Comput Optim Appl
DOI 10.1007/s10589-013-9537-8
Optimal error estimates for finite element discretization
of elliptic optimal control problems with finitely many
pointwise state constraints
Dmitriy Leykekhman · Dominik Meidner ·
Boris Vexler
Received: 10 October 2012
© Springer Science+Business Media New York 2013
Abstract In this paper we consider a model elliptic optimal control problem with
finitely many state constraints in two and three dimensions. Such problems are challenging due to low regularity of the adjoint variable. For the discretization of the
problem we consider continuous linear elements on quasi-uniform and graded meshes
separately. Our main result establishes optimal a priori error estimates for the state,
adjoint, and the Lagrange multiplier on the two types of meshes. In particular, in three
dimensions the optimal second order convergence rate for all three variables is possible only on properly refined meshes. Numerical examples at the end of the paper
support our theoretical results.
Keywords Optimal control · Finite elements · Error estimates · State constraints
1 Introduction
In this paper we consider the following optimal control problem
Minimize
1
α
u − ud 2 + q2
2
2
P. Leykekhman was supported by NSF grant DMS-1115288.
D. Leykekhman
Department of Mathematics, University of Connecticut, 196 Auditorium Road, Storrs,
CT 06269-3009, USA
e-mail: [email protected]
D. Meidner · B. Vexler ()
Lehrstuhl für Mathematische Optimierung, Fakultät für Mathematik,
Technische Universität München, Boltzmannstraße 3, 85748 Garching b. München, Germany
e-mail: [email protected]
D. Meidner
e-mail: [email protected]
(1a)
D. Leykekhman et al.
Table 1 Proved orders of
convergence on quasi-uniform
meshes
Dimension
q̄ − q̄h ū − ūh |μ̄ − μ̄h |
d =2
1
2
2
d =3
1
2
1
1
subject to the state equation
−u = q
in Ω,
u = 0 on ∂Ω
(1b)
and to finitely many pointwise state constraints
u(xi ) = bi
i = 1, 2, . . . , n,
(1c)
with mutually distinct points x1 , x2 , . . . , xn ∈ Ω. Here q and u denote the control and
the state variable respectively, α > 0 is the regularization parameter and · denotes
the L2 -norm on the domain Ω, see Sect. 2 for a precise functional analytic setting.
Such optimal control problems with finitely many state constraints are motivated by
technological applications and are analyzed in, e.g., [6–9] and also in [23] for the
case of finite dimensional control variable. In addition, the investigation of finitely
many state constraints is also helpful in the context of optimal control problems with
pointwise constraints on the whole domain of type u(x) ≤ b almost everywhere in Ω.
The corresponding active set may consists only of finitely many points, see, e.g., [24].
The main difference is that the active points are not known a priori in the contrast to
the problem considered here.
The focus of the paper is the a priori error analysis for the finite element discretization of (1a), (1b), (1c). We discretize the state and the control by linear finite
elements, see Sect. 3 for details. We denote by q̄, ū and μ̄ the optimal control, optimal state and the Lagrangian multiplier for the state constrains. The corresponding
discrete solutions are denoted by q̄h , ūh and μ̄h and the aim is to analyze the errors
q̄ − q̄h , ū − ūh and |μ̄ − μ̄h |. We make the analysis for two types of families of
meshes: quasi-uniform meshes and graded meshes, which are locally refined towards
the points x1 , x2 , . . . , xn . The number of elements of such meshes with respect to the
mesh size h are asymptotically the same. They are of order h−2 and h−3 for two and
three dimensions, respectively, see, e.g., [3].
The orders of convergence with respect to the mesh size h for the case of quasiuniform meshes are shown in Table 1 up to logarithmic terms, see Theorem 4 and
Theorem 5 for precise results. These results improve for the problem (1a), (1b), (1c)
the known estimates for the state and the multiplier, see [9], where the convergence
only of the same order as for the control variable is shown. This improved convergence is also observed in the numerical examples, see Sect. 7. The main instrument
to prove these improved error estimates is a duality argument for the whole optimality system. To our knowledge such a duality argument is only recently used in [22]
and [20].
The numerical examples illustrate that these estimates can not be further improved
by using quasi-uniform meshes. This can be explained by the lack of regularity of the
Optimal error estimates for finite element discretization of elliptic
Table 2 Proved orders of
convergence on graded meshes
Dimension
q̄ − q̄h ū − ūh |μ̄ − μ̄h |
d =2
2
2
2
d =3
2
2
2
adjoint variable, which fulfills an elliptic equation with a linear combination of Dirac
measures on the right-hand side, see Sect. 2 for details. However, the solutions to
elliptic problems with such irregular data can be approximated with almost optimal
order (up to a logarithmic term) if using graded meshes, which are locally refined
towards the singularities, see [4, 11, 14]. Our main contribution for the optimal control problem under consideration is showing that the optimal second order can be
achieved on properly graded meshes for the errors in the control, state and in the Lagrangian multiplier in two and three dimensions, see Table 2 and the precise results
in Theorem 7 and Theorem 8.
The proof of this result is on one hand based on a duality argument for the whole
optimality system and on the other hand on a pointwise error estimate for the state
equation of optimal order O(h2 |ln h|) on properly graded meshes, see Theorem 6.
An important feature of this result is the fact that only the L2 (Ω)-norm of the righthand side q enters the estimate, whereas on uniform meshes only the first order O(h)
can be expected for the pointwise error and a general right-hand side from L2 (Ω).
A direct application of this theorem results in an estimate of order O(h2 |ln h|) for the
finite element approximation of a Green’s function with respect to the L2 (Ω)-norm,
see Corollary 2. These estimates (Theorem 6 and Corollary 2) extend the results
from [4] also for the three dimensional case. The proof is different from [4]. It uses
the techniques from [28] and covers simultaneously the two and the three dimensional
cases.
The paper is structured as follows. In the next section we provide the functional
analytic setting for the problem (1a), (1b), (1c) and discuss the optimality conditions
as well as regularity issues. Section 3 is devoted to the finite element discretization of
(1a), (1b), (1c) and to the corresponding optimality conditions on the discrete level. In
Sect. 4 we introduce a dual problem, which is essential for our error analysis, which
is provided in Sect. 5 for quasi-uniform and in Sect. 6 for properly graded meshes. In
Sect. 7 we discuss numerical examples illustrating our error estimates.
2 Continuous problem
Let Ω ⊂ Rd for d ∈ {2, 3} be a convex domain with smooth boundary ∂Ω and let s
be a real number with
2d
d
<s<
.
d +2
d −1
The dual s of s defined by 1s + s1 = 1 fulfills s > d.
We define the control space as Q := L2 (Ω) and employ the usual notation for
Lebesgue, Hilbert, and Sobolev spaces.
D. Leykekhman et al.
Then, the weak formulation of the state equation (1b) for a given control q ∈ Q
reads as follows: Find a state u ∈ W01,s (Ω) satisfying
(∇u, ∇ϕ) = (q, ϕ)
∀ϕ ∈ W01,s (Ω).
(2)
By standard arguments, one obtains the existence and uniqueness of a solution u ∈
H 2 (Ω) ∩ H01 (Ω) of (2). By embedding, this implies u ∈ W01,s (Ω). Since s > d, we
have by the embedding W01,s (Ω) → C(Ω̄) that the pointwise state constraints given
by (1c) are well-defined for the state u ∈ W01,s (Ω).
Remark 1 The solution u of (2) does not depend on the value of s and it holds u ∈
H 2 (Ω) with the estimate
2 ∇ u ≤ Cq.
(3)
By introducing the operator G : C(Ω̄) → Rn given by
T
G(u) = u(x1 ), u(x2 ), . . . , u(xn ) ,
the state constraint (1c) can be formulated for b = (b1 , b2 , . . . , bn )T ∈ Rn as
G(u) = b.
(4)
With the cost functional J : Q × W01,s (Ω) → R defined as
α
1
J (q, u) := u − ud 2 + q2 ,
2
2
the weak formulation of the optimal control problem (1a), (1b), (1c) reads as
Minimize J (q, u) for (q, u) ∈ Q × W01,s (Ω) subject to (2) and (4),
(5)
where ud ∈ L2 (Ω) is the target state and α > 0 is the regularization parameter.
Lemma 1 The admissible set
Wad := (q, u) ∈ Q × W01,s (Ω)|u solves (2) for the given q and fulfills (4)
is nonempty.
Proof We construct an admissible pair (q, u) as follows. Since the points x1 , x2 , . . . ,
xn ∈ Ω are mutually distinct, we have
min dist(xi , xj ) = η1 > 0 and
i=1,2...,n
i=j
min dist(xi , ∂Ω) = η2 > 0.
i=1,2...,n
Optimal error estimates for finite element discretization of elliptic
Let η = min(η1 , η2 ). Since η > 0, it is possible to construct cut-off functions ωi ∈
C ∞ (Ω) such that
0, on Ω \ B η (xi )
2
ωi =
1, on B η (xi ).
4
By construction, we have that the function
u=
n
bi ωi ∈ C0∞ (Ω)
i=1
satisfies the state constraint (4). Hence the pair (q, u) with q = −u ∈ C0∞ (Ω) is
in Wad .
By standard arguments, the existence of an admissible point for problem (5) from
Lemma 1 ensures the existence and uniqueness of an optimal solution to the considered problem.
The existence result for the state equation ensures the existence of a control-tostate mapping S : q → u = u(q) defined through (2). By means of this mapping, we
introduce the reduced cost functional j : Q → R as
j (q) := J q, S(q) .
We now formulate the necessary and sufficient optimality conditions for the problem under the consideration:
Theorem 1 A control q̄ ∈ Q with associated state ū = u(q̄) ∈ W01,s (Ω) is an optimal solution to problem (5) if and only if G(ū) = b and there exists an adjoint state
z̄ ∈ W01,s (Ω) and a Lagrange multiplier μ̄ ∈ Rn such that
∀ϕ ∈ W01,s (Ω),
(∇ ū, ∇ϕ) = (q̄, ϕ)
(∇ϕ, ∇ z̄) = (ϕ, ū − ud ) +
n
μ̄i ϕ(xi )
(6)
∀ϕ ∈ W01,s (Ω),
(7)
∀ϕ ∈ Q.
(8)
i=1
(α q̄ + z̄, ϕ) = 0
Proof We note that linearity and surjectivity of the constraints implies Robinson’s
constraint qualification, see [19]. Since both the state equation and the state constraint
are linear, linearity is trivially fulfilled. The surjectivity of the state equation and of
the state constraint is equivalent to the existence of an admissible pair (q, u) for any
given b ∈ Rn , which was already established in Lemma 1. Using the formal Lagrange
technique, this implies the existence of z̄ ∈ W01,s (Ω) and μ̄ ∈ Rn fulfilling (6)–(8). Remark 2 Note that (8) is equivalent to the condition
1
q̄ = − z̄.
α
D. Leykekhman et al.
Consequently, the optimal control exhibits an improved regularity q̄ ∈ W01,s (Ω).
We denote by z̄0 ∈ W01,s (Ω) the solution of
∀ϕ ∈ W01,s (Ω)
(∇ϕ, ∇ z̄0 ) = (ϕ, ū − ud )
(9)
and by z̄i ∈ W01,s (Ω) for i = 1, 2, . . . , n the solutions of
(∇ϕ, ∇ z̄i ) = ϕ(xi )
∀ϕ ∈ W01,s (Ω).
(10)
With this notation, we have the following splitting of the adjoint state z̄:
z̄ = z̄0 +
n
μ̄i z̄i .
i=1
Remark 3 The solutions z̄i for i = 0, 1, . . . , n of (9) and (10) do not depend on the
value of s. Since ū − ud ∈ L2 (Ω) we have z̄0 ∈ H 2 (Ω) with the following estimate
2 ∇ z̄0 ≤ Cū − ud .
(11)
Lemma 2 The solutions z̄i ∈ W01,s (Ω) of (10) for i = 1, 2, . . . , n satisfy the following the estimates
C
,
−d
where the constant C depends only on the domain Ω and the points xi .
z̄i ≤ C
and ∇ z̄i Ls (Ω) ≤
s
Proof The first assertion follows immediately from the embedding W01,s (Ω) →
L2 (Ω). To prove the second assertion, we note that for v ∈ W01,s (Ω) → C(Ω̄) and
y ∈ W01,s (Ω) the estimates
vL∞ (Ω) ≤
s
C
∇vLs (Ω)
−d
and ∇yLs (Ω) ≤ C
sup
ϕ∈W01,s (Ω)
(∇y, ∇ϕ)
∇ϕLs (Ω)
hold, see [2] and [1]. By means of these estimates and (10), we obtain
∇zi Ls (Ω) ≤ C
sup
ϕ∈W01,s (Ω)
(∇zi , ∇ϕ)
=C
∇ϕLs (Ω)
sup
ϕ∈W01,s (Ω)
ϕ(xi )
C
.
≤ ∇ϕLs (Ω) s − d
Based on this result, we can prove the boundedness of the remaining quantities in
the optimality system:
Lemma 3 It holds
q̄ + ū + z̄0 + |μ̄| ≤ C ud + |b| ,
Optimal error estimates for finite element discretization of elliptic
where b = (b1 , b2 , . . . , bn )T , C is independent of s, and | · | denotes the Euclidian
norm on Rn .
Proof From Remark 2, we have q̄ ∈ W01,s (Ω). Hence, by using the optimality system (6)–(8), we obtain
0 = αq̄2 + (z̄, q̄) = αq̄2 + (∇ ū, ∇ z̄) = αq̄2 + (ū, ū − ud ) +
n
μ̄i ū(xi )
i=1
= αq̄2 + ū2 − (ū, ud ) +
n
μ̄i bi .
i=1
This implies
1
1
αq̄2 + ū2 ≤ ud 2 + |μ̄||b|.
2
2
(12)
Testing the optimality condition (8) with ϕ = z̄i ∈ W01,s (Ω) → Q for i = 1, 2, . . . , n,
we get for μ̄ the system of linear equations
Z μ̄ = f,
with Z ∈ Rn×n and f ∈ Rn given by
Zij = (z̄i , z̄j )
for i, j = 1, 2, . . . , n and
fi = −(α q̄ + z̄0 , z̄i ) for i = 1, 2, . . . , n.
Since the points x1 , x2 , . . . , xn are mutually distinct, the set of Green’s functions
{z̄1 , z̄2 , . . . , z̄n } is linearly independent in L2 (Ω). Consequently, the Gramian matrix Z of this set is regular and the norm of its inverse Z −1 depends only on the
position of x1 , x2 , . . . , xn and Ω. It follows by means of the stability estimate
z̄0 ≤ C ū + ud (13)
and Lemma 2 that
|μ̄| ≤ C|f | ≤ C αq̄ + z̄0 ≤ C αq̄ + ū + ud .
Inserting this into (12) and applying Young’s inequality yields
1
1
αq̄2 + ū2 ≤ ud 2 + C|b| αq̄ + ū + ud 2
2
1
α
1
≤ ud 2 + Cα|b|2 + q̄2 + C|b|2 + ū2
2
2
4
1
+ C|b|2 + ud 2 .
2
Hence, we get the first part of the assertion
q̄ + ū ≤ C ud + |b| .
(14)
D. Leykekhman et al.
Using this estimate in (13) and (14) implies the remaining estimate
|μ̄| + z̄0 ≤ C ud + |b| .
3 Discrete problem
To define the Galerkin finite element discretization, we consider a family of polygonal
approximations Ωh ⊂ Ω of the computational domain Ω such that all corners of Ωh
lie on the boundary ∂Ω.
Remark 4 Note that this condition implies that |Ω \ Ωh | ≤ Ch2 .
On each Ωh , we construct two or three dimensional meshes covering Ω̄h , which
satisfy the usual regularity conditions such as conformity and shape regularity (see,
e.g., [10]). The meshes consist of triangular or tetrahedral cells K and are denoted
by Th = {K}, where we define the discretization parameter h as a cellwise constant
function by setting h|K = hK with the diameter hK of the cell K. We use the symbol
h also for the maximum cell size, i.e., h = max hK .
In what follows, we will use two different kind of meshes: in Sect. 5, we derive
error estimates on quasi-uniform and in Sect. 6 on graded meshes.
On the mesh Th , we construct a conforming finite element space Vh for the state
variable in a standard way:
Vh = v ∈ C(Ω̄)|v|K ∈ P1 (K) for K ∈ Th and v|Ω\Ωh = 0 .
Here, P1 (K) consists of shape functions obtained via linear transformations of linear
polynomials defined on the reference cell.
Remark 5 Note, that by construction there holds Vh ⊂ W01,s (Ω) and Vh ⊂ W01,s (Ω)
2d
d
.
, d−1
for any value of s ∈ d+2
Then, the conforming Galerkin discretization of the state equation for a given
control q ∈ Q has the following form: Find a state uh = uh (q) ∈ Vh satisfying
(∇uh , ∇ϕ) = (q, ϕ) ∀ϕ ∈ Vh .
(15)
The state constraint on the discrete level is given as in Sect. 2 by
G(uh ) = b.
(16)
For discretizing the controls, we employ the same discretization as for the state
variable. That is, we set Qh = Vh . Then, the corresponding fully discrete optimal
control problem reads as follows:
Minimize J (qh , uh ) for (qh , uh ) ∈ Qh × Vh subject to (15) and (16),
(17)
and the existence and uniqueness of solutions is implied by the following lemma:
Optimal error estimates for finite element discretization of elliptic
Lemma 4 For h small enough, the discrete admissible set
Wad,h := (qh , uh ) ∈ Qh × Vh |uh solves (15) for the given qh and fulfills (16)
is nonempty.
Proof Each element uh ∈ Vh can be expressed by means of the nodal basis functions
ωh,i ∈ Vh as
uh =
dim
Vh
ξj ωh,j .
j =1
In order that uh fulfills the state constraints, the coefficients ξi have to satisfy the
following linear system of equations:
dim
Vh
ξj ωh,j (xi ) = bi ,
i = 1, 2, . . . , n.
(18)
j =1
Choosing h sufficiently small implies that dim Vh ≥ n and also that the n equations
above are linearly independent. Hence, system (18) admits at least one solution. Using the discrete Laplacian h : Vh → Vh given by
(−h uh , ϕ) = (∇uh , ∇ϕ) ∀ϕ ∈ Vh ,
the pair (qh , uh ) with qh = −h uh ∈ Qh belongs to Wad,h .
Also in the discrete setting, the existence of an admissible point for problem (17)
from Lemma 4 ensures the existence and uniqueness of an optimal solution to the
considered problem. Furthermore, we introduce the discrete reduced cost functional
jh : Q → R by
jh (q) := J q, uh (q) ,
where uh (q) denotes the discrete control-to-state mapping.
We now formulate the necessary and sufficient optimality conditions for the discrete problem (17):
Theorem 2 A control q̄h ∈ Qh with associated state ūh = uh (q̄h ) ∈ Vh is an optimal
solution of problem (17) if and only if G(ūh ) = b and there exists an adjoint state
z̄h ∈ Vh and a Lagrange multiplier μ̄h ∈ Rn such that
(∇ ūh , ∇ϕ) = (q̄, ϕ)
(∇ϕ, ∇ z̄h ) = (ϕ, ūh − ud ) +
n
μ̄h,i ϕ(xi )
∀ϕ ∈ Vh ,
(19)
∀ϕ ∈ Vh ,
(20)
∀ϕ ∈ Qh .
(21)
i=1
(α q̄h + z̄h , ϕ) = 0
D. Leykekhman et al.
Proof The assertion of the theorem can be proved in the same way as the continuous
analogue using Lemma 4 to ensure the surjectivity of the constraints.
Remark 6 Note that (21) is equivalent to the condition
1
q̄h = − z̄h ,
α
where the equality has to be understood in Qh = Vh .
As in the continuous setting, we denote by z̄0,h ∈ Vh the solution of
(∇ϕ, ∇ z̄0,h ) = (ϕ, ūh − ud )
∀ϕ ∈ Vh
(22)
and by z̄h,i ∈ Vh for i = 1, 2, . . . , n the solutions of
(∇ϕ, ∇ z̄h,i ) = ϕ(xi )
∀ϕ ∈ Vh
(23)
which yields the representation
z̄h = z̄0,h +
n
μ̄h,i z̄h,i .
i=1
As stated in Lemma 3 for the continuous case, we have in the discrete setting the
following estimate:
Lemma 5 For h sufficiently small on quasi-uniform or graded mashes, the solutions
z̄h,i ∈ Vh of (23) for i = 1, 2, . . . , n are uniformly bounded:
z̄h,i ≤ C.
Proof For both quasi-uniform refined and graded meshes, we have by Lemma 9 and
Corollary 2 (see below), respectively, that z̄i − z̄h,i → 0 for h → 0. Lemma 2
provides the boundedness of the continuous solutions z̄i for i = 1, 2, . . . , n. Hence,
we have for h sufficiently small that
z̄h,i ≤ z̄h,i − z̄i + z̄i ≤ C.
Lemma 6 There holds
q̄h + ūh + z̄0,h + |μ̄h | ≤ C ud + |b| .
Proof As in the proof of Lemma 3, we obtain
1
1
αq̄h 2 + ūh 2 ≤ ud 2 + |μ̄h ||b|
2
2
and we get the following system of linear equations for μ̄h :
Zh μ̄h = fh
(24)
Optimal error estimates for finite element discretization of elliptic
with Zh ∈ Rn×n and fh ∈ Rn given by
Zh,ij = (z̄h,i , z̄j,h )
for i, j = 1, 2, . . . , n
fh,i = −(α q̄h + z̄0,h , z̄h,i )
and
for i = 1, 2, . . . , n.
Since the points x1 , x2 , . . . , xn are assumed to be mutually distinct, the set of discrete
Green’s functions {z̄h,1 , z̄h,2 , . . . , z̄h,n } is linearly independent in L2 (Ω). From this
and the fact that zh,i → zi in L2 (Ω) for h → 0, we obtain using the estimate
−1 Z ≤
h
Z −1 1 − Z −1 (Zh − Z)
that the Gramian matrix Zh of this set is regular and the norm of its inverse Zh−1 can be bounded independently of h. It follows by means of the stability estimate
z̄0,h ≤ C ūh + ud and Lemma 5 that
|μ̄h | ≤ C|fh | ≤ C αq̄h + z̄0,h ≤ C αq̄h + ūh + ud .
Inserting this into (24) yields as in the proof of Lemma 3 the desired estimates.
Lemma 7 For the optimal state ū ∈ H 2 (Ω) ∩ H01 (Ω) of (5) and the optimal discrete
state ūh ∈ Vh of (17), it holds
ū − ūh ≤ Cq̄ − q̄h + Ch2 ud + |b| ,
∇(ū − ūh ) ≤ Cq̄ − q̄h + Ch ud + |b| .
Proof By inserting the solution u(q̄h ) of (2) with q = q̄h , stability estimates for the
state equations, and standard finite element error estimates, we get
ū − ūh ≤ ū − u(q̄h ) + u(q̄h ) − ūh = u(q̄) − u(q̄h ) + u(q̄h ) − uh (q̄h )
≤ Cq̄ − q̄h + Ch2 ∇ 2 u(q̄h ) ≤ Cq̄ − q̄h + Ch2 q̄h .
By Lemma 6, we obtain the first assertion. The second assertion can be proved similarly.
Lemma 8 For the solution z̄0 ∈ H 2 (Ω) ∩ H01 (Ω) of (9) and the discrete solution
z̄0,h ∈ Vh of (22), it holds
z̄0 − z̄0,h ≤ Cq̄ − q̄h + Ch2 ud + |b| ,
∇(z̄0 − z̄0,h ) ≤ Cq̄ − q̄h + Ch ud + |b| .
Proof By inserting the solution z0 (ūh ) ∈ H01 (Ω) of
∇ϕ, ∇z0 (ūh ) = (ϕ, ūh − ud ) ∀ϕ ∈ H01 (Ω),
D. Leykekhman et al.
stability estimates for the state equations, and the standard finite element error estimates, we get
z̄0 − z̄0,h ≤ z̄0 − z0 (ūh ) + z0 (ūh ) − z̄0,h ≤ ū − ūh + Ch2 ∇ 2 z0 (ūh ) ≤ ū − ūh + Ch2 ūh + ud .
By the Lemmas 6 and 7, we obtain the first assertion. The second assertion follows
in the same manner.
4 Dual problem
To derive optimal error estimates, we will make use of an optimal control problem
which is dual to the considered problem (5) in the following sense: We define for
x = (q, u, z, μ) ∈ X = Q × W01,s (Ω) × W01,s (Ω) × Rn and ϕ = (ϕq , ϕu , ϕz , ϕμ ) ∈ X
the bilinear form A : X × X → R as
A(x, ϕ) = (q, ϕz ) − (∇u, ∇ϕz ) + (u, ϕu ) +
n
μi ϕu (xi ) − (∇z, ∇ϕu )
i=1
+ α(q, ϕq ) + (z, ϕq ) +
n
u(xi )ϕμi .
i=1
Then, the optimality system of (5) from Theorem 1 for x̄ = (q̄, ū, z̄, μ̄) ∈ X is equivalent to
n
A(x̄, ϕ) = (ud , ϕu ) +
bi ϕμi ∀ϕ ∈ X.
i=1
Let x̃ = (q̃, ũ, z̃, μ̃) ∈ X be the solution of the dual optimal control problem given
as
A(ϕ, x̃) = (ûd , ϕu ) +
n
b̂i ϕμi
∀ϕ ∈ X
(25)
i=1
with
ûd =
ū − ūh
ū − ūh and b̂ =
μ̄ − μ̄h
.
|μ̄ − μ̄h |
Since A is self-adjoint, this characterization is equivalent to x̃ being the solution of
the optimality system for the following problem:
Minimize
1
α
u − ûd 2 + q2
2
2
(26a)
subject to the constraints
−u = q
in Ω,
u = 0 on ∂Ω
(26b)
Optimal error estimates for finite element discretization of elliptic
and
u(xi ) = b̂i
i = 1, 2, . . . , n.
(26c)
Hence, the dual problem (26a) (26b), (26c) is identical to the original optimal control
problem (5) for the particular choices ud = ûd and b = b̂.
As for the primal problem, we split z̃ in the regular part z̃0 and the irregular parts
z̃i defined as solutions of
(∇ϕ, ∇ z̃0 ) = (ϕ, ũ − ûd ) ∀ϕ ∈ W01,s (Ω),
1,s (∇ϕ, ∇ z̃i ) = ϕ(xi )
∀ϕ ∈ W0 (Ω), i = 1, 2, . . . , n.
(27)
(28)
Using this notation, we have as for z̄ the following splitting of z̃:
z̃ = z̃0 +
n
μ̃i z̃i .
i=1
Remark 7 By definition, it holds z̃i = z̄i for i = 1, 2, . . . , n.
Additionally, we define the solution x̃h ∈ Xh = Qh × Vh × Vh × Rn of the discrete
analog of (26a), (26b), (26c) by
A(x̃ − x̃h , ϕ) = 0 ∀ϕ ∈ Xh .
(29)
Remark 8 Because of the same structure of the primal and dual problems (5)
and (26a), (26b), (26c) and the boundedness of ûd and b̂, the proved assertions on
the primal quantities q̄, ū, z̄, μ̄, and q̄h , ūh , z̄h , μ̄h are also valid for the dual quantities q̃, ũ, z̃, μ̃, and q̃h , ũh , z̃h , μ̃h . That is, it holds for i = 1, 2, . . . , n
q̃ + ũ + z̃0 + z̃i + |μ̃| ≤ C,
(30)
q̃h + ũh + z̃h,0 + z̃h,i + |μ̃h | ≤ C.
(31)
5 Error estimates on quasi-uniform meshes
Throughout this section, we assume the family of meshes Th for h → 0 to be quasiuniform. That is, there exists a constant C independent of h such that
max hK ≤ C min hK .
K∈Th
K∈Th
5.1 Error estimate for the state and adjoint equations
We start with the following estimate of the pointwise error between the continuous
state u(q) and its Ritz projection uh (q).
D. Leykekhman et al.
Theorem 3 Let q ∈ W 1,s (Ω), u = u(q) ∈ W01,s (Ω) be the solution of (2), and
uh = uh (q) ∈ Vh be the solution of (15) for a control q ∈ Q. Then, it holds for
i = 1, 2, . . . , n
2
d
u(xi ) − uh (xi ) ≤ C ds
h3− s ∇qLs (Ω) ,
d −s
where the constant C does not depend on s.
Proof Since by assumption, Ω is smooth and convex, it holds for 1 < t < ∞
2 ∇ u t
≤ Ct qLt (Ω) ,
L (Ω)
(32)
1
where Ct ∼ t−1
for t → 1 and Ct ∼ t for t → ∞. In particular, there exists a constant
C independent of t such that Ct ≤ Ct holds for t ≥ 2. The exact form of the constant
can be traced for example from [17, Theorem 9.9]. For q ∈ W01,s (Ω), we have by
ds
> 2 and by (32), it follows
embedding that q ∈ Lp (Ω) for p = d−s
2 ∇ u
Lp (Ω)
≤ CpqLp (Ω) ≤ Cp∇qLs (Ω) .
In [26, (1.9), p. 438] it was proved for d = 2 for 2 ≤ t < ∞ that
u − uh Lt (Ω) ≤ Ct h2 ∇ 2 uLt (Ω) ,
(33)
(34)
where Ct is the constant in (32). Using the stability of the Ritz projection in W01,t
from [5, Theorem 8.5.3], the proof in [26] can be repeated also for d = 3 yielding (34)
for d ∈ {2, 3}.
Let now be xi ∈ K∗ for some cell K∗ ∈ Th . Then, by the triangle inequality
u(xi ) − uh (xi ) ≤ u − uh L∞ (K ) ≤ u − ih uL∞ (K ) + ih u − uh L∞ (K ) ,
∗
∗
∗
where ih denotes the usual Lagrange interpolant. For u − ih uL∞ (K∗ ) , we have by
well-known interpolation estimates
∇ 2 uLp (K ) .
2− pd u − ih uL∞ (K∗ ) ≤ Ch
∗
For ih u − uh L∞ (K∗ ) , it follows using an inverse estimate and estimate (34) for t = p
that
− pd
ih u − uh L∞ (K∗ ) ≤ Ch
ih u − uh Lp (K∗ )
−d ≤ Ch p ih u − uLp (K∗ ) + u − uh Lp (K∗ )
2− d ≤ Cph p ∇ 2 uLp (K ) .
∗
Finally, by means of (33), it follows by the definition of p
d
u(xi ) − uh (xi ) ≤ Cp 2 h2− p ∇qLs (K ) ≤ Cp 2 h3− ds ∇qLs (Ω) ,
∗
Optimal error estimates for finite element discretization of elliptic
which concludes the proof.
Lemma 9 For the solutions z̄i of (28), it holds
d
z̄i − z̄h,i ≤ Ch2− 2 .
Proof Let i be arbitrary but fixed and let v ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of
(∇v, ∇ϕ) = (z̄i − z̄h,i , ϕ)
∀ϕ ∈ H01 (Ω)
and vh ∈ Vh be its discrete analog defined by
∇(v − vh ), ∇ϕ = 0 ∀ϕ ∈ Vh .
Then, we have by the Galerkin orthogonality
z̄i − z̄h,i 2 = (z̄i − z̄h,i , z̄i ) − (z̄i − z̄h,i , z̄h,i ) = (∇v, ∇ z̄i ) − (∇vh , ∇ z̄h,i )
= v(xi ) − vh (xi ).
By the suboptimal L∞ error estimate
d
v − vh L∞ (Ω) ≤ Ch2− 2 ∇ 2 v from [10, p. 168] and (3) applied to v, we get
d
d
z̄i − z̄h,i 2 ≤ v − vh L∞ (Ω) ≤ Ch2− 2 ∇ 2 v ≤ Ch2− 2 z̄i − z̄h,i ,
which completes the proof.
5.2 Error estimate for the optimal control problem
Throughout this section, let s = sε with
sε =
d +ε
d +ε−1
and ε from now on be chosen as ε = |log h|−1 . It follows sε = d + ε and for h sufficiently small, it holds 0 < ε < 3 and consequently
d
2d
< sε <
.
d +2
d −1
Corollary 1 Let u(q̄) and u(q̄h ) be the solution of (2) and uh (q̄) and uh (q̄h ) be
the solution of (15) for the optimal controls q = q̄ and q = q̄h , respectively. Then, it
holds for i = 1, 2, . . . , n
u(q̄)(xi ) − uh (q̄)(xi ) ≤ Ch4−d |log h|7−2d ud + |b| ,
u(q̄h )(xi ) − uh (q̄h )(xi ) ≤ Ch4−d |log h|7−2d ud + |b| .
D. Leykekhman et al.
dsε
Proof Let pε = d−s
. To prove the first assertion, we note that q̄ ∈ W01,sε (Ω). Hence,
ε
we get from Theorem 3 and (8) that
n
d
3−
u(q̄)(xi ) − uh (q̄)(xi ) ≤ Cp 2 h sε ∇ z̄0 Lsε (Ω) + |μ̄|
∇ z̄i Lsε (Ω) .
ε
i=1
Using Lemma 3, it follows
∇ z̄0 Lsε (Ω) ≤ Cū − ud ≤ C ud + |b| ,
which leads to
n
d
u(q̄)(xi ) − uh (q̄)(xi ) ≤ Cp 2 h3− sε ud + |b| + |μ̄|
∇ z̄i Lsε (Ω) .
ε
i=1
The Lemmas 2 and 3 and the estimates
−
ε
ε
d
>1−d −
=1−d −
sε
d +ε
d
pε =
d +ε
d − 2 + (1 −
1
d )ε
≤
d +ε
d −2+
and
ε
2
<
C
ε 3−d
yield
u(q̄)(xi ) − uh (q̄)(xi ) ≤
ε
C
h4−d− d ud + |b| .
ε 7−2d
−1
The choice ε = |log h|−1 finally implies the first assertion by using that h−|log h|
= e.
To prove the second assertion, we apply Theorem 3 for q = q̄h and obtain using (21) that
n
2 3− sdε
|u(q̄h )(xi ) − uh (q̄h )(xi )| ≤ Cpε h
∇ z̄h,i Lsε (Ω) .
∇ z̄h,0 Lsε (Ω) + |μ̄h |
i=1
Using the stability of the Ritz projection in W01,sε (Ω) (cf. [5, Theorem 8.5.3])
∇ z̄h,i Lsε (Ω) ≤ C∇ z̄i Lsε (Ω) ,
i = 1, 2, . . . , n,
the stability estimate
∇ z̄h,0 Lsε (Ω) ≤ Cūh − ud for zh,0 and Lemma 6, it follows
n
d
3−
u(q̄h )(xi ) − uh (q̄h )(xi ) ≤ Cp 2 h sε ud + |b| + |μ̄h |
∇ z̄i Lsε (Ω) .
ε
i=1
As above for the first assertion, this implies the second assertion.
Optimal error estimates for finite element discretization of elliptic
Theorem 4 Let q̄ ∈ Q and q̄h ∈ Qh be the solutions of (5) and (17). Then, it holds
d
7
q̄ − q̄h ≤ Ch2− 2 |log h| 2 −d ud + |b| .
Proof We adapt the lines of the proof of Theorem 3.6 in [12]. We observe using the
optimality conditions (6)–(8)
u(q̄h ) − u(q̄), u(q̄) − ud = ∇ z̄, ∇ u(q̄h ) − u(q̄)
+
n
μ̄i u(q̄)(xi ) − u(q̄h )(xi )
i=1
= (z̄, q̄h − q̄) +
n
μ̄i u(q̄)(xi ) − u(q̄h )(xi )
i=1
= −α(q̄, q̄h − q̄) +
n
μ̄i u(q̄)(xi ) − u(q̄h )(xi ) .
i=1
Using this equality, we obtain for the difference j (q̄h ) − j (q̄)
2 α
1
j (q̄h ) − j (q̄) = u(q̄h ) − u(q̄) + q̄h − q̄2 + u(q̄h ) − u(q̄), u(q̄) − ud
2
2
+ α(q̄, q̄h − q̄)
2 α
1
= u(q̄) − u(q̄h ) + q̄ − q̄h 2
2
2
n
+
μ̄i u(q̄)(xi ) − u(q̄h )(xi ) .
i=1
(35)
Similarly for jh , we obtain
2 α
1
jh (q̄) − jh (q̄h ) = uh (q̄h ) − uh (q̄) + q̄h − q̄2
2
2
n
+
μ̄h,i uh (q̄h )(xi ) − uh (q̄)(xi ) .
i=1
By adding (35) and (36), we obtain
αq̄ − q̄h 2 ≤ j (q̄h ) − j (q̄) + jh (q̄) − jh (q̄h )
−
n
i=1
n
μ̄i bi − u(q̄h )(xi ) −
μ̄h,i b − u(q̄)(xi )
i=1
≤ |j (q̄) − jh (q̄)| + |j (q̄h ) − jh (q̄h )|
(36)
D. Leykekhman et al.
n
uh (q̄h )(xi ) − uh (q̄)(xi )
+ |μ̄h |
1
2
i=1
n
u(q̄)(xi ) − u(q̄h )(xi )
+ |μ̄|
1
2
.
(37)
i=1
For |j (q̄) − jh (q̄)| it holds by standard estimates
j (q̄) − jh (q̄) = 1 u(q̄) − ud 2 − 1 uh (q̄) − ud 2
2
2
1
= u(q̄) − uh (q̄), u(q̄) − ud + uh (q̄) − ud
2
1
≤ u(q̄) − uh (q̄) u(q̄) + uh (q̄) + 2ud 2
2
≤ Ch2 ud + |b| .
In the same manner, we obtain for |j (q̄h ) − jh (q̄h )|
j (q̄h ) − jh (q̄h ) ≤ Ch2 ud + |b| 2 .
For |uh (q̄h )(xi ) − uh (q̄)(xi )| we obtain using the first assertion of Corollary 1
uh (q̄h )(xi ) − uh (q̄)(xi ) = bi − uh (q̄)(xi ) = |u(q̄)(xi ) − uh (q̄)(xi )
≤ Ch4−d |log h|7−2d ud + |b|
and for |u(q̄)(xi ) − u(q̄h )(xi )| using the second assertion of Corollary 1
u(q̄)(xi ) − u(q̄h )(xi ) = bi − u(q̄h )(xi ) = uh (q̄h )(xi ) − u(q̄h )(xi )
≤ Ch4−d |log h|7−2d ud + |b| .
Inserting the last four estimates into (37) finally yields the assertion.
Using the standard techniques, the proved order of convergence for the error in the
control carries over to the errors in the remaining variables. However, for ū − ūh and |μ̄ − μ̄h | it is possible to show the following improved convergence order.
Theorem 5 For the optimal states ū ∈ H 2 (Ω) and ūh ∈ Vh and the Lagrangian
multipliers μ̄ ∈ Rn and μ̄h ∈ Rn of the continuous and discrete problem (5) and (17),
respectively, it holds
ū − ūh + |μ̄ − μ̄h | ≤ Ch4−d |log h|7−2d ud + |b| .
Proof Choosing ϕ = x̄ − x̄h in (25) yields by Galerkin orthogonality
ū − ūh + |μ̄ − μ̄h | = A(x̄ − x̄h , x̃) = A(x̄ − x̄h , x̃ − x̃h ),
Optimal error estimates for finite element discretization of elliptic
where x̃h ∈ Xh = Vh × Vh × Vh × Rn is the discrete analog of x̃ defined by
A(x̃ − x̃h , ϕ) = 0 ∀ϕ ∈ Xh .
By inspection of the definition of A we have
A(x̄ − x̄h , x̃ − x̃h ) = (q̄ − q̄h , z̃ − z̃h ) − ∇(ū − ūh ), ∇(z̃ − z̃h )
+ (ū − ūh , ũ − ũh ) +
n
(μ̄i − μ̄h,i ) ũ(xi ) − ũh (xi )
i=1
− ∇(z̄ − z̄h ), ∇(ũ − ũh ) + α(q̄ − q̄h , q̃ − q̃h )
+ (z̄ − z̄h , q̃ − q̃h )
n
ū(xi ) − ūh (xi ) (μ̃i − μ̃h,i ).
+
(38)
i=1
Now, we estimate the terms on the right-hand side of (38) separately:
(i) (q̄ − q̄h , z̃ − z̃h ): Using Theorem 4 and the Lemmas 9 and 8 applied to the dual
problem, we get
(q̄ − q̄h , z̃ − z̃h ) ≤ q̄ − q̄h z̃ − z̃h ≤ Ch4−d |log h|7−2d ud + |b| .
(ii) (∇(ū − ūh ), ∇(z̃ − z̃h )): We split
∇(ū − ūh ), ∇(z̃ − z̃h ) = ∇(ū − ūh ), ∇(z̃0 − z̃h,0 )
+
n
∇(ū − ūh ), ∇(μ̃i z̃i − μ̃h,i z̃h,i ) . (39)
i=1
For the first term in (39), we have by the Lemmas 7 and 8 together with Theorem 4 applied to the primal and the dual problem
∇(ū − ūh ), ∇(z̃0 − z̃h,0 ) ≤ ∇(ū − ūh )∇(z̃0 − z̃h,0 )
≤ Ch4−d |log h|7−2d ud + |b| .
For the second term in (39), we proceed
∇(ū − ūh ), ∇(μ̃i z̃i − μ̃h,i z̃h,i ) = (μ̃i − μ̃h,i ) ∇(ū − ūh ), ∇ z̃i
+ μ̃h,i ∇(ū − ūh ), ∇(z̃i − z̃h,i ) . (40)
For the first term in (40), we have due to ū(xi ) = bi = ūh (xi ) that
∇(ū − ūh ), ∇ z̃i = ū(xi ) − ūh (xi ) = 0.
For the second term in (40), we have by the Galerkin orthogonality and Corollary 1
∇(ū − ūh ), ∇(z̃i − z̃h,i ) = ∇ ū − uh (q̄) , ∇(z̃i − z̃h,i )
D. Leykekhman et al.
= ∇ ū − uh (q̄) , ∇ z̃i
= u(q̄)(xi ) − uh (q̄)(xi )
≤ Ch4−d |log h|7−2d ud + |b| .
Collecting the previous estimates and using (31), we obtain
∇(ū − ūh ), ∇(z̃ − z̃h ) ≤ Ch4−d |log h|7−2d ud + |b| .
(iii) (ū − ūh , ũ − ũh ): Using Lemma 7 together with Theorem 4 for ū and ũ, we
estimate
d
(ū − ūh , ũ − ũh ) ≤ ū − ūh ũ − ũh ≤ Ch4− 2 |log h|7−2d ud + |b| .
(iv) (μ̄i − μ̄h,i )(ũ(xi ) − ũh (xi )): Since ũ(xi ) = b̂i = ũh (xi ), we have that
(μ̄i − μ̄h,i ) ũ(xi ) − ũh (xi ) = 0.
(v) (∇(z̄ − z̄h ), ∇(ũ − ũh )): This term is treated as the term in (ii) leading to
∇(z̄ − z̄h ), ∇(ũ − ũh ) ≤ Ch4−d |log h|7−2d ud + |b| .
(vi) α(q̄ − q̄h , q̃ − q̃h ) + (z̄ − z̄h , q̃ − q̃h ): By the continuous and discrete optimality
conditions α q̄ + z̄ = 0 and α q̄h + z̄h = 0, which are fulfilled pointwise in Ω,
we get
α(q̄ − q̄h , q̃ − q̃h ) + (z̄ − z̄h , q̃ − q̃h ) = 0.
(vii) (ū(xi ) − ūh (xi ))(μ̃i − μ̃h,i ): Since ū(xi ) = bi = ūh (xi ), we have that
ū(xi ) − ūh (xi ) (μ̃i − μ̃h,i ) = 0.
By collecting all the estimates from (i)–(vii), we end up with the assertion.
6 Error estimates on graded meshes
To derive an improved estimate not only for ū and μ̄ but also for q̄, we need a mesh
that is graded towards the set of points {x1 , x2 , . . . , xn }. Let
rK = dist K, {x1 , x2 , . . . , xn } = min dist(K, xi )
i=1,2,...,n
denote the distance from a cell K ∈ Th to the set {x1 , x2 , . . . , xn }. We consider the
graded mesh given in term of the cell size as
hK =
h2d−2 ,
d
4
hrK ,
if rK = 0,
if rK > 0,
(41)
Optimal error estimates for finite element discretization of elliptic
that corresponds to the mesh grading parameters 12 for d = 2 and 14 for d = 3. Notice
that the number of elements of such a triangulation is of order h−d , see, e.g., [3].
Such meshes can be constructed via a coordinate transformation, see [25], by dyadic
refinement, see, e.g., [15], or a combination of both, see [21].
6.1 Error estimate for the state equation
In our argument, we will often deal with functions that are harmonic on some parts
of the domain. The following inverse-type inequality significantly simplifies many
arguments.
Lemma 10 Let D ⊂ Ω and for r > 0 let Dr = {x ∈ Ω| dist(x, D) ≤ r}. Assume that
v : Dr → R is harmonic on Dr , i.e.
(∇v, ∇ϕ) = 0
Then, it holds
∀ϕ ∈ H01 (Dr ).
2 ∇ v ≤ Cr −1 ∇vD .
r
D
Proof We first note that a partial derivative of a harmonic function is also a harmonic
function. Let vm = ∂m v for m = 1, 2, . . . , d and let ω ∈ C ∞ (Ω) be a cutoff function
such that
0, on Ω \ Dr ,
ω=
1, on D,
and |∇ω| ≤ Cr −1 . Using that vm is harmonic on Dr , we have
ω∇vm 2 = (ω∇vm , ω∇vm ) = ∇vm , ∇ ω2 vm − ∇vm , vm ∇ ω2
= −2(ω∇vm , vm ∇ω) ≤ Cω∇vm vm ∇ω ≤ Cr −1 ω∇vm vm Dr ,
which implies the assertion by
2 ∇ v ≤ C max∇vm D ≤ Cω∇vm ≤ Cr −1 vm D ≤ Cr −1 ∇vD .
r
r
D
m
As in Sect. 5, we start with the derivation of an estimate of the pointwise error
between the state u(q) and its Ritz projection uh (q). In [4], the following estimate
was established for d = 2,
u(q)(x0 ) − uh (q)(x0 ) ≤ Ch2 |log h| 32 q.
In the following theorem, we derive a pointwise error estimate for d ∈ {2, 3}. In addition we obtain a slight improvement in terms of the power of the logarithmic term.
Theorem 6 Let u(q) ∈ H 2 (Ω) be the solution of (2) and uh (q) be the solution
of (15) for a given q ∈ Q. Then, it holds for i = 1, 2, . . . , n
u(q)(xi ) − uh (q)(xi ) ≤ Ch2 |log h|q.
D. Leykekhman et al.
Proof Let for simplicity i = 1 be fixed and K∗ ∈ Th be a cell containing x1 . For
abbreviation, we set u = u(q) and uh = uh (q). Denote rj = 2−j diam(Ω) for j =
0, 1, . . . and define
Ω∗ = x ∈ Ω |x − x1 | ≤ C∗ h∗ ,
where h∗ = hK∗ and for C∗ to be determined later. We also define
Ωj = x ∈ Ω rj +1 < |x − x1 | ≤ rj .
Let J be chosen such that rJ +1 ≤ C∗ h∗ ≤ rJ . Note that by construction diam(Ωj ) ≤
rj and J ≤ C|log h|. Furthermore, it holds for k > j + 1 that rj > rk and
1
rj ≤ dist(Ωk , Ωj ) ≤ rj .
2
(42)
Then we have the decomposition
J
Ω = Ω∗ ∪
Ωj .
j =0
d
The setting h∗ = h2d−2 and hj = hrj4 implies
hj
=
rj
h
1− d
rj 4
≤
h
1− d
rJ 4
≤
2h
1− d4
(C∗ h∗ )
=
2
1− d
C∗ 4
,
(43)
since (2d − 2)(1 − d4 ) = 1 for d ∈ {2, 3}.
Let δ̃ ∈ C0∞ (K∗ ) be an approximative delta function fulfilling
(δ̃, ϕ) = ϕ(x1 )
∀ϕ ∈ Vh
and
−d(1− p1 )
δ̃Lp (Ω) ≤ Ch∗
,
1 ≤ p ≤ ∞.
(44)
The explicit construction of such a function is given for instance in the Appendix
of [28].
Then, we define g ∈ H 2 (Ω) ∩ H01 (Ω) as the solution of
(∇g, ∇ϕ) = (δ̃, ϕ)
∀ϕ ∈ H01 (Ω)
and the Ritz projection gh ∈ Vh as the solution of
(∇gh , ∇ϕ) = (δ̃, ϕ) ∀ϕ ∈ Vh .
Note that by construction (∇gh , ∇ϕ) = ϕ(x1 ) holds for all ϕ ∈ Vh .
Optimal error estimates for finite element discretization of elliptic
Using the properties of g and gh , the nodal interpolation ih , and Galerkin orthogonality, we have the estimate
(u − uh )(x1 ) = (u − ih u)(x1 ) + (ih u − uh )(x1 )
= (u − ih u)(x1 ) + ∇g, ∇(ih u − uh )
= (u − ih u)(x1 ) + ∇g, ∇(ih u − u) + ∇(g − gh ), ∇(u − uh )
= (u − ih u)(x1 ) + (δ̃, u − ih u) + ∇(g − gh ), ∇(u − ih u)
≤ Cu − ih uL∞ (K∗ ) {1 + δ̃L1 (Ω) } + ∇(g − gh ), ∇(u − ih u) .
For the first term, by the interpolation estimate on K∗ , estimate (44) for p = 1, and
the elliptic regularity, we have
∇ 2 u
2− d2 u − ih uL∞ (K∗ ) δ̃L1 (Ω) ≤ Ch∗
K∗
≤ Ch2 ∇ 2 u ≤ Ch2 q.
We split the second term as
∇(g − gh ), ∇(u − ih u) = ∇(g − gh ), ∇(u − ih u) Ω
∗
+ ∇(g − gh ), ∇(u − ih u) Ω\Ω
∗
= I1 + I2 .
For I1 , we have by the Cauchy-Schwarz inequality and the interpolation estimate
I1 ≤ ∇(g − gh )Ω ∇(u − ih u)Ω ≤ CC∗ h∗ ∇g∇ 2 uΩ .
∗
∗
For all 1 ≤ p < ∞ in 2D and 1 ≤ p ≤ 6 in 3D and
∗
1
p
= 1 − p1 , we have that
2
∇g = (g, δ̃) ≤ gLp (Ω) δ̃ p
L (Ω∗) ≤ Cp∇gδ̃Lp (Ω∗) .
Consequently, by (44), we get
−d
∇g ≤ Cpδ̃Lp (Ω∗) ≤ Cph∗ p .
For d = 2, we choose p = |log h∗ | and obtain
∇g ≤ C|log h∗ | ≤ C|log h|.
For d = 3, we obtain by choosing p = 6
−1
∇g ≤ Ch∗ 2 ≤ Ch−2 .
Inserting these estimates and using the definition h∗ = h2d−2 yields
I1 ≤ Ch2 |log h|∇ 2 u ≤ Ch2 |log h|q.
D. Leykekhman et al.
For I2 , we have
I2 ≤
J
J
∇(g − gh ) ∇(u − ih u)
∇(g − gh ), ∇(u − ih u) Ω ≤
Ω
Ω
j
j =0
≤
J
j =0
j
j =0
j
J
d
Chj ∇(g − gh )Ω ∇ 2 uΩ ≤ Ch∇ 2 uΩ
rj4 ∇(g − gh )Ω
j
j
j
j =0
j
≤ CShq,
where S is given by
S=
J
j =0
d
rj4 ∇(g − gh )Ω .
(45)
j
Our goal is now to establish S ≤ Ch|log h|. Let 0 < θ < 14 be some fixed constant.
We define the following sets:
Ωj = x ∈ Ω| 1 − θ 2 rj +1 < |x − x1 | ≤ rj 1 + θ 2 ,
Ωj = x ∈ Ω|(1 − θ )rj +1 < |x − x1 | ≤ rj (1 + θ ) .
This implies Ωj ⊂ Ωj ⊂ Ωj and by the local energy estimate from [13, Thm. 3.4]
we have
∇(g − gh ) ≤ C ∇(g − ih g) + r −1 g − ih gΩ + r −1 g − gh Ω . (46)
j
j
Ω
Ω
j
j
j
j
d
By interpolation estimates, hj ≤ Crj from (43), and hj = hrj4 , we get for the first
two terms
∇(g − ih g)
Ωj
d
+ rj−1 g − ih gΩj ≤ Chj ∇ 2 g Ω = Chrj4 ∇ 2 g Ω .
j
(47)
j
Because of the local support of δ̃, we have that g is harmonic on Ωj and by Lemma 10
we have
2 ∇ g ≤ Cr −1 ∇gΩ .
(48)
j
Ω
j
j
Using the pointwise estimates of the derivatives of Green’s function G(·, ·) for the
Laplacian, see, e.g., [18] for d ≥ 3 and [16] for d = 2
∇x G(x, y) ≤ C dist(x, y)1−d
(49)
and that dist(K∗ , Ωj ) ≥ Crj for C∗ large enough, we have for x ∈ Ωj that
∇g(x) =
K∗
∇x G(x, y)δ̃(y) dy ≤ Crj1−d δ̃L1 (Ω) ≤ Crj1−d .
Optimal error estimates for finite element discretization of elliptic
This implies
1
1− d
∇gΩj ≤ C Ωj 2 rj1−d = Crj 2 .
(50)
Combining the estimates (46)–(50), we have
∇(g − gh )
− d4
Ωj
+ Crj−1 g − gh Ωj .
≤ Chrj
Inserting this in (45) implies by using J ≤ C|log h|
S≤C
J
J
d
d
−1
−1
h + rj4 g − gh Ωj ≤ Ch|log h| +
rj4 g − gh Ωj .
j =0
(51)
j =0
To treat the last term in (51), we employ a duality argument to show that
J
d
rj4
−1
j =0
g − gh Ωj ≤ Ch|log h| +
S
2
for C∗ sufficiently large.
It holds
g − gh Ωj =
sup
ψ∈C0∞ (Ωj )
(g − gh , ψ)
.
ψ
(52)
To estimate the right-hand side (g − gh , ψ), we introduce the following dual problem:
For a given ψ ∈ C0∞ (Ωj ), let v ∈ H 2 (Ω) ∩ H01 (Ω) be the solution of
(∇v, ∇ϕ) = (ψ, ϕ)
∀ϕ ∈ H01 (Ω).
Then, it holds
(g − gh , ψ) = ∇(g − gh ), ∇(v − ih v)
= ∇(g − gh ), ∇(v − ih v) Ω + ∇(g − gh ), ∇(v − ih v) Ω\Ω
∗
∗
= I3 + I4 .
For I3 , we get as for I1 before
I3 ≤ Ch2 |log h|∇ 2 v Ω ≤ Ch2 |log h|ψ.
(53)
∗
For I4 , we have
I4 ≤
J
J
∇(g − gh ), ∇(v − ih v) Ω ≤ C
hk ∇(g − gh )Ω ∇ 2 v Ω
k
k=0
=
j −2
k=0
k
k=0
... +
j +1
k=j −1
... +
J
k=j +2
. . . = M1 + M2 + M3 .
k
D. Leykekhman et al.
To estimate M2 , we notice that here k ≈ j and hence rk ≈ rj . Thus,
M1 ≤ Chj ∇(g − gh )Ω̃ ∇ 2 v ≤ Chj ∇(g − gh )Ω̃ ψ
j
(54)
j
with Ω̃j = Ωj −1 ∪ Ωj ∪ Ωj +1 . To estimate M1 and M3 , we notice that v is harmonic
on Ωk . Thus, we have by Lemma 10 that
2 ∇ v ≤ Cr −1 ∇vΩ .
k
Ω
k
k
Using estimate (49) for Green’s function G(x, y) for x ∈ Ωk and y ∈ Ωj
∇v(x) =
Ωj
1−d
∇x G(x, y)ψ(y) dy ≤ C dist Ωk , Ωj
ψL1 (Ω )
j
1−d d2
≤ C dist Ωk , Ωj
rj ψ,
we get
d
d
d
2 1
∇ v ≤ Cr −1 Ω 2 dist Ω , Ω 1−d r 2 ψ ≤ Cr 2 −1 dist Ω , Ω 1−d r 2 ψ.
k
k
j
k
j
j
j
k
k
Ω
k
For M1 , we notice that j > k + 1. Thus, it holds by (42) that dist(Ωk , Ωj ) ≥ Crk ,
which implies
d d
2 ∇ v ≤ Cr − 2 r 2 ψ
k
j
Ω
k
and consequently
d
M1 ≤ Crj2 ψ
j −2
−d
rk 2 hk ∇(g − gh )Ω .
(55)
k
k=0
For M3 , we notice that k > j + 1. Thus, it holds by (42) that dist(Ωk , Ωj ) ≥ Crj ,
which implies
d
d
2 ∇ v ≤ Cr 2 −1 r 1− 2 ψ
k
j
Ω
k
and consequently
1− d2
M3 ≤ Crj
ψ
J
d
rk2
−1
k=j +2
hk ∇(g − gh )Ω .
(56)
k
Using (52), we have by collecting the estimates (53)–(56) that
d
g − gh Ωj ≤ Ch2 |log h| + Crj2
j −2
−d
rk 2 hk ∇(g − gh )Ω
k=0
+ Chj ∇(g − gh )Ω̃
j
k
Optimal error estimates for finite element discretization of elliptic
1− d2
+ Crj
J
d
rk2
−1
hk ∇(g − gh )Ω .
(57)
k
k=j +2
d
To estimate the last term in (51), we have to sum up these terms weighted by rj4
The contribution of the first term on the right-hand side of (57) is
J
d
rj4
−1 2
d
−1
d
h |log h| ≤ CrJ4+1 h2 |log h| ≤ CC∗4
j =0
−1
d
h∗4
C
−1 2
h |log h| ≤
1− d
C∗ 4
−1
:
h|log h|,
since (2d − 2)( d4 − 1) = −1 for d ∈ {2, 3}. For the third term in (57), we get by (43)
J
d
rj4
−1
j =0
J
d
hj
hj ∇(g − gh )Ω̃ ≤ C
rj4 ∇(g − gh )Ω
≤
j
j r
j
j =0
C
1− d
C∗ 4
S.
For the second term in (57), it follows by changing the order of summation and using
the properties of the geometric series that
J
d
rj4
−1
j −2
d
rj2
j =0
−d
rk 2 hk ∇(g − gh )Ω
k
k=0
J
j
3 d −1 − d2
rk hk ∇(g
j =0
k=0
≤C
J
≤C
rj 4
−d
rk 2 hk ∇(g
J
− d 3 d −1
rk 2 rk 4
k=0
≤
C
1− d
C∗ 4
k
J
3 d −1
− gh ) Ω
rj 4
k
k=0
≤C
− gh )Ω
j =k
J
d
hk
hk ∇(g − gh )Ω = C
rk4 ∇(g − gh )Ω
k
k r
k
k=0
S.
Finally, for the last term in (57), we obtain similarly
J
d
rj4
J
−1 1− d2
rj
j =0
−1
k=j +2
≤C
J
− d4
rj
j =0
≤C
d
rk2
J
k=0
J
d
rk2
−1
hk ∇(g − gh )Ω
hk ∇(g − gh )Ω
k
k=j
d
rk2
−1
k
−d
hk ∇(g − gh )Ω
rj 4
k
j =0
k
D. Leykekhman et al.
≤C
J
d
rk2
−1 − d4
rk hk ∇(g
k=0
C
≤
1− d4
J
d
hk
− gh )Ω = C
rk4 ∇(g − gh )Ω
k
k r
k
k=0
S.
C∗
This implies for C∗ large enough that
J
j =0
d
rj4
−1
g − gh Ωj ≤
C
1− d
C∗ 4
h|log h| +
C
1− d
C∗ 4
1
S ≤ Ch|log h| + S.
2
Inserting this into (51) yields
S ≤ Ch|log h|,
which completes the proof.
Corollary 2 For the solutions z̄i of (10), it holds
z̄i − z̄h,i ≤ Ch2 |log h|.
Proof The assertion can be proved as Lemma 9 using Theorem 6 instead of the sub
optimal L∞ error estimate.
6.2 Error estimate for the optimal control problem
The following Corollary is the counterpart of Corollary 1 for uniform meshes.
Corollary 3 Let u(q̄) and u(q̄h ) ∈ H 2 (Ω) be the solution of (2) and uh (q̄) and
uh (q̄h ) be the solution of (15) for q = q̄ and q = q̄h , respectively. Then, it holds for
i = 1, 2, . . . , n
u(q̄)(xi ) − uh (q̄)(xi ) ≤ Ch2 |log h| ud + |b| ,
u(q̄h )(xi ) − uh (q̄h )(xi ) ≤ Ch2 |log h| ud + |b| .
Proof From Theorem 6, we have
u(q̄)(xi ) − uh (q̄)(xi ) ≤ Ch2 |log h|q̄.
Then, by Lemma 3 we directly obtain the first assertion. The second assertion can be
proved in the same way.
In the following Lemma, we prove a first estimate for the error in the control,
which will be improved later on.
Lemma 11 Let q̄ ∈ Q and q̄h ∈ Qh be the solutions of (5) and (17). Then, it holds
1
q̄ − q̄h ≤ Ch|log h| 2 ud + |b| .
Optimal error estimates for finite element discretization of elliptic
Proof We proceed as in the proof of Theorem 4 to derive (37). The first two terms of
the right-hand side of (37) are estimated as before yielding
j (q̄) − jh (q̄) + j (q̄h ) − jh (q̄h ) ≤ Ch2 ud + |b| 2 .
For |u(q̄)(xi ) − uh (q̄)(xi )| and |uh (q̄h )(xi ) − u(q̄h )(xi )|, we obtain by Corollary 3
u(q̄)(xi ) − uh (q̄)(xi ) + uh (q̄h )(xi ) − u(q̄h )(xi ) ≤ Ch2 |log h| ud + |b| .
The rest of the proof goes along the line of the proof of Theorem 4.
Theorem 7 For the optimal states ū ∈ H 2 (Ω) ∩ H01 (Ω) and ūh ∈ Vh and the Lagrangian multipliers μ̄ ∈ Rn and μ̄h ∈ Rn of the continuous and discrete problem (5)
and (17), respectively, it holds on a mesh graded as defined by (41)
ū − ūh + |μ̄ − μ̄h | ≤ Ch2 |log h| ud + |b| .
Proof As in the proof of Theorem 5, we get
ū − ūh + |μ̄ − μ̄h | = (q̄ − q̄h , z̃ − z̃h ) − ∇(ū − ūh ), ∇(z̃ − z̃h )
+ (ū − ūh , ũ − ũh ) +
n
(μ̄i − μ̄h,i ) ũ(xi ) − ũh (xi )
i=1
− ∇(z̄ − z̄h ), ∇(ũ − ũh ) + α(q̄ − q̄h , q̃ − q̃h )
+ (z̄ − z̄h , q̃ − q̃h ) +
n
ū(xi ) − ūh (xi ) (μ̃i − μ̃h,i ).
i=1
(58)
Now, we estimate the terms on the right-hand side of (58) separately:
(i) (q̄ − q̄h , z̃ − z̃h ): Using the Lemmas 11 and 8 and Corollary 2 applied to the
dual problem, we get
(q̄ − q̄h , z̃ − z̃h ) ≤ q̄ − q̄h z̃ − z̃h ≤ Ch2 |log h| ud + |b| .
(ii) (∇(ū − ūh ), ∇(z̃ − z̃h )): We split as in the proof of Theorem 5:
∇(ū − ūh ), ∇(z̃ − z̃h ) = ∇(ū − ūh ), ∇(z̃0 − z̃h,0 )
+
n
∇(ū − ūh ), ∇(μ̃i z̃i − μ̃h,i z̃h,i ) . (59)
i=1
For the first term in (59), we have by the Lemmas 7 and 8 together with
Lemma 11
∇(ū − ūh ), ∇(z̃0 − z̃h,0 ) ≤ ∇(ū − ūh )∇(z̃0 − z̃h,0 )
D. Leykekhman et al.
≤ Ch2 |log h| ud + |b| .
For the second term in (59), we proceed as in the proof of Theorem 5:
∇(ū − ūh ), ∇(μ̃i z̃i − μ̃h,i z̃h,i ) = (μ̃i − μ̃h,i ) ∇(ū − ūh ), ∇ z̃i
+ μ̃h,i ∇(ū − ūh ), ∇(z̃i − z̃h,i ) . (60)
For the first term in (60), we have
∇(ū − ūh ), ∇ z̃i = ū(xi ) − ūh (xi ) = 0.
For the second term in (60), we have by Galerkin orthogonality and Corollary 3
as in the proof of Theorem 5
∇(ū − ūh ), ∇(z̃i − z̃h,i ) = u(q̄)(xi ) − uh (q̄)(xi ) ≤ Ch2 |log h| ud + |b| .
Collecting the previous estimates and using (31), we obtain
∇(ū − ūh ), ∇(z̃ − z̃h ) ≤ Ch2 |log h| ud + |b| .
(iii) (ū − ūh , ũ − ũh ): Using Lemma 7 together with Lemma 11 for ū and ũ, we
estimate
(ū − ūh , ũ − ũh ) ≤ ū − ūh ũ − ũh ≤ Ch2 |log h| ud + |b| .
(iv) (μ̄i − μ̄h,i )(ũ(xi ) − ũh (xi )): Since ũ(xi ) = b̂i = ũh (xi ), we have that
(μ̄i − μ̄h,i ) ũ(xi ) − ũh (xi ) = 0.
(v) (∇(z̄ − z̄h ), ∇(ũ − ih ũ)): This term is treated as the term in (ii) leading to
∇(z̄ − z̄h ), ∇(ũ − ũh ) ≤ Ch2 |log h| ud + |b| .
(vi) α(q̄ − q̄h , q̃ − q̃h ) + (z̄ − z̄h , q̃ − q̃h ): By the continuous and discrete optimality
conditions α q̄ + z̄ = 0 and α q̄h + z̄h = 0, we get
α(q̄ − q̄h , q̃ − q̃h ) + (z̄ − z̄h , q̃ − q̃h ) = 0.
(vii) (ū(xi ) − ūh (xi ))(μ̃i − μ̃h,i ): Since ū(xi ) = bi = ūh (xi ), we have that
ū(xi ) − ūh (xi ) (μ̃i − μ̃h,i ) = 0.
Collecting all estimates from (i)–(vii), we end up with the assertion.
Finally, the proved order of convergence for ū and μ̄ can on graded meshes be
used to prove the following optimal estimate for the error in the control variable.
Theorem 8 Let q̄ ∈ Q and q̄h ∈ Qh be the solutions of (5) and (17). Then, it holds
on a mesh graded as defined by (41)
q̄ − q̄h ≤ Ch2 |log h| ud + |b| .
Optimal error estimates for finite element discretization of elliptic
Proof For any p ∈ Q, it holds
αq̄ − q̄h 2
≤ jh (p)(q̄ − q̄h , q̄ − q̄h ) = jh (q̄)(q̄ − q̄h ) − jh (q̄h )(q̄ − q̄h )
= α q̄ + zh,0 (q̄), q̄ − q̄h − (α q̄h + z̄h,0 , q̄ − q̄h )
=−
n
n
μ̄i (z̄i , q̄ − q̄h ) − z̄0 − zh,0 (q̄), q̄ − q̄h +
μ̄h,i (z̄h,i , q̄ − q̄h )
i=1
i=1
≤ z̄0 − zh,0 (q̄)q̄ − q̄h +
n
|μ̄h,i − μ̄i |z̄i q̄ − q̄h i=1
+
n
|μ̄h,i |z̄h,i − z̄i q̄ − q̄h .
i=1
We estimate z̄0 − zh,0 (q̄) in standard way by inserting the Ritz projection Rh z̄0 ∈
Vh of z̄0 as
z̄0 − zh,0 (q̄) ≤ z̄0 − Rh z̄0 + Rh z̄0 − zh,0 (q̄)
≤ Ch2 ū − ud + ū − uh (q̄) ≤ Ch2 ū − ud + q̄
≤ Ch2 ud + |b| .
Using this, Theorem 7, and Corollary 2 complete the proof by
q̄ − q̄h ≤ C z̄0 − zh,0 (q̄) + |μ̄ − μ̄h | + max z̄i − z̄h,i ≤ Ch2 |log h| ud + |μ̄| .
i=1,2,...,n
7 Numerical results
In this section, we are going to validate the a priori error estimates for the error in
the control and state variable as well as in the Lagrangian multiplier. To this end,
we consider the following concrete optimal control problem (5) with known exact
solution on Ω = B1 (0) ⊂ Rd with d ∈ {2, 3} with n = 1 state constraints and x1 =
0 ∈ Rd . For this choice of Ω one can directly calculate z̄1 as
2 ln|x|,
for d = 2,
1
· z̄1 (x) = −
1
4π
1 − |x|
, for d = 3,
where |x| denotes the Euclidean norm of x ∈ Rd . We further choose μ̄1 = 1 and
ud = ū. This implies z̄ = z̄1 and q̄ = − α1 z̄. A direct calculation shows that
1
(1 − |x|2 + |x|2 ln|x|), for d = 2,
· 2
ū(x) = −
1
2
8πα
for d = 3
3 − |x| + 3 |x| ,
D. Leykekhman et al.
Fig. 1 Discretization errors on uniform meshes
fulfills the state equation for the chosen q̄. The value b1 = ū(0) for the state constraint
is then
1
− 8πα
, for d = 2,
b1 =
1
− 12πα , for d = 3.
For the computations, we choose α = 1.
The optimal control problems are solved by the optimization library RO D O B O
[27] and the finite element toolkit G ASCOIGNE [29] using a two-step procedure. For
the first step, we write the discrete optimality system (19)–(21) in the case n = 1 in
terms of the discrete control-to-state-operator Sh : q → uh as
ūh = Sh q̄h ,
z̄h = Sh (ū − ud + μ̄h,1 δx1 ),
q̄h = −α −1 z̄h .
Combining these equations yields
αI + Sh2 ūh = Sh2 ud − μ̄h,1 Sh2 δx1 .
Noting that Hh := (αI + Sh2 ) coincides with the Hessian of the discrete reduced functional jh , we obtain
b1 = ūh (x1 ) = Hh−1 Sh2 ud (x1 ) − μ̄h,1 Hh−1 Sh2 δx1 (x1 ),
from which the unknown Lagrangian multiplier μ̄h,1 can directly be computed.
Thereafter, using the computed value of μ̄h,1 the optimal control q̄h and the associated state ūh are computed classically by a conjugate gradient method solving the
reduced optimality system.
Optimal error estimates for finite element discretization of elliptic
Fig. 2 Discretization errors on graded meshes
Figure 1 depicts the development of the errors in the control and state variables
and in the Lagrangian multiplier under refinement of the cell size h using uniform
mesh refinement for d = 2 (a) and d = 3 (b). Here, N = O(h−d ) denotes the number
of nodes. The discretization error q̄ − q̄h and ū − ūh clearly exhibit the proved
d
orders of convergence O(h2− 2 ) and O(h4−d ), respectively. For d = 2, the error in
term of the Lagrangian multiplier decreases much faster than predicted, whereas for
d = 3 the proved O(h) convergence can be observed not until the mesh gets finer
than approx 104 nodes.
In Fig. 2 the development of the errors in the control, state and Lagrangian multiplier on a series of graded meshes for d = 2 (a) and d = 3 (b) are shown. Here, for
all errors the expected order O(h2 ) is observed.
References
1. Alkhutov, Y.A., Kondrat’ev, V.A.: Solvability of the Dirichlet problem for second-order elliptic equations in a convex domain. Differ. Equ. 28, 650–662 (1992)
2. Alt, H.W.: Lineare Funktionalanalysis: Eine Anwendungsorientierte Einführung, 6th edn. Springer
Lehrbuch. Springer, Berlin (2012)
3. Apel, T., Sändig, A.M., Whiteman, J.R.: Graded mesh refinement and error estimates for finite element
solutions of elliptic boundary value problems in non-smooth domains. Math. Methods Appl. Sci. 19,
63–85 (1996)
4. Apel, T., Benedix, O., Sierch, D., Vexler, B.: A priori mesh grading for an elliptic problem with Dirac
right-hand side. SIAM J. Numer. Anal. 49(3), 992–1005 (2011)
5. Brenner, S.C., Scott, L.R.: The Mathematical Theory of Finite Element Methods, 3rd edn. Texts in
Applied Mathematics, vol. 15. Springer, Berlin (2007)
D. Leykekhman et al.
6. Casas, E.: Error estimates for the numerical approximation of semilinear elliptic control problems
with finitely many state constraints. ESAIM Control Optim. Calc. Var. 8, 345–374 (2002) (electronic).
A tribute to J.L. Lions
7. Casas, E.: Necessary and sufficient optimality conditions for elliptic control problems with finitely
many pointwise state constraints. ESAIM Control Optim. Calc. Var. 14(3), 575–589 (2008)
8. Casas, E., Mateos, M.: Second order optimality conditions for semilinear elliptic control problems
with finitely many state constraints. SIAM J. Control Optim. 40(5), 1431–1454 (2002) (electronic)
9. Casas, E., Mateos, M.: Numerical approximation of elliptic control problems with finitely many pointwise constraints. Comput. Optim. Appl. 51(3), 1319–1343 (2012). doi:10.1007/s10589-011-9394-2
10. Ciarlet, P.G.: The Finite Element Method for Elliptic Problems. Classics Appl. Math., vol. 40. SIAM,
Philadelphia (2002)
11. D’Angelo, C.: Finite element approximation of elliptic problems with Dirac measure terms in
weighted spaces: applications to one- and three-dimensional coupled problems. SIAM J. Numer. Anal.
50(1), 194–215 (2012). doi:10.1137/100813853
12. Deckelnick, K., Hinze, M.: Convergence of a finite element approximation to a state-constrained
elliptic control problem. SIAM J. Numer. Anal. 45(5), 1937–1953 (2007)
13. Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on
sharply varying grids. Math. Comput. 80(273), 1–9 (2011). doi:10.1090/S0025-5718-2010-02353-1
14. Eriksson, K.: Improved accuracy by adapted mesh-refinements in the finite element method. Math.
Comput. 44(170), 321–343 (1985). doi:10.2307/2007955
15. Fritzsch, R.: Optimale Finite-Elemente-Approximationen für Funktionen mit Singularitäten. Ph.D.
thesis, TU Dresden, Dresden, Germany (1990)
16. Fromm, S.J.: Potential space estimates for Green potentials in convex domains. Proc. Am. Math. Soc.
119, 225–233 (1993)
17. Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001). Reprint of the 1998 edition
18. Grüter, M., Widman, K.O.: The Green function for uniformly elliptic equations. Manuscr. Math. 37,
303–342 (1982)
19. Hinze, M., Pinnau, R., Ulbrich, M., Ulbrich, S.: Optimization with PDE Constraints. Mathematical
Modelling: Theory and Applications, vol. 23. Springer, Berlin (2008)
20. Leykekhman, D.: Investigation of commutative properties of discontinuous Galerkin methods in pde
constrained optimal control problems. J. Sci. Comput. 53(3), 483–511 (2012)
21. Li, H., Mazzucato, A., Nistor, V.: Analysis of the finite element method for transmission/mixed boundary value problems on general polygonal domains. Electron. Trans. Numer. Anal. 37, 41–69 (2010)
22. May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems (2010, submitted)
23. Merino, P., Tröltzsch, F., Vexler, B.: Error estimates for the finite element approximation of a semilinear elliptic control problem with state constraints and finite dimensional control space. Modél. Math.
Anal. Numér. 44(1), 167–188 (2010). doi:10.1051/m2an/2009045
24. Merino, P., Neitzel, I., Tröltzsch, F.: On linear-quadratic elliptic control problems of semi-infinite
type. Appl. Anal. 90(6), 1047–1074 (2011). doi:10.1080/00036811.2010.489187
25. Oganesyan, L.A., Rukhovets, L.A.: Variational-difference schemes for linear second-order elliptic
equations in a two-dimensional region with piecewise smooth boundary. USSR Comput. Math. Math.
Phys. 8, 129–152 (1968)
26. Rannacher, R., Scott, R.: Some optimal error estimates for piecewise linear finite element approximations. Math. Comput. 38, 437–445 (1982)
27. RO D O B O: A C++ library for optimization with stationary and nonstationary PDEs with interface to
GASCOIGNE [29]. http://www.rodobo.uni-hd.de
28. Schatz, A.H., Wahlbin, L.B.: Interior maximum-norm estimates for finite element methods, part II.
Math. Comput. 64(211), 907–928 (1995)
29. The finite element toolkit GASCOIGNE: http://www.gascoigne.uni-hd.de