discretization of the control problem

We consider the following discretization of Problem (CP2), where only the state variables are discretized:

(CP3) min F

_h

(u) = J(S

_h

(Bu + e), u) s.t. u ∈ U

ad

.

As in the case of Problem (CP2) it is well-known that Problem (CP3) has a unique solution ¯ u

_h

which is characterized by the pointwise variational inequality

(8)

B

^T

p ¯

_h

(t) + ν ¯ u

_h

(t)

T

(u − ¯u

h

(t)) ≥ 0 ∀u ∈ U

for a.a. t ∈ [0, T ], where U = {u ∈ R

^m

| a ≤ u ≤ b}, ¯ p

_h

= S

_h^∗

(¯ z

_h

− z

d

) = S

h

(¯ z

h

− z

^d

) is the solution of the discretized adjoint equation and ¯ z

h

= S

_h

(B ¯ u

_h

+ e) is the solution of the discretized state equation (compare e.g., Malanowski [7], Tr¨oltzsch [11]).

Since the optimal control ¯ u is feasible for (CP3) we can use (8) to prove the following estimate in the L

^∞

-norm (see Hinze [6], Theorem 3.6):

k¯u − ¯u

^h

k

∞

≤ C h

k(S

^∗

− S

h^∗

)(S(B ¯ u + e) − z

^d

)k

∞

+ h

²

kB¯uk

2

i .

Combining this result with Theorem 3 and using the fact that S

^∗_h

= S

_h

and S

^∗

= S we obtain

Theorem 4. For the solution ¯ u to Problem (CP2) and the solution ¯ u

_h

to Problem (CP3) the error estimate

k¯u − ¯u

^h

k

^∞

≤ c h

²

holds with a constant c ≥ 0 independent of h.

(6)

5. Full discretization of the control problem

For a full discretization we use the grid G defined by (5) and we denote by U

_h

the finite-dimensional space of piecewise constant polynomials on the grid G. Further we define

s

_i

= 1

2 (t

_i

+ t

_i+1

) , i = 0, . . . , N − 1 . In this way we obtain the fully discretized problem (CP4) min F

_h

(u) = J(S

_h

(Bu + e), u)

s.t. u ∈ U

ad

∩ U

h

.

As in the case of Problems (CP2) and (CP3) it is well-known that Prob- lem (CP4) has a unique solution ¯ u

h

which is characterized by the pointwise variational inequality

(9)

B

^T

p ¯

_h

(s

i

) + ν ¯ u

_h

(s

i

)

_T

(u − ¯u

h

(s

i

)) ≥ 0 ∀u ∈ U

for i = 0, . . . , N − 1, where U = {u ∈ R

^m

| a ≤ u ≤ b}, ¯p

h

= S

_h^∗

(¯ z

_h

− z

d

) = S

h

(¯ z

h

− z

^d

) is the solution of the discretized adjoint equation and ¯ z

h

= S

_h

(B ¯ u

_h

+ e) is the solution of the discretized state equation.

For y ∈ L

²

(0, T ; R

ⁿ

) we define Y (t) =

Z

t 0

y(s) ds , t ∈ [0, T ] .

In view of estimates for the discretization error P

_h

u − ¯u ¯

h

we need the following auxiliary result.

Lemma 5. Let y ∈ L

²

(0, T ; R

ⁿ

) and let z

h

= S

h

(y) be the solution of (6).

Then we have

(10) kz

h

k

∞

≤ c kY k

1

with some constant c ≥ 0 independent of y.

P roof. We denote by z

⁰_h

∈ V

h,0

the solution of (6) for A = 0. Then Z

T

0

( ˙z

_h⁰

(t) + Y (t))

^T

˙v

_h

(t) dt = 0 ∀v

h

∈ V

h,0

.

(7)

Choosing the basis functions v

_h

= v

^(j)_h

, j = 1, . . . , N − 1, defined by (7), we obtain

− z

_h⁰

(t

j+1

) − 2z

⁰h

(t

j

) + z

⁰_h

(t

j−1

)

h + 1

h Y

_j−1

− 1

h Y

_j

= 0 , for j = 1, . . . , N − 1, where

Y

_j

= 1 h

Z

_t_j+1

tj

Y (t) dt , j = 0, . . . , N − 1 .

This shows that the derivative of the continuous, piecewise linear function w

h

defined by

w

_h

(t

_j

) = −z

h⁰

(t

_j+1

) + z

_h⁰

(t

_j

) − Y

j

, j = 0, . . . , N − 1 ,

vanishes. Hence, with some constant q we have w

h

≡ q on [t

0

, t

_N−1

], i.e., z

_h⁰

(t

j+1

) − z

h⁰

(t

j

) + Y

j

= q , j = 0, . . . , N − 1 ,

From z

⁰_h

(0) = z

_h⁰

(T ) = 0 we obtain

0 = z

_h⁰

(T ) = h

N−1

X

j=0

z

⁰_h

(t

j+1

) − z

h⁰

(t

j

)

h = N q −

N−1

X

j=0

Y

_j

,

and therefore

q = 1 N

N−1

X

j=0

Y

_j

= 1 N

Z

T 0

Y (t) dt .

Again with z

⁰_h

(0) = 0 this implies

z

⁰_h

(t

i

) = h

i−1

X

j=0

z

_h⁰

(t

j+1

) − z

h⁰

(t

j

)

h = iq −

i−1

X

j=0

Y

j

= i

N − 1 X

ⁱ⁻¹

j=0

Y

_j

+ i N

N−1

X

j=i

Y

_j

,

(8)

for i = 1, . . . , N − 1. This further implies

|z

h⁰

(t

_i

)| ≤

N−1

X

j=0

|Y

j

| , i = 1, . . . , N − 1 .

Using the boundary conditions z

_h⁰

(0) = z

_h⁰

(T ) = 0 again we obtain

kz

h⁰

k

^∞

≤

N−1

X

j=0

|Y

^j

| ≤ Z

T

0

|Y (t)| dt = kY k

¹

.

Observing that z

h

− z

⁰h

solves the discretized variational equation ( ˙z, ˙v

_h

) + (Az, v

_h

) = (Az

_h⁰

, v

_h

) ∀v

h

∈ V

h,0

it finally follows from Theorem 1 that

kz

h

k

∞

≤ kz

h

− z

h⁰

k

∞

+ kz

h⁰

k

∞

≤ c

2

kAk kz

h⁰

k

∞

+ kz

h⁰

k

∞

≤ (c

2

kAk + 1) kY k

1

, which proves the assertion.

In the following, we use the notation BV (0, T ; R

^m

) for the linear space of functions u : [0, T ] → R

^m

of bounded variation and V

_τ^σ

(u) for the variation of the function u on the interval [τ, σ]. Further, we define z

_h

(u) = S

_h

(Bu+e) and we approximate a control function u by the piecewise constant function P

_h

u defined by

(P

h

u)(t) = u(s

i

) ∀t ∈ [t

ⁱ

, t

_i+1

[ , i = 0, . . . , N − 1 .

Lemma 6. Let ¯ u be the solution to Problem (CP2), and suppose that ˙¯ u ∈ BV (0, T ; R

^m

). Then

(11) kz

h

(¯ u) − z

h

(P

_h

u)k ¯

∞

≤ c (V

0^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

, with some constant c ≥ 0 independent of ¯u and h.

P roof. The piecewise linear function z

h

(¯ u) − z

^h

(P

h

u) ∈ V ¯

h,0

solves the discretized variational equation (6) for y = B(¯ u − P

h

u). Therefore, with ¯

Y (t) = B Z

t

0

(¯ u − P

h

u)(s) ds ¯

(9)

it follows from Lemma 5 that

kz

h

(¯ u) − z

h

(P

_h

u)k ¯

∞

≤ c

1

kY k

1

= c

₁

N−1

X

i=0

Z

ti+1

ti

|Y (t)| dt .

By the definition of Y we have Y (t) = Y (t

_i

) + B

Z

t ti

(¯ u − P

h

u)(s) ds ¯ ∀t ∈ [t

i

, t

_i+1

] . and therefore

(12)

N−1

X

i=0

Z

ti+1

ti

|Y (t)| dt =

N−1

X

i=0

Z

ti+1

ti

Y (t

ⁱ

) + Z

t

ti

B(¯ u − P

h

u)(s) ds ¯ dt.

Because of Y (0) = 0 the first term on the right hand side of (12) can be estimated by

Z

ti+1

ti

|Y (t

i

)| ds = h

i−1

X

j=0

Y (t

j+1

) − Y (t

j

) = h

i−1

X

j=0

Z

tj+1

tj

Y (t) dt ˙

≤ h kBk

i−1

X

j=0

Z

tj+1

tj

u(t) − ¯u(s ¯

^j

) dt

≤ h kBk

i−1

X

j=0

Z

^h₂

0

(¯ u(s

_j

+ t) − ¯u(s

j

)) − (¯u(s

j

) − ¯u(s

j

− t) dt

≤ h kBk

N−1

X

j=0

Z

^h₂

0

Z

t

0

| ˙¯u(s

j

+ s) − ˙¯u(s

j

+ s − t)| ds dt .

Assuming ˙¯ u ∈ BV (0, T ; R

^m

) we obtain Z

ti+1

ti

|Y (t

i

)| ds ≤ h

³

4 kBk

N−1

X

j=0

V

_t^t_j^j+1

( ˙¯ u) = h

³

4 kBk V

0^T

( ˙¯ u) , and therefore

(13)

N−1

X

i=0

Z

ti+1

ti

|Y (t

ⁱ

)| dt ≤ T

4 kBk V

0^T

( ˙¯ u) h

²

.

(10)

The second term on the right hand side of (12) can be estimated by

N−1

X

i=0

Z

ti+1

ti

Z

t ti

(¯ u − P

h

u)(s) ds ¯ dt ≤

N−1

X

i=0

Z

ti+1

ti

Z

ti+1

ti

|(¯u − P

h

u)(s)| ds dt ¯

= h

N−1

X

i=0

Z

ti+1

ti

|¯u(s) − ¯u(s

i

)| ds ≤ h k ˙¯uk

∞ N−1

X

i=0

Z

ti+1

ti

|s − s

i

| ds

≤ h k ˙¯uk

∞ N−1

X

i=0

h

²

4 = T

4 k ˙¯uk

∞

h

²

. Together with (12) and (13) we obtain

Z

T

0

|Y (t)| dt ≤ T

4 kBk (V

0^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

. Choosing c =

^T₄

kBk we get (11).

Defining p

h

(u) = S

^∗_h

(S

h

(Bu + e) − z

^d

) we obtain from Lemma 6:

Corollary 7. Let ¯ u be the solution to Problem (CP2), and suppose ˙¯ u ∈ BV (0, T ; R

^m

). Then

(14) kp

h

(¯ u) − p

h

(P

_h

u)k ¯

∞

≤ c

1

(V

₀^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

and

(15) k¯ p − p

h

(P

_h

u)k ¯

∞

≤ c

2

(kB¯u + ek

∞

+ kz

d

k

∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

with nonnegative constants c

1

and c

2

independent of ¯ u and h.

P roof. The estimate (14) is an immediate consequence of Theorem 1 and Lemma 6. Since

k¯ p − p

h

(P

_h

u)k ¯

∞

≤ k¯ p − p

h

(¯ u)k

∞

+ kp

h

(¯ u) − p

h

(P

_h

u)k ¯

∞

the second estimate (15) follows from Theorem 3 and (14).

Now we prove a result on discrete convergence of the optimal controls, which

is similar to that of Theorem 2.3 in Meyer/R¨osch [8]. However, Assump-

tion (A3) used in [8] is replaced by a weaker assumption that the derivative

of the optimal control ˙¯ u has bounded variation.

(11)

Theorem 8. Let ¯ u be the solution to Problem (CP2), ¯ u

_h

the solution to the discretized Problem (CP4) and suppose ˙¯ u ∈ BV (0, T ; R

^m

). Then

(16) k¯u

h

− P

h

uk ¯

2

≤ c (kB¯u + ek

∞

+ kz

d

k

∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

with a constant c ≥ 0 independent of ¯u and h.

P roof. Let p

_h

(u) = S

_h^∗

(z

_h

(u)−z

d

) be the solution of the discretized adjoint equation and z

_h

(u) = S

_h

(Bu + e) the solution of the discretized state equation. Based on the optimality conditions for Problems (CP2) and (CP4) we obtain with the help of the same arguments as in Meyer/R¨osch [8]

ν k¯u

h

− P

h

uk ¯

²2

≤ (P

h

p − p ¯

h

(¯ u

_h

), B(¯ u

_h

− P

h

u)) . ¯ For the right hand side we find

(P

_h

p − p ¯

h

(¯ u

_h

), B(¯ u

_h

− P

h

u)) = (p ¯

_h

(P

_h

u) − p ¯

h

(¯ u

_h

), B(¯ u

_h

− P

h

u)) ¯ +(¯ p − p

^h

(P

h

u), B(¯ ¯ u

h

− P

^h

u)) ¯ +(P

_h

p − ¯p, B(¯u ¯

h

− P

h

u) . ¯

The first and third term on the right hand side can be estimated according to Meyer/R¨osch [8], Section 4. For the second term we obtain from (15)

(¯ p − p

h

(P

_h

u), B(¯ ¯ u

_h

− P

h

u)) ¯

≤ kBk k¯ p − p

h

(P

_h

u)k ¯

2

k¯u

h

− P

h

uk ¯

2

≤ ˜ck¯u

^h

− P

^h

uk ¯

²

(kB¯u + ek

^∞

+ kz

^d

k

^∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

^∞

) h

²

with a constant ˜ c ≥ 0 independent of h. Now we can proceed in the same way as in Meyer/R¨osch [8], Section 4.

Example 9. In order to illustrate the result of Theorem 8, we consider the same problem as in Hinze [6], Section 4.2, for which the optimal solution is known. In this problem, we choose T = 1, z

d

≡ 2, ν = 0.1, a = −∞, b = 2.5( √

2 − 1), and e(t) = −2 + t

²

− t − min(−

¹_ν

(t

²

− t), b). The optimal control is

u(t) = min(− ¯

¹ν

(t

²

− t), b) .

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Figure 1 shows the optimal control ¯ u and the discrete optimal control ¯ u

_h

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.5 1 1.5 2 2.5

t

Figure 1. Controls ¯ u (thin line) and ¯ u

^h

(thick line).

Based on the projection formula (4) we use the solution ¯ u

h

of the discretized Problem (CP4) to define a new control ˜ u ∈ U

ad

by

(17) u = P r ˜

_[a,b]

− 1

ν B

^T

S

_h^∗

(S

_h

(B ¯ u

_h

+ e) − z

d

)

.

Meyer/R¨osch [8] have shown a superconvergence estimate for this control in the L

²

-norm. We prove a corresponding result using the L

^∞

-norm.

Theorem 10. Let ¯ u be the solution to Problem (CP2), ¯ u

h

the solution to the discretized Problem (CP4), and suppose ˙¯ u ∈ BV (0, T ; R

^m

). If ˜ u is defined by (17), then we have

(18) k¯u − ˜uk

∞

≤ c (kB¯u + ek

∞

+ kz

d

k

∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

with a constant c ≥ 0 independent of ¯u and h.

P roof. With the notations of Theorem 8 we have

k¯ p − p

^h

(¯ u

h

)k

^∞

≤ k¯ p − p

^h

(P

h

u)k ¯

^∞

+ kp

^h

(P

h

u) − p ¯

^h

(¯ u

h

)k

^∞

.

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For the first term on the right hand side we obtain from Corollary 7 k¯ p − p

^h

(P

h

u)k ¯

∞

≤ c

1

(kB¯u + ek

∞

+ kz

^d

k

∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

∞

) h

²

. For the last term we have by Theorem 1, Lemma 2, and Theorem 8

kp

h

(P

_h

u) − p ¯

h

(¯ u

_h

)k

∞

≤ ˜c

2

kP

h

u − ¯u ¯

h

k

2

≤ c

²

(kBu + ek

^∞

+ kz

^d

k

^∞

+ V

₀^T

( ˙¯ u) + k ˙¯uk

^∞

) h

²

. Combining these estimates and using the fact that the operator P r

_[a,b]

is Lipschitz continuous, we get the desired result.

Example 11. In order to illustrate the result of Theorem 10 we consider the same problem as in Example 9. Figure 2 shows the function

w = − 1

ν B

^T

S

_h^∗

(S

_h

(B ¯ u

_h

+ e) − z

d

)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

0 0.5 1 1.5 2 2.5

t

Figure 2. Control ˜ u (thick line) and function w (thin line).

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and the control ˜ u = P r

_[a,b]

(w) defined by (17). Column 3 of Table 1 and Figure 3 show the linear order of convergence w.r.t. the mesh size h of the error k¯u − ¯u

^h

k

^∞

, while columns 4 and 5 of Table 1 and Figure 4 show quadratic convergence w.r.t. the mesh size h of the error k¯u − ˜uk

∞

.

h F

_h

(¯ u

_h

) k¯u − ¯u

h

k

∞

k¯u − ˜uk

∞

k¯u − ˜uk

∞

/h

²

1/3 2.6535 1.0355 0.2532 2.2787

1/4 2.5621 0.9382 0.1550 2.4797

1/6 2.4654 0.6952 0.0686 2.4712

1/10 2.4017 0.4502 0.0248 2.4843 1/20 2.3710 0.2375 0.0062 2.4953 1/50 2.3618 0.0980 9.9840 · 10

⁻⁴

2.4963 1/100 2.3604 0.0495 2.4985 · 10

⁻⁴

2.4985

Table 1. Error as a function of mesh size h.

0 0.05 0.1 0.15 0.2 0.25

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

h

Figure 3. Error k¯u − ¯u

^h

k

^∞

as a function of mesh size h.

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0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0

0.05 0.1 0.15 0.2 0.25 0.3 0.35

h

Figure 4. Error k¯u − ˜uk

^∞

as a function of mesh size h.

References

[1] J.P. Aubin, Behaviour of the error of the approximate solution of boundary value problems for linear elliptic operators by Galerkin’s and finite difference methods, Ann. Scoula Norm. Sup. Pisa 21 (1967), 599–637.

[2] N. Br¨ autigam, Diskretisierung elliptischer Steuerungsprobleme, Ph.D. Thesis, Jena 2006.

[3] N. Br¨ autigam, Discretization of Elliptic Control Problems by Finite Elements, Technical Report, Jena 2006.

[4] P.G. Ciarlet, The Finite Element Method for Elliptic Problems, North Holland 1987.

[5] C. Großmann and H.-G. Roos, Numerik partieller Differentialgleichungen, Teubner 2005.

[6] M. Hinze, A Variational Discretization Concept in Control Constrained Opti- mization: The Linear-Quadratic Case, Computational Optimization and Ap- plications 30 (2005), 45–61.

[7] K. Malanowski, Convergence of Approximations vs. Regularity of Solutions for Convex, Control-Constrained Optimal Control Problems, Appl. Math. Optim.

8 (1981), 69–95.

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[8] C. Meyer and A. R¨ osch, Superconvergence Properties of Optimal Control Prob- lems, SIAM J. Contr. Opt. 43 (2004), 970–985.

[9] J.A. Nitsche, Ein Kriterium f¨ ur die Quasioptimalit¨ at des Ritzschen Verfahrens, Numerische Mathematk 11 (1968), 346–348.

[10] B. Sendov and V.A. Popov, The Averaged Moduli of Smoothness, Wiley- Interscience 1988.

[11] F. Tr¨ oltzsch, Optimale Steuerung partieller Differentialgleichungen, Vieweg 2005.

Received 27 February 2006