mass lumping for the optimal control of elliptic partial ......and wachsmuth,2011, (4.13)] and to...

Mass lumping for the optimal controlof elliptic partial differential

equations

Arnd Rösch∗ Gerd Wachsmuth †

April 13, 2017

The finite element discretization of a control constrained elliptic optimalcontrol problem is studied. Control and state are discretized by higher orderfinite elements. The inequality constraints are only posed in the Lagrangepoints. The computational effort is significantly reduced by a new mass lump-ing strategy. The main contribution is the derivation of new a priori errorestimates up to order h4 on locally refined meshes. Moreover, we propose anew algorithmic strategy to obtain such highly accurate results. The theo-retical findings are illustrated by numerical examples.

Keywords: optimal control, control constraint, higher order finite elements, masslumping, a priori error estimates.

MSC: 49K20, 49M25, 65N30

1 Introduction

The discretization of optimal control problems by finite elements is nowadays a standardtool. A series of papers investigates a priori discretization error estimates in particularfor control constrained problems. In this sense, the theory for the optimal control oflinear, elliptic equations seems to be nearly completed. A closer look shows that theknown approaches are limited due to regularity issues. A standard discretization with

∗Faculty of Mathematics, University of Duisburg-Essen, Thea-Leymann-Straße 9, 45127 Essen, Ger-many; email: [email protected]

†Faculty of Mathematics, Technische Universität Chemnitz, Reichenhainer Str. 41, 09126 Chemnitz,Germany; email: [email protected].

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http://www.ams.org/mathscinet/msc/msc2010.html?t=49K20

http://www.ams.org/mathscinet/msc/msc2010.html?t=49M25

http://www.ams.org/mathscinet/msc/msc2010.html?t=65N30

Mass lumping for the optimal control of PDEs Arnd Rösch, Gerd Wachsmuth

piecewise constant controls is limited to the rate of h, which is the mesh size, see Aradaet al. [2002], Falk [1973], Geveci [1979]. Piecewise linear approximations are limited toh3/2, see Rösch [2006], Casas and Mateos [2008]. The superconvergence approach yieldsa numerical approximation of order h2, Meyer and Rösch [2004].

By using adaptive mesh refinement, the approximation order in two dimensions is limitedby N−3/2, where N is the number of unknowns, (this corresponds to h3) because ofthe required number of cells close to the kinks of the optimal control, see [Schneiderand Wachsmuth, 2015, Eq. (4)] for a discussion of this limitation and Schneider andWachsmuth [2016] for results showing convergence of order N−3/2.

The only known (non-adaptive) approach where the accuracy is not limited by h2 is thevariational discretization by Hinze [2005]. Of course, at least quadratic finite elementsare necessary to obtain an accuracy higher than h2 in L2(Ω). The main challenge is thenumerical computation of the optimal control and the evaluation of the scalar product ofthe optimal control with a finite element function. This can be done exactly for piecewiselinear finite elements, but the usage of higher order finite elements is computationallychallenging, see Sevilla and Wachsmuth [2010].

We also mention the work Springer and Vexler [2013], in which the variational discretiza-tion together with the post-processing approach is used to obtain convergence of orderthree (w.r.t. the temporal discretization) for the time-dependent control of a parabolicequation.

This is the starting point of our new method which has convergence order up to h4.We propose a new fully discrete approach with higher accuracy and low computationaleffort. In order to compensate for the low regularity of the control, we use locally refinedmeshes. As a model problem, we consider the optimal control of an elliptic PDE withhomogeneous Neumann boundary conditions. That is, we discuss the problem

Minimize1

2‖y − yd‖2L2(Ω) +

α

2‖u‖2L2(Ω)

w.r.t. y ∈ H1(Ω), u ∈ L2(Ω)

s.t. −∆y + y = u in Ω

∂

∂ny = 0 on ∂Ω

and ua ≤ u ≤ ub in Ω.

(P)

We assume that yd is continuous and sufficiently smooth. Moreover, we assume that thedomain Ω ⊂ R2 is convex and polygonal. For simplicity of the presentation, we furtherassume ua, ub ∈ R. Note that the state equation in (P) has to be understood in the weaksense, that is, y satisfies

a(y, v) :=

∫Ω∇y · ∇v + y v dx =

∫Ωu v dx ∀v ∈ H1(Ω). (1.1)

2


2 Motivation

It is well-known that the (necessary and sufficient) optimality condition for an optimalcontrol u of (P) with associated optimal state y is given by the pointwise projectionformula

u(x) = Proj[ua,ub]p(x)

αfor a.a. x ∈ Ω, (2.1)

where p is the (weak) solution of the adjoint equation

−∆p+ p = yd − y in Ω,

∂

∂np = 0 on ∂Ω.

(2.2)

Although the adjoint state p may possesses high regularity (e.g., H2(Ω) on convex do-mains), the regularity of the control is limited by the projection formula (2.1). In-deed, the gradient of the optimal control u is discontinuous in neighborhoods of the setx ∈ Ω : p(x)/α ∈ ua, ub. This limits the regularity of u and we do not, in general,have u ∈ H2(Ω). Nevertheless, one has u ∈W 1,∞(Ω) and u ∈ H3/2−ε(Ω) for all ε > 0 bystandard results on Nemytskii operators (induced by Lipschitz continuous scalar func-tions) in Sobolev spaces, see, e.g., [Bourdaud and Meyer, 1991, Théorème 2], [Oswald,1992, Theorem 1], and [Runst and Sickel, 1996, Theorem 5.4.1]. Finally, we mention thatthe set x ∈ Ω : p(x)/α ∈ ua, ub, on which the control might posses kinks, is typicallya (finite union of) arc(s).

Now, we consider a discretized version of (P). Let T be a quasi-uniform triangulation ofΩ, cf. [Brenner and Scott, 2008, Definition (4.4.13)]. We will work with a finite elementspace

Vh = vh ∈ C(Ω) : vh|T ∈ P(T ) ∀T ∈ T ,

where P denotes a certain polynomial space of higher order. We will specify the detailslater, cf. Section 4. We do not distinguish between the function vh ∈ Vh and the associ-ated coefficient vector (both are related by a bijective mapping induced by the Lagrangeinterpolation). We denote by M and K the usual mass and stiffness matrix associatedwith the inner products of L2(Ω) and H1(Ω), respectively. That is,

Mij =

∫Ωφi φj dx, Kij =

∫Ω∇φi · ∇φj + φi φj dx,

for all i, j = 1, . . . , N and φiNi=1 is the nodal basis of Vh. By ‖uh‖M := (u>M u)1/2 wedenote the norm induced by the mass matrix and the operator Ih : C(Ω) → Vh in (P′h)is the (usual) nodal Lagrange interpolation, see (3.1) below.

A possibility to discretize the optimal control problem (P) would be

Minimize1

2‖yh − Ih yd‖2M +

α

2‖uh‖2M

s.t. K yh = M uh

and ua ≤ uh ≤ ub.

(P′h)

3


Since Vh consists of polynomials of higher order, only a coefficient-wise interpretation ofthe control constraints is practicable from a computational point of view. That is, thecontrol constraint in (P′h) is to be understood as ua ≤ uh(xL) ≤ ub for all Lagrange-nodes xL of the triangulation or, equivalently, as ua ≤ uih ≤ ub for all coefficients uih ofuh. Note that such a discretization is not conforming if the polynomial degree is largerthan one.

We denote by ProjM[ua,ub](·) the projection onto [ua, ub]N w.r.t. the norm ‖·‖M . Conse-

quently, vh = ProjM[ua,ub](wh) if and only if

(vh − wh)>M (uh − vh) ≥ 0 ∀uh ∈ Vh : ua ≤ uh ≤ ub.

This projection can be evaluated coefficient-wise if and only if the mass matrix M isdiagonal, otherwise nonlocal effects appear.

It is easy to see that the (necessary and sufficient) optimality condition for an optimalcontrol uh of (P′h) can be written as

uh = ProjM[ua,ub]phα, (2.3)

where the discretized adjoint state ph is defined by the equations

K ph = M (Ih yd − yh), K yh = M uh.

The main drawback of this approach is that the projection formula (2.3) cannot beevaluated in a coefficient-wise manner.

The superconvergence approach introduced in Meyer and Rösch [2004] works with dif-ferent discrete spaces for state and control. In particular, the controls are discretizedby piecewise constant functions and, thus, the mass matrix becomes a diagonal matrixwhich is heavily exploited in the derivation of the approximation results.

We will propose here a new approach with a mass matrixM and a diagonal lumped-massmatrix ML. The mass matrix is used in the tracking term and the lumped-mass matrixis used twice: in the right-hand side of the state equation and in the control cost termin the objective.

Hence, we propose to use

Minimize1

2‖yh − Ih yd‖2M +

α

2‖uh‖2ML

s.t. K yh = ML uh

and ua ≤ uh ≤ ub.

(Ph)

with a diagonal, positive semidefinite approximationML of the mass matrix. As in (P′h),the control constraints are to be understood coefficient-wise.

Mass lumping is a standard tool for the numerical solution of time dependent partialdifferential equations. Until now, only a few papers are devoted to mass lumping in

4


optimal control. Mass lumping is used in the computation of an L1 term, see [Wachsmuthand Wachsmuth, 2011, (4.13)] and to obtain a discrete projection formula, see [Casaset al., 2012, Lemma 3.4]. However, only convergence of order h was proved. This wasimproved by Pieper [2015], who also considered the lumped-mass matrix in the right-hand side of the discrete PDE, see [Pieper, 2015, Section 4.5.4] and convergence of orderh2 for a piecewise linear discretization was obtained. Similar ideas are also used for thecontrol of ordinary differential equations, see for instance Alt et al. [2007].

It is easy to check, that the optimality conditions for (Ph) are given by

uh = Proj[ua,ub]phα, (2.4a)

K ph = M (Ih yd − yh), (2.4b)K yh = ML uh. (2.4c)

If the diagonal ML is not strictly positive, the solution of (Ph) is not unique. Indeed,entries of uh corresponding to a zero diagonal entry of ML do not enter the objective orthe state equation in (Ph). We fix these entries by the optimality condition (2.4a). Weemphasize that the projection in (2.4a) is to be understood coefficient-wise.

We mention that there are two possible interpretations of mass lumping. The first oneis from a linear algebra point-of-view and the lumped-mass matrix is understood as adiagonal approximation of the mass matrix, by defining, e.g.,

(ML)ii =N∑j=1

Mij .

From a numerical analysis point-of-view, mass lumping can be understood as follows.For the triangulation T we have the (global) Lagrange nodes xiL, i = 1, . . . , N and theassociated (global) basis functions φi satisfy φi(x

jL) = δij . Then, we choose a quadrature

formula whose nodes are exactly the Lagrange nodes and the weights ωi are non-negative.Now, we can define a lumped-mass matrix by approximating the L2(Ω) inner product bythe quadrature formula. Indeed, the matrix ML defined via

(ML)ij :=

N∑k=1

φi(xkL)φj(x

kL)ωk =

ωi if i = j,

0 else(2.5)

is diagonal.

3 Basic error estimate for the control problem

In this section we will derive the basic error estimate. Here, we do not need a specificform of the finite element space Vh. Only the following three properties are required:

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• The coefficient vector uh corresponds to a nodal basis, i.e., for every coefficient uihthere is a Lagrange point xiL ∈ Ω with

uih = uh(xiL).

Moreover, the interpolation operator Ih : C(Ω)→ Vh satisfies

(Ih v)i = v(xiL) ∀v ∈ C(Ω), i = 1, . . . , N. (3.1)

• The control constraints in (Ph) are understood coefficient-wise, i.e., they are equiv-alent to

ua ≤ uih ≤ ubfor all coefficients associated with the nodal basis.

• The lumped-mass matrix ML is diagonal and all entries are non-negative.

Lemma 3.1. Let us assume that the above assumptions are satisfied by the finite elementspace Vh. Moreover, we assume u, p ∈ C(Ω). Then we have

α ‖Ih u− uh‖2ML≤ (ph − Ih p, uh − Ih u)ML

, (3.2)

where (yh, uh, ph) is the unique solution of the optimality system (2.4).

In (3.2), we used the usual notations

(a, b)ML:= a>ML b and ‖a‖2ML

:= (a, a)ML.

Proof. Together with (2.4a) the assumptions on Vh imply

[α uh(xiL)− ph(xiL)] [v − uh(xiL)] ≥ 0 ∀v ∈ [ua, ub].

We choose v = u(xiL), which is possible due to u ∈ C(Ω) and ua ≤ u ≤ ub a.e. in Ω. Thisyields

[α uh(xiL)− ph(xiL)] [u(xiL)− uh(xiL)] ≥ 0.

Since u and p are continuous, the projection formula (2.1) holds everywhere, and weobtain

[α u(xiL)− p(xiL)] [u− u(xiL)] ≥ 0 ∀u ∈ [ua, ub].

Thus,[α u(xiL)− p(xiL)] [uh(xiL)− u(xiL)] ≥ 0,

since uh(xiL) ∈ [ua, ub]. Now, we weight both inequalities by the i-th diagonal entry ofML and sum over all indices i to obtain

(α Ih u− α uh + ph − Ih p)>ML (uh − Ih u) ≥ 0.

Here, we used (Ih u)i = u(xiL). This implies the assertion.

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In the following theorem, we estimate the right-hand side of (3.2) by approximation errorsfor the state and adjoint equation.

Theorem 3.2. Under the assumptions of Lemma 3.1, the error estimate

α

2‖Ih u− uh‖2ML

+1

2‖Ih y − yh‖2M +

1

2‖K−1ML Ih u− yh‖2M

≤ 1

2‖Ih y −K−1ML Ih u‖2M +

1

2α‖Ih p−K−1M (Ih yd − Ih y)‖2ML

(3.3)

is valid.

Proof. We use the law of cosines in the Hilbert space (Vh, ‖·‖M ), the discrete stateequation (2.4c), an algebraic manipulation and the discrete adjoint equation (2.4b) (inthis order) to obtain

1

2‖Ih y − yh‖2M +

1


=1

2‖Ih y −K−1ML Ih u‖2M + (Ih y − yh, K−1ML Ih u− yh)M

=1

2‖Ih y −K−1ML Ih u‖2M + (Ih y − yh, K−1ML Ih u−K−1ML uh)M

=1

2‖Ih y −K−1ML Ih u‖2M + (K−1M (Ih y − yh), Ih u− uh)ML

=1

2‖Ih y −K−1ML Ih u‖2M + (K−1M (Ih y − Ih yd) + ph, Ih u− uh)ML

.

Together with the inequality (3.2), we obtain

α ‖Ih u− uh‖2ML+

1

2‖Ih y − yh‖2M +

1


≤ 1

2‖Ih y −K−1ML Ih u‖2M + (Ih p−K−1M (Ih yd − Ih y), Ih u− uh)ML

.

Finally, we use Young’s inequality to obtain

α


+1

2‖Ih y − yh‖2M +

1


≤ 1

2‖Ih y −K−1ML Ih u‖2M +

1

2α‖Ih p−K−1M (Ih yd − Ih y)‖2ML

,

and this is the assertion.

We briefly interpret the terms appearing in the inequality (3.3). The first term onthe right-hand side describes the approximation error of the state equation caused bydiscretization and mass lumping. The second term measures the discretization error ofthe adjoint equation in the mass lumping norm. In the next section we will estimatethese two terms on the right-hand side of the inequality (3.3).

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Finally, we discuss the terms on the left-hand side of (3.3). The first term is an errorestimate for the approximation of the control. However, since the lumped-mass matrix isinvolved, only the error in the Lagrange nodes is measured. The second term is equal to12 ‖Ih y − yh‖

2L2(Ω) and this is, up to the interpolation error, the L2(Ω)-error in the state

variable.

4 Error estimates for the equations

Let us now give a precise description of the discrete spaces Vh. We require the followingproperties:• The convex polygonal domain Ω is discretized exactly by the triangulation T , i.e.,

we have Ω = Ωh := interior(⋃T∈T T ), where interior(A) denotes the interior of the

set A ⊂ R2.• The triangulation T is quasi-uniform in the sense of [Brenner and Scott, 2008,

Definition (4.4.13)]. Hanging nodes are not allowed. There is no restriction on thediameter hT := maxx1,x2∈T ‖x1−x2‖ of a single element T ∈ T . Later we will havethe mild restriction (4.7). We denote by h := maxT∈T hT the global mesh size.• The discrete space Vh is generated by a family of finite elements which is affine

equivalent to a common reference element (independent of h), cf. [Brenner andScott, 2008, Section 3.4]. A basis of Vh is given by the continuous and piecewisepolynomial basis functions φi, i = 1, . . . , N , corresponding to Lagrange nodes xiL ∈Ω. We postulate the following assumptions on the basis functions.– We require φi(x

jL) = δij , for 1 ≤ i, j ≤ N .

– All basis functions have nonnegative integrals, i.e.,∫

Ω φi dx ≥ 0.

– The basis functions build a partition of unity, i.e.,∑N

i=1 φi ≡ 1 on Ω.– There exists two positive numbers k and k′ with Pk ⊂ Vh ⊂ Pk′ . Later, we

will need the inequalities k ≥ 2 and k′ ≤ k + 1. Here, Pk (Pk′) is the usualspace of continuous functions which are piecewise polynomials of degree atmost k (k′).

– The mass matrixM of the finite element space Vh and the lumped mass matrixML are connected by the relation

(ML)ii =

N∑j=1

Mij .

– The quadrature rule with points xiLNi=1 and weights ωiNi=1 correspondingwith the lumped-mass matrix, cf. (2.5), is exact for (piecewise) polynomialsof degree k + k′ − 2, that is∫

Ωϕ dx =

N∑i=1

ωi ϕ(xiL)

8


for all ϕ ∈ C(Ω) for which ϕ|T is a polynomial of degree at most k + k′ − 2for all T ∈ T . Note that this directly implies∫

Ωφi dx =

N∑j=1

ωj φi(xjL) = ωi ≥ 0,

since the local degree of φi is at most k′ ≤ k + k′ − 2. Hence, the quadratureweights ωi are uniquely determined by the Lagrange nodes xiL. Moreover, wehave

(ML)ii =N∑j=1

Mij =N∑j=1

∫Ωφi φj dx =

∫Ωφi

N∑j=1

φj dx =

∫Ωφi dx = ωi.

Let us describe two particular finite elements for which the associated finite elementspaces Vh satisfy the above assumptions: standard P2 elements and enriched P3 elementsas constructed in Cohen et al. [2001].

We denote by T = conv(0, 0), (1, 0), (0, 1) the reference cell. The standard P2 elementis given by (T , P2(T ),N2), where P2(T ) denotes the quadratic polynomials on T andN2 are the degrees of freedom given by the point evaluations in the standard Lagrangenodes, i.e., in the points (0, 0), (1

2 , 0), (1, 0), (12 , 0), (1

2 ,12), (1, 1), cf. Figure 4.1 (left). The

Figure 4.1: Degrees of freedom on the reference cell T for standard P2 elements (left) andthe enriched P3 elements of Cohen et al. [2001] (right).

quadrature rule (on the reference cell) associated with mass lumping possesses the weights1/3 (for the midpoints of the edges) and 0 (for the vertices), and is exact for polynomialsof degree 2. This is caused by the fact that the local basis function associated with avertex has mean value 0. Hence, the lumped mass matrixML is singular for P2 elements.This is not a obstruction for the analysis, since we only required that ML is positivesemidefinite. To the contrary, this is beneficial for the numeric implementation, since thediscretized problem (Ph) does not depend on any of the vertex values of uh and, thus,we can work with less degrees of freedom for uh. To summarize, the standard P2 elementsatisfies the above assumptions with k = k′ = 2.

The enriched P3 elements consists of the standard cubic finite element space P3(T ) andthis space is enriched by all bubble functions of degree at most four. That is, the localansatz space is given by

P3(T ) := ϕ ∈ P4(T ) : ϕ|E ∈ P3(E) for all edges E of T.

9


For the degrees of freedom N3, one takes point evaluations at cleverly chosen points onT , cf. Figure 4.1 (right), and we refer to [Cohen et al., 2001, Lemma 4.3] for their preciselocations. The associated quadrature rule has positive weights in all 12 points and is exactfor polynomials of degree at most 5, see again [Cohen et al., 2001, Lemma 4.3]. Hence,a finite element space constructed from the reference element (T , P3(T ), N3) satisfies theabove assumptions with k = 3, k′ = 4.

We denote the space of continuous, piecewise linear functions by V (1)h and by I(1)

h thenodal interpolation operator to the space V (1)

h .

4.1 Error estimates for the adjoint equation

In this section we will estimate the second term ‖Ih p−K−1M (Ih yd − Ih y)‖MLon the

right-hand side of (3.3). We start with an auxiliary result.

Lemma 4.1. There is a constant c > 0 (depending only on the reference element), suchthat

‖vh‖2ML≤ c ‖vh‖2M = c ‖vh‖2L2(Ω) (4.1)

holds for arbitrary vh ∈ Vh.

Proof. The result follows from a simple transformation argument. Let us investigate thenorm on a single element T ∈ T . After transformation to the reference element we canestimate the lumped-mass matrix semi-norm by the mass matrix norm. Retransformationyields the assertion.

Lemma 4.2. The following estimate is valid

‖Ih p−K−1M (Ih yd− Ih y)‖ML≤ c (‖Ih p− p‖L2(Ω) + ‖p−K−1M (Ih yd− Ih y)‖L2(Ω)).

(4.2)

Proof. We obtain the desired result immediately from the last lemma:

‖Ih p−K−1M (Ih yd − Ih y)‖ML

≤ c (‖Ih p−K−1M (Ih yd − Ih y)‖L2(Ω))

≤ c (‖Ih p− p‖L2(Ω) + ‖p−K−1M (Ih yd − Ih y)‖L2(Ω)).

Let us remark that the second term on the right-hand side of this inequality is the usualfinite element error for the adjoint equation evaluated for the optimal state y. The firstterm represents an interpolation error.

10


4.2 Error estimates for the state equation

Next, we estimate the first term ‖Ih y −K−1ML Ih u‖M on the right-hand side of (3.3).Let us introduce the notation

y := K−1ML Ih u.

This auxiliary discrete state is just the Galerkin solution of the state equation in whichthe quadrature rule corresponding to the lumped-mass matrix is used to evaluate theright-hand side u. Indeed,

v>hK y = v>hML Ih u =N∑i=1

ωi vh(xiL) u(xiL) ≈∫

Ωvh u dx

for all vh ∈ Vh. Here, “≈” indicates that the sum is the evaluation of the integral byusing the quadrature rule associated with the lumped mass matrix.

The first addend on the right-hand side of (3.3) can be estimated by the triangle inequality

‖Ih y −K−1ML Ih u‖M = ‖Ih y −K−1ML Ih u‖L2(Ω)

≤ ‖Ih y − y‖L2(Ω) + ‖y −K−1ML Ih u‖L2(Ω).

Owing to the regularity of y, we can estimate the interpolation error ‖Ih y− y‖L2(Ω) andthe second term is addressed in the following lemma.

Lemma 4.3. The following a priori error estimate holds

‖y − y‖L2(Ω) ≤ c h

(‖y − Ih y‖H1(Ω) + sup

wh∈Vh\0

(wh, u)L2(Ω) − w>hML Ih u

‖wh‖H1(Ω)

)

+ c supwh∈V

(1)h \0

(wh, u)L2(Ω) − w>hML Ih u

‖wh‖H1(Ω).

(4.3)

Proof. Using the first Lemma of Strang, see, e.g., [Ciarlet, 1978, Thm. 4.1.1], we obtain

‖y − y‖H1(Ω) ≤ c ‖y − Ih y‖H1(Ω) + c supwh∈Vh\0

(wh, u)L2(Ω) − whML Ih u

‖wh‖H1(Ω).

Now, we use the Nitsche trick to estimate ‖y − y‖L2(Ω). We define ϕ as the solution ofthe dual problem with right-hand side y − y, i.e.

a(ϕ, v) = (y − y, v)L2(Ω) ∀ϕ ∈ H1(Ω),

where a is the bilinear form from the state equation (1.1). Since Ω is assumed to beconvex and polygonal, we have ϕ ∈ H2(Ω) and ‖ϕ‖H2(Ω) ≤ C ‖y − y‖L2(Ω).

11


Now, we find

‖y − y‖2L2(Ω) = a(ϕ, y − y)

= a(ϕ− I(1)h ϕ, y − y) + a(I

(1)h ϕ, y − y)

= a(ϕ− I(1)h ϕ, y − y) + (I

(1)h ϕ, u)L2(Ω) − (I

(1)h ϕ)>ML Ih u

≤ ‖ϕ− I(1)h ϕ‖H1(Ω) ‖y − y‖H1(Ω)

+ supwh∈V

(1)h \0


‖wh‖H1(Ω)‖I(1)h ϕ‖H1(Ω)

≤(h ‖y − y‖H1(Ω) + sup

wh∈V(1)h \0


‖wh‖H1(Ω)

)‖ϕ‖H2(Ω)

≤(h ‖y − y‖H1(Ω) + sup

wh∈V(1)h \0


‖wh‖H1(Ω)

)‖y − y‖L2(Ω).

Here, we used [Brenner and Scott, 2008, Theorem (4.4.4)] for the interpolation error andthe stability of the interpolation in H2(Ω). Together with the error estimate for theenergy norm we obtain the assertion.

This result is the key to estimate the first term in (3.3). Hence, the main contributionto this error term is addressed in (4.3). We emphasize that the terms containing the supin (4.3) are just the (normalized) quadrature errors

(wh, u)L2(Ω) − w>hML Ih u =

∫Ωwh u dx−

N∑i=1

ωiwh(xiL) u(xiL). (4.4)

It remains to estimate these quadrature errors. The global quadrature error can be splitinto elementwise error contributions.

4.3 Quadrature error on a single element

In this section we will study the error caused by the mass lumping. Let us define theorder r := k+ k′− 1. Let us briefly recall the two finite element spaces of interest, whichwere defined at the beginning of Section 4.

1. quadratic polynomials for which we have Vh = Pk, i.e., k = k′ = 2, r = 3,

2. enriched cubic polynomials for which we have P3 ⊂ Vh ⊂ P4, i.e., k = 3, k′ = 4,r = 6.

In both cases, the quadrature rule associated with mass lumping is exact for polynomialsof degree up to order r − 1, see Section 4. Let us discuss a single triangle T of thetriangulation T . In the following discussions, c > 0 will denote a generic constant, which

12


will not depend on the triangle T , but only on its shape regularity, and on the referenceelement.

To allow for a local analysis, we introduce the local quadrature formulas. We denote byxT,jL , j = 1, . . . , NL, the Lagrange nodes of the cell T ∈ T and we set

ωTj =

∫TφT,j dx,

where φT,j is the local basis function associated with the Lagrange node xT,jL . Notethat local the quadrature rule on T (given by the points xT,jL j=1,...,NL and the weightsωTj j=1,...,NL) is obtained by an affine transformation of the associated quadrature ruleon the reference element. Each local Lagrange node xT,jL corresponds to a global Lagrangenode xiL and, similarly, all global basis functions φi are obtained by a sum of local basisfunctions φT,j . Thus, it can be checked that the global and local quadrature rules arerelated via

N∑i=1

v(xiL)ωi =∑T∈T

NL∑j=1

v(xT,jL )ωTj ∀v ∈ C(Ω). (4.5)

For φ ∈ C(T ) and vh ∈ Vh|T we define the quadrature error

ET (φ, vh) :=∣∣∣∫Tφ vh dx−

NL∑j=1

φ(xT,jL ) vh(xT,jL )ωTj

∣∣∣.The sum over all elements yields the desired estimate, cf. (4.4) and (4.5).

Theorem 4.4. There exist a constant c such that for each cell T ∈ T , we have theestimates

ET (φ, vh) ≤ c hk+1T ‖φ‖Wk+1,2(T ) ‖vh‖W 2,2(T ) ∀φ ∈W k+1,2(T ), vh ∈ Vh|T , (4.6a)

ET (φ, vh) ≤ c h2T ‖φ‖L∞(T ) ‖vh‖L∞(T ) ∀φ ∈ C(T ), vh ∈ Vh|T . (4.6b)

We emphasize that the constant c in Theorem 4.4 does not depend on the triangulationT , but only on its shape regularity.

Proof. By standard arguments (transformation to reference cell and the Bramble-HilbertLemma), we obtain the estimate

ET (φh, vh) ≤ c hrT∣∣φh vh∣∣W r,1(T )

for φh, vh ∈ Vh|T . Next, we use the nodal interpolant φh := Ihφ ∈ Vh|T . Of course wehave φ(xiL) = (Ihφ)(xiL). Hence, Ihφ − φ is zero in the Lagrange points and these are

13


precisely the quadrature nodes xiL. Consequently, we find

ET (φ, vh) ≤ ET (φ− Ihφ, vh) + ET (Ihφ, vh)

=∣∣∣∫T

(φ− Ihφ) vh dx∣∣∣+ ET (Ihφ, vh)

≤ c ‖φ− Ihφ‖L2(T ) ‖vh‖L2(T ) + c hrT∣∣Ihφ vh∣∣W r,1(T )

,

which implies

ET (φ, vh) ≤ c ‖φ− Ihφ‖L2(T ) ‖vh‖L2(T ) + c hrT ‖Ihφ‖W r,2(T ) ‖vh‖W r,2(T ).

Since Ihφ, vh are polynomials of degree k′, all derivatives of order k′ + 1, . . . , r are zero.Together with an inverse estimate (see [Brenner and Scott, 2008, Lemma (4.5.3)]) weget

ET (φ, vh) ≤ c ‖φ− Ihφ‖L2(T ) ‖vh‖L2(T ) + c hrT ‖Ihφ‖Wk′,2(T ) ‖vh‖Wk′,2(T )

≤ c ‖φ− Ihφ‖L2(T ) ‖vh‖L2(T ) + c hk+1T ‖Ihφ‖Wk′,2(T ) ‖vh‖W 2,2(T ).

Next we use the stability of the Lagrange interpolant, see [Brenner and Scott, 2008,Theorem (4.4.4)] (note that this requires k′ ≤ k + 1), and the interpolation estimate[Brenner and Scott, 2008, Theorem (4.4.4)] to obtain

ET (φ, vh) ≤ c ‖φ− Ihφ‖L2(T ) ‖vh‖L2(T ) + c hk+1T ‖φ‖Wk′,2(T ) ‖vh‖W 2,2(T )

≤ c hk+1T ‖φ‖Wk+1,2(T ) ‖vh‖L2(T ) + c hk+1

T ‖φ‖Wk′,2(T ) ‖vh‖W 2,2(T )

≤ c hk+1T ‖φ‖Wk+1,2(T ) ‖vh‖W 2,2(T ).

This shows (4.6a). The estimate (4.6b) follows from

ET (φ, vh) ≤(∫

T1 dx+

N∑i=1

ωi

)‖φ‖L∞(T ) ‖vh‖L∞(T )

and the shape regularity of T .

4.4 Quadrature error on the entire mesh

Now, we use the estimate from Theorem 4.4 in order to bound the quadrature error termin the estimate (4.3) for the state equation from Lemma 4.3.Our idea is to work with two different mesh sizes. That is, we use the mesh width hgoodfor cells on which the optimal control u is quite regular (these cells will be called goodcells) and a finer mesh width hbad for cells on which the optimal control u possesses kinks(these cells will be called bad cells). That is, we use hgood ≥ hbad with

| lnhgood| ∼ | lnhbad|, (4.7)

14


i.e., there are constants c, C > 0 with c | lnhgood| ≤ | lnhbad| ≤ C | lnhgood|. We willfirst derive an error estimate containing both mesh sizes. Later we will balance the errorterms.More precisely, our assumptions on the mesh widths hgood and hbad are as follows.• Cells T with smooth behavior of the solution u have a diameter less than hgood,

that is, we require that u = p or u = ua or u = ub hold on each of these cells. Inparticular, this implies u ∈ W k+1,2(T ) if p ∈ W k+1,2(T ). The set of all these cellsis denoted by Tgood.• The remaining elements are denoted by Tbad and have a diameter less than hbad.

These are cells where the optimal control has a kink. By Nbad we denote thenumbers of cells in Tbad.

Theorem 4.5. We assume that the optimal adjoint state p belongs to the space X =W k+1,2(Ω). Together with the above assumptions, we have

∑T∈T

ET (u, wh) ≤ cC(hk+1

good + (1 + |lnhgood|)1/2Nbad h3bad)√∑

T∈T‖wh‖2H2(T )

and ∑T∈T

ET (u, wh) ≤ cC(hkgood + (1 + |lnhgood|)1/2Nbad h

3bad)‖wh‖H1(Ω),

where C = max‖p‖X , |ua|, |ub|

.

Proof. On the cells belonging to Tgood we use (4.6a) and obtain∑T∈Tgood

ET (u, wh) ≤ c hkgood

∑T∈Tgood

hT ‖u‖Wk+1,2(T ) ‖wh‖W 2,2(T )

≤ c hkgood

√∑T∈T‖u‖2

Wk+1,2(T )

√∑T∈T

h2T ‖wh‖2H2(T )

(4.8)

≤ c hk+1good max

‖p‖X , |ua|, |ub|

√∑T∈T‖wh‖2H2(T )

(4.9)

with X = W k+1,2(Ω).

Using the inverse estimate [Brenner and Scott, 2008, Eq. (4.5.15)] in (4.8), we find∑T∈Tgood

ET (u, wh) ≤ c hkgood max‖p‖X , |ua|, |ub|

‖wh‖H1(Ω). (4.10)

Let us now investigate the second type of cells T ∈ Tbad. We start with the identity

ET (u, wh) = α−1ET (p, wh) + ET (u− α−1 p, wh).

15


The first term contains only smooth terms and can be estimated in the same way as onthe first type of cells. The crucial term is the second one where we use (4.6b) to obtain

ET (u− α−1 p, wh) ≤ c h2bad ‖u− α−1 p‖L∞(T ) ‖wh‖L∞(T )

≤ c h3bad ‖p‖W 1,∞(T )‖wh‖L∞(T ). (4.11)

Here we used the Lipschitz continuity of u, p and the fact u = α−1 p for at least onepoint in the element. This implies ‖u − α−1 p‖L∞(T ) ≤ c hT ‖p‖W 1,∞(T ). Summing upthe error terms we find∑

T∈Tbad

ET (u− α−1 p, wh) ≤ cNbad h3bad ‖p‖W 1,∞(Ω) ‖wh‖L∞(Ω)

≤ cNbad h3bad ‖p‖X ‖wh‖L∞(Ω),

where we used X → W 3,1(Ω) → W 1,∞(Ω). Since our mesh parameters hgood and hbadsatisfy (4.7), we can use [Brenner and Scott, 2008, Lemma (4.9.2)] and find the discreteSobolev embedding

‖wh‖L∞(Ω) ≤ c (1 + | lnhgood|)1/2‖wh‖H1(Ω).

This shows∑T∈Tbad

ET (u− α−1 p, wh) ≤ cNbad (1 + |lnhgood|)1/2 h3bad ‖p‖X ‖wh‖H1(Ω).

Together with (4.9) and (4.10), we obtain the assertion.

Plugging these estimates into (4.3), we obtain the following corollary.

Corollary 4.6. Under the assumptions of Theorem 4.5, we have

‖y − y‖L2(Ω) ≤ c(hgood ‖y − Ih y‖H1(Ω) + hk+1

good +Nbad h3bad (1 + |lnhgood|)1/2

),

where the constant c depends on ‖p‖X .

Proof. The estimate follows from Lemma 4.3 and Theorem 4.5.

5 Error estimates for the optimal control problem

In this section we will combine the results of Section 4.4 and our main estimate (3.3).The number of refined cells Nbad plays a crucial role in Corollary 4.6. We will require

Nbad ≤ c h−1bad,

16


which is reasonable if the kinks are located on a finite number of regular curves. A similarassumption is commonly used for error estimates of control constrained problems, seeRösch [2006], Pieper [2015].

Theorem 5.1. Let us assume p ∈W k+1,2(Ω), Nbad ≤ c h−1bad. Then the error estimate

α


+1

2‖Ih y − yh‖2M +

1


≤ c(hk+1

good + (1 + |lnhgood|)1/2)h2bad + hgood ‖y − Ih y‖H1(Ω)

)(5.1)

is satisfied.

This statement follows immediately from Corollary 4.6 combined with (3.3).

Corollary 5.2. The choice hbad = h2good gives the estimate

α


+1

2‖Ih y − yh‖2M +

1


≤ c (h3good + hgood ‖y − Ih y‖H1(Ω)) (5.2)

in the case of P2-elements (k = k′ = 2) and

α


+1

2‖Ih y − yh‖2M +

1


≤ c ((1 + |lnhgood|)1/2 h4good + hgood ‖y − Ih y‖H1(Ω)) (5.3)

in the case of enriched P3-elements (k = 3, k′ = 4).

The logarithmic term in (5.2) can be dropped due to Corollary 4.6. Further, for theverification of (5.2), hbad = h

3/2+εgood , ε > 0, is enough. By using ε = 0, one obtains an

additional logarithmic term.

Remark 5.3. Our assumption on the adjoint state p is quite strong since one has toexpect corner singularities due to the polygonal domain. However, it is possible tocombine our approach with a mesh grading at the corners of the polygon. We refer toApel [1999, 2004] for graded meshes in combination with higher order finite elements.The adjoint state p belongs to a corresponding weighted Sobolev space of higher order.The optimal state y has the same regularity if one stays away from the kinks of theoptimal control u. Some details concerning mesh grading are given in Section 7.2.

Remark 5.4. Let us analyze the interpolation error ‖y − Ih y‖H1(Ω), which appears inthe different error estimates. Here two different effects are of interest. The first effectis connected with possible corner singularities that reduce the order of the interpolation

17


error. This effect can be again compensated by mesh grading. The second effect is causedby kinks in the optimal control. Then, the control belongs to H1.5−ε(Ω), for any ε > 0,but not to H2(Ω). Hence, we can expect (up to corner singularities) H3.5−ε(Ω)-regularityof the optimal state y but not H4(Ω)-regularity.

Consequently, we have no restriction for P2-elements (k = k′ = 2) to obtain

α


+1

2‖Ih y − yh‖2M +

1


≤ c h3good (5.4)

The situation is more difficult for the enriched P3-elements (k = 3, k′ = 4). The elementscontaining the kink have the smaller mesh size hbad. Consequently, in the regular casein which

‖Ihy − y‖H1(Ω) ≤ c h3good

is satisfied, Corollary 5.2 leads to the error estimate

α


+1

2‖Ih y − yh‖2M +

1


≤ c (1 + |lnhgood|)1/2 h4good. (5.5)

By standard arguments, we obtain the same rates for the approximation of the adjointstates.

6 An algorithmic approach

In the last section we derived an optimal convergence order. Until now, this is not apractical approach, since it was assumed that the regions with high and low regularityof the control are known and refined appropriately. However, the presented results canserve as a benchmark for algorithmic approaches.

Let us remember that we assumed a high regularity of the adjoint state p. This regularityis partially transferred to the optimal control via the projection formula (2.1). Hence,the regions of low regularity of the optimal control are kinks caused by the pointwiseprojection. Consequently, a successful algorithm has to detect these kinks where a finemesh is needed to get highly accurate numerical solutions. In our opinion, two generalapproaches are possible and reasonable.

Computational Approach 11. Compute a numerical solution for a quasi-uniform mesh with mesh size hgood.

2. Determine the region where kinks of u may occur. (We will explain this step indetail below.)

3. Refine the mesh in the region where kinks of u may occur with a mesh size hbadand compute on the new mesh an improved numerical solution.

18


Computational Approach 21. Compute a numerical solution for a quasi-uniform mesh with mesh size hgood.

2. Repeat as often it is necessary: Refine all elements which contain a kink of uh andpossess a diameter larger then hbad.

Next we will analyze Computational Approach 1 and the regular case addressed in Re-mark 5.4. The Computational Approach 2 is rather heuristically and will not be analyzed.

Let us denote the numerical solution of the discretized problem (Ph) of the first stepwith (yh, uh, ph). Since the mesh is quasi-uniform, we have hgood ∼ hbad. We candirectly apply the error estimate (5.1) to obtain

‖Ih y − yh‖L2(Ω) = ‖Ih y − yh‖M ≤ c (1 + |lnhgood|)1/2 h2good.

Since the interpolation error ‖Ih y − y‖L2(Ω) is of higher order, we get

‖y − yh‖L2(Ω) ≤ c (1 + |lnhgood|)1/2 h2good.

The regularity of the optimal adjoint state and the above inequality imply

‖p− ph‖L∞(Ω) ≤ c (1 + |lnhgood|)1/2 h2good. (6.1)

Kinks in the optimal control u may occur at x ∈ Ω if p(x)/α ∈ ua, ub. Because of (6.1)kinks cannot appear at x ∈ Ω if

|ph(x)− αua| > c (1 + |lnhgood|)1/2 h2good and

|ph(x)− αub| > c (1 + |lnhgood|)1/2 h2good

(6.2)

hold. On the set of elements on which (6.2) is satisfied, no mesh refinement is necessary.Next, we discuss how many elements are needed if the remaining part is refined by themesh size hbad.

To do this, we need an additional assumption on the behavior of the adjoint state p. Letus define the family of sets

K(ε) = x ∈ Ω : |p− αua| ≤ ε or |p− αub| ≤ ε.

Let us assume that the measure of this set can be bounded by

|K(ε)| ≤ γε for all ε ∈ (0, ε)

with a certain positive γ and a fixed ε > 0. We remark that a similar property isfrequently used, in particular for the approximation of bang-bang controls, see Deckelnickand Hinze [2012], Wachsmuth and Wachsmuth [2011].

Now, the relation (6.2) can be guaranteed outside of the setK(2c (1+|lnhgood|)1/2 h2good).

Due to our assumption we know

|K(2c (1 + |lnhgood|)1/2 h2good)| ≤ 2cγ(1 + |lnhgood|)1/2 h2

good).

19


Until now, we investigated sets of points x ∈ Ω. To get an estimate we need that theunion of cells T ∈ T containing such points has the same behavior. Let us assume thatthe area of all elements which intersect this set, i.e., the area of

TK =⋃

T∈T :T∩K(c (1+|lnhgood|)1/2 h2good)6=∅

T

can be bounded by|TK | ≤ c (1 + |lnhgood|)1/2 h2

good)

with a fixed constant c.

This region is discretized in the second step with the mesh size hbad. The number ofelements needed is proportional to (1 + |lnhgood|)1/2 h2

good)/h2bad. Hence, the number of

needed element is not essentially increasing if

(1 + |lnhgood|)1/2 h4good ≤ c h2

bad.

For the quadratic finite elements we can choose hbad ∼ h3/2good. In the case of enriched

cubic elements we use hbad ∼ (1 + |lnhgood|)1/4 h2good.

On the refined mesh, a new finite element solution (yh, uh, ph) will be computed. We canapply directly (5.4) and (5.5) to the new discrete solution to obtain

α


+1

2‖Ih y − yh‖2M +

1


≤ c (1 + |lnhgood|)1/2h3good (6.3)

in the case of P2-elements (k = k′ = 2) and

α


+1

2‖Ih y − yh‖2M +

1


≤ c (1 + |lnhgood|)h4good (6.4)

in the case of enriched P3-elements (k = 3, k′ = 4).

7 Numerical experiments

In this section, we present numerical examples. First, we consider a regular case in whichno mesh grading is needed and this illustrates the convergence results (5.4) and (5.5),see Section 7.1. Afterwards, we discuss the application of mesh grading to our situationand present an example in which mesh grading is necessary to obtain the optimal rates,see Section 7.2.

20


7.1 Results for smooth solutions

First, we consider the optimal control problem (P) with the data

Ω = (0, 1)2, α = 0.05, yd(x1, x2) = expx1 sin(x2), ua = −1.5, ub = 1.0. (7.1)

In the numerical implementation, we used an algorithm similar to Computational Ap-proach 2 in Section 6 which is additionally coupled with a nested iteration. We startwith a coarse initial mesh and perform the following in each iteration:

• We set hgood = maxT∈T diam(T ) and hbad = h3/2good (P2 elements) or hbad = h2

good(P3 elements).

• If there is any cell T with diam(T ) > hbad and on which Proj[ua,ub](ph/α) has akink, we refine all those cells (local refinement).

• Otherwise, we refine all cells T with diam(T ) > hgood/2 (global refinement).

For the mesh refinement, we use the longest-edge-bisection algorithm, similar to Rivara[1991]. The above algorithm is implemented in the finite element toolbox FEniCS, cf.Alnæs et al. [2015], Logg et al. [2012], by using the geometric multigrid implementationfrom Ospald [2012].

The computational results are shown in Figure 7.1 (for P2 elements) and Figure 7.2 (forenriched P2 elements). Since an analytical solution for the problem under consideration

10−2 10−1

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

10−2

hgood

‖yh − yh∗‖L2(Ω)

‖uh − uh∗‖L2(Ω)

‖uh − Ihuh∗‖ML

h2.25

h3

Figure 7.1: Errors in the control and state for the discretization with P2 elements.

21


10−2 10−1

10−11

10−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

hgood




h3

h4

Figure 7.2: Errors in the control and state for the discretization with enriched P3

elements.

is not known, we used a fine grid solution as a reference for the computed errors. In fact,we used

yh∗ = yh′ and uh∗ = Proj[ua,ub]ph′

α, (7.2)

where (uh′ , yh′ , ph′) is the solution of (Ph) on a finer grid with mesh sizes h′ = 5.5 · 10−3

in case of P2 elements and h′ = 1.6 · 10−2 in case of enriched P3 elements. As predictedin (5.4) and (5.5), we see convergence of order hk+1 for the errors

‖yh − yh∗‖L2(Ω) and ‖uh − Ihuh∗‖ML.

Note that the error of the control ‖uh − uh∗‖L2(Ω) converges significantly slower. Forquasiuniform meshes and P1-elements one knows a convergence of order h3/2, cf. Rösch[2006], Casas and Mateos [2008]. Clearly, these techniques can be extended to moregeneral meshes and larger classes of finite elements. However, the best approximationon our mesh is of order h3/2

bad due to the presence of the kink. Exactly this order isobserved in our numerical tests. Because of the coupling of hgood and hbad, we obtainh

3/2bad = (h

3/2good)3/2 = h2.25

good for the case of P2 elements and h3/2bad = (h2

good)3/2 = h3good for

the case of enriched P3 elements. This is essentially smaller as the convergence order ofour new method (3 respectively 4).

Let us give a possible explanation for the fact that mesh grading seems not to be nec-essary for problem (P) with data given in (7.1). The homogeneous Neumann boundary

22


condition in the state equation can be seen as a symmetry boundary condition. In fact,we can extend the problem to, e.g., (0, 1)× (0, 2) by a reflection at the line R× 1. Onthis extended domain, there are no corners at (0, 1) and (1, 1). This might be a reasonthat no corner singularities (see (7.4) below) appear for this problem and we benefit fromhigher regularity of the solution.

7.2 Graded meshes

As already mentioned in the theoretical part corner singularities of state and adjoint statemay influence the accuracy of the numerical results. Here we will sketch some theoreticalaspects of mesh grading. Moreover, we will present numerical experiments for gradedmeshes. The theory works for convex as well as for nonconvex polyhedral domains.

Let us discuss the following Neumann boundary value problem

−∆z + z = f on Ω,∂

∂nz = 0 on Γ. (7.3)

Even for for convex domains and smooth data the regularity of the solution of the Poissonequation is limited due to corner singularities. The behavior close to a corner is dominatedby the so-called singular part zsing of the solution z. This singular part is given by

zsing = rλ cos(λϕ) ξ(r), (7.4)

where (r, ϕ) are the polar coordinates with the origin located in the corner and ξ denotesa smooth cut-off function, i.e., ξ ∈ C∞0 ([0,∞)) with ξ(0) = 1. The singular exponent λis given by the formula λ = π/ω, where ω denotes the interior angle of the investigatedcorner. A simple computation shows that H3-regularity or H4-regularity near the cornerrequires λ > 2 or λ > 3 which is only satisfied for very small angles ω.

An established way is to work with weighted Sobolev spaces, which we introduce next.Let m be the number of corners of the domain Ω. At each corner xj , we introduce acircular sector Ωj := B(xj , Rj)∩Ω, where B(x,Rj) is the closed ball with center xj andradius Rj . The radii Rj > 0 are chosen in such a way that the domains Ωj are disjoint.

We set

Ω0 := Ω \m⋃j=1

B(xj , Rj/2).

Note that the decomposition Ω =⋃mj=0 Ωj is overlapping.

For a natural number k, an integrability parameter p ∈ [1,∞) and weight vector β =

(β1, .., βm)> ∈ [0,∞)m we define the weighted Sobolev spaceW k,pβ (Ω) as set of all (equiv-

alence classes) of functions v on Ω with finite norm

‖v‖Wk,pβ (Ω)

:= ‖v‖Wk,p(Ω0) +m∑j=1

‖v‖Wk,pβj

(Ωj),

23


where the Sobolev space W k,p(Ω0) is defined as usual and the weighted part of the normis defined by

‖v‖Wk,pβj

(Ωj):=

(∑|α|≤k

‖rβjj Dαv‖pLp(Ωj)

)1/p

with the standard notation for the multiindex α, and rj(x) := ‖x − xj‖ is the distanceto the corner xj .In the case that the elements of weight vector β are strictly positive, the weight functionrβjj vanishes at the corner xj and is bounded on Ω. Therefore, we have W k,p(Ω) ⊂W k,pβ (Ω) and the space W k,p

β (Ω) allows for stronger singularities in the neighborhoodof the corners compared to the space W k,p(Ω). In particular, we still obtain W k,2

β (Ω)-regularity of solutions of (7.3) under an assumption on β, cf. Lemma 7.1 below.The idea behind graded meshes is to benefit from the regularity of solutions in suchweighted Sobolev spaces. Graded meshes are refined towards the corners of the domain.Assume that the element T is located in one of the domains ΩRj . Then, one requiresthat the diameter hT of T satisfies

c1 h1/µj ≤ hT ≤ c2 h

1/µj for rT,j = 0, (7.5a)

c1 h r1−µjT,j ≤ hT ≤ c2 h r

1−µjT,j for rT,j ∈ (0, Rj/2), (7.5b)

c1 h ≤ hT ≤ c2 h for rT,j > Rj/2 (7.5c)

with certain positive constants c1, c2 and rT,j denotes the distance of the element T tothe corner j, i.e., rT,j = minx∈T rj(x). Note that the number of finite elements in thegraded mesh has still the order O(h−2).Following the argumentation in [Apel et al., 2012, Lemma 4.1], we obtain the followingresult.

Lemma 7.1. We denote by z the solution of (7.3). Let us assume that 2 − λj < βj ≤2(1−µj) holds for all j ∈ 1, . . . ,m and that the right-hand side satisfies f ∈W 1,2

β (Ω).Moreover, we consider a quasi-uniform family Th of triangulations obeying (7.5). Wedenote by zh the solution of the Galerkin discretization of (7.3) with continuous P2

elements on the mesh Th. Then, the following error estimates are satisfied for the finiteelement solution zh

‖z − zh‖L2(Ω) + h ‖z − zh‖H1(Ω) ≤ c h3 ‖z‖W 3,2β (Ω)

≤ c ‖f‖W 1,2β (Ω)

. (7.6)

Similarly, we obtain

‖z − zh‖L2(Ω) + h ‖z − zh‖H1(Ω) ≤ c h4 ‖z‖W 4,2β (Ω)

≤ c ‖f‖W 2,2β (Ω)

(7.7)

for the discretization with enriched P3-elements if 3 − λj < βj ≤ 3(1 − µj) holds for allj ∈ 1, . . . ,m and f ∈W 2,2

β (Ω).

24


This result can be directly used to substitute corresponding results for the adjoint statep in previous sections.

A more complicate issue is the derivation of error estimates for the state y. Here, we haveto modify the derivations of Theorem 4.5. For graded meshes we set X = W k+1,2

β (Ω).

Since the diameter hT of cells near the corner contains now the factor h r1−µjT,j , one can

derive corresponding error estimates in weighted norms. Moreover, as usual for gradedmeshes, the elements located in the corner (rT,j = 0) have to discussed separately. Weabstain from performing this technical discussion in detail. At the end, the results forSection 5 are still true (even in the nonconvex case) with graded meshes. Note, thatin the nonconvex case the mesh grading is very strong. The standard L-shape domainrequires a mesh grading factor µ < 2/9 for P3-elements and µ < 1/3 for P2-elements.

The following experiments will provide some numerical evidence that mesh grading isnecessary for obtaining the optimal order of convergence. The domain Ω is the interior ofthe quadrilateral conv(0, 0), (1, 0), (0.7, 0.7), (0, 1). The remaining data of the problem(P) is chosen as in (7.1).

The interior angle at the upper-right vertex x1 = (0.7, 0.7) is approximately ω1 ≈ 2.381and the interior angles at the upper-left and lower-right vertices are approximately ω2 ≈1.167. Hence, we have λ1 ≈ 1.320 for the upper-right vertex and λ2 ≈ 2.695. Accordingto Lemma 7.1 we would have to choose the mesh-grading parameters

µ1 < 0.66 and µ2 < 1.35

for optimal convergence rates with P2 elements and

µ1 < 0.44 and µ2 < 0.90

for optimal convergence rates with enriched P3 elements. Note that this means that wealso need mesh grading in the convex vertices (1, 0) and (0, 1) for P3 elements.

The numerical solution is obtained as described in Section 7.1. That is, for a givenmesh, we solve the discrete problem (Ph). Afterwards, we refine all cells T , on which uhpossesses a kink and hT ≥ hbad, and all cells T which violate the grading condition (7.5).

In order to present the effect of mesh grading, we solved the optimal control problemonce without mesh grading (i.e., µ1 = µ2 = 1) and once with sufficiently strong meshgrading.

The computed errors for P2 elements are shown in Figure 7.3 (µ1 = µ2 = 1) and Figure 7.4(µ1 = 0.6, µ2 = 1). The reference solutions yh∗ , uh∗ are obtained as described inSection 7.1, with h′ = 4 · 10−3, see (7.2). It can be clearly seen that the estimated errorsare significantly smaller if mesh grading is used. We mention that the dimension of theansatz space only increases from 208162 to 211852 due to the mesh grading for the finestmesh width h = 8 · 10−3 which is shown in the error plots.

The estimated errors for the enriched P3 elements are shown in Figure 7.5 (µ1 = µ2 = 1)and Figure 7.6 (µ1 = 0.4, µ2 = 0.8). The reference solutions yh∗ , uh∗ are obtained

25


10−2 10−1

10−8

10−7

10−6

10−5

10−4

hgood




h2.25

h2.5

Figure 7.3: Errors in the control and state for the discretization with P2 elements withoutmesh grading.

10−2 10−1

10−9

10−8

10−7

10−6

10−5

10−4

10−3

hgood




h2.25

h3

Figure 7.4: Errors in the control and state for the discretization with P2 elements withmesh grading.

26


with the mesh size h′ = 1.2 · 10−2 according to (7.2). Again, the estimated errors aresignificantly smaller if mesh grading is used and this improvement is even more significantas in case of P2 elements. The dimension of the ansatz space only increases from 210385to 220403 due to the mesh grading for the finest mesh width h = 2.5 · 10−2 which isshown in the error plots.

8 Generalizations

In this section we shortly comment on possible generalizations. The application of ourtheory to varying control bounds is not connected with essential difficulties. We onlyhave to require

ua, ub ∈ X, and minx∈Ω

ub(x)− ua(x) > 0,

where X = W k+1,2(Ω). It is also possible to apply our theory to non-convex domains.The main features were already addressed in Section 7.2. Note that a quite strong meshgrading is required to obtain the required error estimates for the Galerkin solutions, cf.Lemma 7.1.

The generalization to three-dimensional domains seems to be challenging. In our deriva-tion we used the discrete Sobolev inequality

‖wh‖L∞(Ω) ≤ c (1 + | lnhgood|)1/2‖wh‖H1(Ω),

see [Brenner and Scott, 2008, Lemma (4.9.2)]. In the three-dimensional case we wouldloose half an order of hgood. Let us mention that specific properties of two-dimensionaldomains were used in several other derivations, too.

A discussion of more general elliptic equations or different boundary conditions can bedone by the presented techniques. Of course, the smoothness requirements of the optimalcontrol, state and adjoint state have to be satisfied. Varying coefficients can be directlyused in the quadrature formulas caused by the mass lumping.

Our theory can also be modified to problems where the control acts only on a subdomainor if the defect term in the objective is only observed on a subdomain.

A more difficult task is the application of our theory to boundary control problems. Thetechniques for deriving optimal error estimates for Neumann boundary control are quitedifferent from the techniques presented in the paper. If a combination of these differenttechniques is possible or not, is not clear at the moment.

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mass lumping for the optimal control of elliptic partial ......and wachsmuth,2011, (4.13)] and to...

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