Abstract
This article deals with error estimates for the finite element approximation of variational normal derivatives and, as a consequence, error estimates for the finite element approximation of Dirichlet boundary control problems with energy regularization. The regularity of the solution is carefully carved out exploiting weighted Sobolev and Hölder spaces. This allows to derive a sharp relation between the convergence rates for the approximation and the structure of the geometry, more precisely, the largest opening angle at the vertices of polygonal domains. Numerical experiments confirm that the derived convergence rates are sharp.
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1 Introduction
The problem investigated in this article is the optimal Dirichlet control problem
where \(u(z)\in H^1(\Omega )\) is the solution of the boundary value problem
The domain \(\Omega \in \mathbb R^2\) is assumed to have a polygonal boundary \(\Gamma \). The function \(u_d\in L^2(\Omega )\) is referred to as desired state and \(f\in L^2(\Omega )\) is a given source term. The parameter \(\nu >0\) is a regularization parameter and the corresponding term in the objective guarantees the existence of a solution in the space \(H^{1/2}(\Gamma )\).
This optimal control problem has first been formulated by Lions [19]. Later, a regularization using the \(L^2(\Gamma )\)-norm of the control became more attention [3, 8, 20, 21]. From the modeling point of view, the \(L^2(\Gamma )\) regularization is reasonable as the regularization term can be interpreted as a measure for control costs, but the disadvantage is that the control has a rather unexpected behavior near the corners. In the general case the control tends to 0 at convex and to infinity at reentrant corners [2]. Thus, the idea of using a regularization in stronger norms was revealed e. g. in [14] where Dirichlet control problems using an \(H^1(\Gamma )\)-regularization are studied and [9, 27] where \(H^{1/2}(\Gamma )\)-regularization is considered. The latter idea, also referred to as energy regularization and is also studied in the present paper. It has to be noted that the behavior near the corners is in this approach just shifted to the tangential derivatives of the control. The physical interpretation of the regularization term using the \(H^{1/2}(\Gamma )\)-norm of the control is, that in case of \(f\equiv 0\) it is equivalent to the energy norm of the corresponding state u(z), which might be, depending on the concrete application, a measure for control costs as well. This becomes clear when defining the seminorm in \(H^{1/2}(\Gamma )\) by
Closely related are the investigations for the Neumann control problem with an \(H^{-1/2}(\Gamma )\)-regularization [6, 32]. Note, that the optimal state is in both approaches equivalent.
Error estimates for approximate solutions of the Dirichlet control problem are discussed already in [27] where all variables are approximated by piecewise linear finite elements. For this approach, and in case of convex computational domains, the convergence rate of 1 for the control in the \(H^{1/2}(\Gamma )\)-norm was proved, but in the numerical experiments a higher convergence rate is observed. The results in the present article will show that the rate 1 is only a worst-case estimate for convex domains, meaning, that if an opening angle of a corner tends to \(180^\circ \), the convergence rate will tend to 1. The same convergence rate is proved in [17] for arbitrary polygonal domains for a discretization using the energy corrected finite element method.
It is the aim of the present paper to prove sharp convergence rates. Depending on the opening angle at the corners one can prove a convergence rate up to 3 / 2 for the control in the \(H^{1/2}(\Gamma )\)-norm. It turns out that this is in general only possible when the opening angles are all less than \(120^\circ \) as the corresponding singularities are mild enough to guarantee \(H^2(\Gamma )\)-regularity of the control.
The difficult part of the convergence proof is to derive an error estimate for a variational normal derivative of the finite element solution of the Poisson and the Laplace equation in the \(H^{-1/2}(\Gamma )\)-norm. Such an error term appears due do the approximation of the Steklov–Poincaré operator \(z\mapsto \partial _n u(z)\) used to realize the \(H^{1/2}(\Gamma )\)-norm, and the approximation for the normal derivative of the adjoint state variable which appears in the optimality condition. A worst-case estimate for variational normal derivatives in the \(H^{-1/2}(\Gamma )\)-norm, as used in [27], can be easily derived when using a trace theorem and standard finite element error estimates. Sharp error estimates require some more effort and will be discussed intensively in the present article. Closely related are the error estimates in the \(L^2(\Gamma )\)-norm for the exact normal derivative of the finite element approximation from [15, 29]. In the latter reference the variational normal derivative used in the present paper is discussed as well. In the present article we consider estimates for the variational normal derivative in \(H^{-1/2}(\Gamma )\). The convergence rate we prove will be related to \(\omega _{\text {max}}\) denoting the largest opening angle of the corners of the domain \(\Omega \). Moreover, \(u\in H^1(\Omega )\) and \(u_h\in V_h\) are the solution of the Poisson or Laplace equation and its finite element approximation, respectively. Throughout this article \(V_h\) is the space of linear and globally continuous finite elements. Under the assumption that the input data are sufficiently smooth, and the normal derivative is continuous in the corners (this is guaranteed e. g. if \(z\equiv 0\) or if u is the optimal state of the Dirichlet control problem (1)–(2)) we show that the variational normal derivative \(\partial _n^h u_h\in V_h^\partial :=\text {tr} V_h\) defined by
for all \(v_h\in V_h\) satisfies the estimate
Here, \(c>0\) is a constant independent of the mesh size h and \(\varepsilon >0\) is an arbitrary but sufficiently small number. The proof is based on an idea developed in [29] where estimates in the \(L^2(\Gamma )\)-norm on a sequence of boundary concentrated meshes is proved.
As an application, we use this result to derive sharp discretization error estimates for the optimal control problem (1)–(2). Therefore, we approximate the control, state and adjoint state by a linear finite element discretization. Under the assumption that \(u_d\) is Hölder continuous in case of convex \(\Omega \), or belongs to \(L^2(\Omega )\) in case of non-convex \(\Omega \), we show the same convergence rate for the control approximation in the \(H^{1/2}(\Gamma )\)-norm, this is,
where z and \(z_h\) are the continuous and discrete optimal control. This confirms the behavior figured out in the numerical experiments from [27] on the unit square, where the rate 3 / 2 was predicted numerically. The conjecture that this rate is achieved on arbitrary convex polygonal domains is obviously wrong. Our theory promises that this rate is obtained unless all opening angles of corners are less than \(2\pi /3\) which is also confirmed by numerical experiments. The worst-case convergence rate of 1 is indeed achieved unless the domain remains convex. If the largest angle tends to \(2\pi \), the convergence rate will tend to 1 / 2.
As a further application of estimates for variational normal derivatives we mention Steklov–Poincaré operators that are frequently used for parallel finite element methods relying on domain decomposition [1, 30, 34]. Closely related are the error estimates from [24]. Therein, the authors derive optimal error estimates for discrete Lagrange multipliers in \(H^{-1/2}(\Gamma )\) defined on the interfaces of the subdomains. The approximation of the multipliers corresponds to some variational approximation of a normal derivative as well.
The article is structured as follows. In Sect. 2 we collect a priori estimates for solutions of the Poisson and Laplace equation in weighted norms involving a regularized boundary distance function. Moreover, we have to carve out the singular behavior near corners of the domain which is done by weighted Sobolev and Hölder spaces. To this end, we provide the required shift theorems. Error estimates for the solution of the Dirichlet boundary value problem in the \(L^2(\Omega )\)- and \(H^1(\Omega )\)-norm as well as for the discrete normal derivatives in the \(H^{-1/2}(\Gamma )\)-norm are derived in Sect. 3. These estimates are applied to the discretization of our optimal control problem in Sect. 4. The results derived therein are confirmed by the numerical experiments in Sect. 5.
2 Auxiliary results
Let us first explain the notation we will use in this paper. The computational domain is denoted by \(\Omega \subset \mathbb R^2\) and is always assumed to have a polygonal boundary \(\Gamma \). By \(W^{k,p}(\Omega )\), \(k\in \mathbb N_0\), \(p\in [1,\infty ]\) we denote the usual Sobolev spaces and write \(H^{k}(\Omega ) := W^{k,2}(\Omega )\), \(L^2(\Omega ):=H^0(\Omega )\). Frequently, we use the space \(H_0^1(\Omega )\) which is the closure of \(C_0^\infty (\Omega )\) with respect to the \(H^1(\Omega )\)-norm. For the corresponding norms and inner products we write \(\Vert \cdot \Vert _{X}\) and \((\cdot ,\cdot )_X\), respectively. The subscript X indicates the related space.
The fractional-order Sobolev space \(H^{1/2}(\Gamma )\) is defined as the set of measurable functions \(u:\Gamma \rightarrow \mathbb R\) with finite Sobolev-Slobodeckij-norm
Later, we will use equivalent definitions, e. g. as the natural trace space of \(H^1(\Omega )\), see [33, §8], or as the interpolation space \([L^2(\Gamma ), H^1(\Gamma )]_{1/2,2}\) [23, Theorem B.11].
Moreover, \(\left<\cdot ,\cdot \right>\) stands for the dual pairing between \(H^{-1/2}(\Gamma )\) and \(H^{1/2}(\Gamma )\).
The aim of this section is to collect some regularity results for the solution of the Laplace and Poisson equation. The weak form reads: Find \(y\in H^1(\Omega )\) satisfying
The functions \(f\in L^2(\Omega )\) and \(z\in H^{1/2}(\Gamma )\) are given input data.
2.1 Weighted regularity
For technical reasons we recall some a priori estimates in weighted norms involving the weight function \(\sigma (x):=\kappa h + {{\,\mathrm{dist}\,}}(x,\Gamma )\) with arbitrary \(\kappa >0\). This is a regularized distance function with respect to the boundary of the domain \(\Omega \). The following result is proved already in [29, Lemma 1].
Lemma 1
Let \(w\in H^1_0(\Omega )\) be the weak solution of \(-\Delta w = f\) in \(\Omega \). Then, the a priori estimate
holds.
Furthermore, we will need an interior regularity result:
Lemma 2
Let \(w\in H^1(\Omega )\) satisfy
with some function \(f\in L^2(\Omega )\). Moreover, let be given \(\Omega _0\subset \subset \Omega _1\subset \Omega \) and denote by \(d:={{\,\mathrm{dist}\,}}(\partial \Omega _1,\partial \Omega _0)\) the distance between the boundaries of \(\Omega _0\) and \(\Omega _1\). Then, the estimate
is valid.
Proof
The estimate (i) can be concluded from the proof of [12, Theorem 8.8] where this assertion is stated with a generic constant depending on the quantity d that we want to carve out exactly. Thus, we repeat the proof for the convenience of the reader. The proof basically relies on [12, Lemma 7.24] which states that a function \(u\in L^2(\Omega )\) belongs to \(H^1(\Omega _0)\) if its difference quotients \(D^h_k u(x):= \frac{1}{h}(u(x+h\mathbf e_k) - u(x))\), \(k\in \{1,2\}\), are bounded in the \(L^2(\Omega _0)\)-norm for all \(h\in \mathbb R\) with |h| sufficiently small such that \(D_k^h\) is well-defined in \(\Omega _1\). Moreover, the inclusion
is valid. To conclude the desired estimate we thus have to confirm that \(\Vert D_k^h \nabla w\Vert _{L^2(\Omega _0)}\) is bounded. For technical reasons we introduce a further set \(\tilde{\Omega }\) satisfying \(\Omega _0\subset \subset \tilde{\Omega }\subset \subset \Omega _1\) and \({{\,\mathrm{dist}\,}}(\Omega _0,\partial \tilde{\Omega })\sim d\). For an arbitrary test function \(v\in H^1_0(\Omega )\) with \({{\,\mathrm{dist}\,}}({{\,\mathrm{supp}\,}}v,\partial \tilde{\Omega })>2h\) we obtain
In the last step we bounded the difference quotient by the first derivative of v. Such an estimate is proved in [12, Lemma 7.23]. Next, we introduce a smooth cut-off function \(\eta \in C_0^\infty (\Omega )\) satisfying \(\eta \equiv 1\) in \(\Omega _0\) and \({{\,\mathrm{supp}\,}}\eta \subset \overline{\tilde{\Omega }}\). Moreover, \(\eta \) is constructed in such a way that \(|\nabla \eta |\le c\, d^{-1}\). For sufficiently small h we obtain from the product rule and (5) for \(v=\eta ^2 D_k^h w\)
Again, we apply [12, Lemma 7.23] to obtain \(\Vert D_k^h w\Vert _{L^2(\tilde{\Omega })}\le c\, \Vert \nabla w\Vert _{L^2(\Omega _1)}\). Moreover, with the product rule we obtain
Insertion of this estimate into (6) yields with Young’s inequality and a kick-back argument for the latter term on the right-hand side
The desired estimate then follows from (4).\(\square \)
2.2 Weighted Sobolev and Hölder spaces
In order to describe the regularity of the solution of boundary value problems in an accurate way we exploit regularity results in weighted Sobolev spaces. These spaces capture the corner singularities contained in the solution and allow us to derive sharp interpolation error estimates. Throughout the paper we denote the corners of \(\Omega \) by \(\varvec{c}_j\), \(j\in \mathcal C:=1,\ldots ,d\). Moreover, denote by \(\Gamma _j\) the boundary edge having endpoints \(\varvec{c}_j\) and \(\varvec{c}_{j+1}\) or \(\varvec{c}_1\) in case of \(j=d\). The interior angle between the edges intersecting in \(\varvec{c}_j\) is \(\omega _j\in (0,2\pi )\).
In order to introduce the weighted Sobolev spaces used for the analysis, we divide the domain into circular sectors \(\Omega _R^j:=\{x\in \Omega :|x-\varvec{c}_j| < R\}\), \(j\in \mathcal C\), with sufficiently small R such that these sectors do not overlap. The remaining sets are denoted by \(\hat{\Omega }_R:=\Omega \setminus \cup \{\Omega _R^j:j\in \mathcal C\}\). For each \(k\in \mathbb N_0\), \(p\in [1,\infty )\) and some weight \(\beta \in \mathbb R_+\) (\(\mathbb R_+:=[0,\infty )\)), we introduce the local norms
and for an analogous definition in case of \(p=\infty \), the sum has to be replaced by the maximum over \(|\alpha |\le k\). For some \(\varvec{\beta }\in \mathbb R_+^d\) the global norms are defined by
in case of \(p\in [1,\infty )\) and with the obvious modification for \(p=\infty \). When replacing V by W in the definition above, we obtain the global norm \(\Vert \cdot \Vert _{W^{k,p}_{\varvec{\beta }}(\Omega )}\). The weighted Sobolev spaces \(V^{k,p}_{\varvec{\beta }}(\Omega )\) and \(W^{k,p}_{\varvec{\beta }}(\Omega )\) are defined as the set of functions whose norms introduced above are finite. The trace spaces are denoted by \(V^{k-1/p,p}_{\varvec{\beta }}(\Gamma )\) and \(W^{k-1/p,p}_{\varvec{\beta }}(\Gamma )\), respectively. Later, we want to derive error estimates using regularity results in the \(V^{k-1/p,p}_{\varvec{\beta }}(\Gamma )\)-norm. To this end, we equip the trace spaces with the equivalent weighted Sobolev-Slobodeckij-norms. Therefore, introduce the boundary segments \(\Gamma _R^j:= \Gamma \cap \partial \Omega _R^j\), \(j\in \mathcal C\), and \(\hat{\Gamma }_R = \Gamma \cap \hat{\Omega }_R\). Moreover, \(\Gamma _R^{j,1}\) and \(\Gamma _R^{j,2}\) are the two legs of \(\Gamma _R^j\). For \(k\in \mathbb N, p\in [1,\infty )\) and \(\varvec{\beta }\in \mathbb R_+^d\) we then define
with
where \(D_t^\alpha \) denotes the tangential derivative of order \(\alpha \). In the obvious way, a norm for the case \(p=\infty \) can be defined as well.
The previous definitions and an intensive discussion on the relation between V- and W-spaces can be found in [26, Chapter 4, §5], [22, Section 6.2].
Later, we will frequently derive error estimates where the convergence rate will depend on the largest weight. Thus, we define
In the next chapter, we will frequently exploit regularity results in these space with \(p=\infty \), but for this case a shift theorem is not valid. As a remedy, weighted Hölder spaces are used and we take the definition from [22, Section 6.7.1]. Again, we define some local norms with parameters \(k\in \mathbb N_0\), \(\sigma \in (0,1]\) and \(\delta \ge \sigma \) by
where the seminorm is defined by
The global norm is then given by
with some vector \(\varvec{\delta }\in [\sigma ,\infty )^d\). Analogously, the norm \(\Vert \cdot \Vert _{C^{k,\sigma }_{\varvec{\delta }}(\Omega )}\) is defined. The corresponding function spaces are defined by
where \(\mathcal S:=\{\varvec{c}_j:j\in \mathcal C\}\). The corresponding trace spaces are endowed with the norm
and analogously for \(C^{k,\sigma }_{\varvec{\delta }}(\Gamma )\).
Next, we establish a regularity result for weighted Sobolev spaces.
Lemma 3
Let \(f\in W^{0,2}_{\varvec{\beta }}(\Omega )\) and \(z\in W^{3/2,2}_{\varvec{\beta }}(\Gamma )\) with \(\varvec{\beta }\in [0,1)^d\) satisfying \(\beta _j > 1-\lambda _j\) for all \(j\in \mathcal C\). Then, the solution u of (3) belongs to \(W^{2,2}_{\varvec{\beta }}(\Omega )\). In case of \(z\in V^{3/2,2}_{\varvec{\beta }}(\Gamma )\) the function u belongs to \(V^{2,2}_{\varvec{\beta }}(\Omega )\).
Proof
The regularity result for V-spaces can be deduced from [18, Theorem 1.4.3]. Note that this result holds even for a larger range of the weights, this is, \(\beta _j\in (1-\lambda _j,1+\lambda _j)\). From this result we infer the solvability in W-spaces as each function \(y\in W^{2,2}_{\beta _j}(\Omega _R^j)\) with \(\beta _j\in (0,1)\) can be decomposed into \(u_0 + p\) with a constant \(p=z(\varvec{c}_j)\) and \(u_0\in V^{2,2}_{\beta _j}(\Omega _R^j)\). This is basically the idea which leads to [26, Theorem 4.§5.11] from which we could conclude the same result.\(\square \)
An analogue of this result is true for the weighted Hölder spaces introduced above. This is used to show boundedness of the solution of (3) in a weighted \(W^{2,\infty }\)-space.
Lemma 4
Assume that \(f\in \Lambda ^{0,\sigma }_{\varvec{\delta }}(\Omega )\) and \(z\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Gamma )\) with \(\sigma \in (0,1]\) and weights \(\varvec{\delta }\in (\sigma ,2+\sigma )^d\) satisfying \(2-\lambda _j>\delta _j-\sigma \) for \(j\in \mathcal C\). Moreover, we exclude the case \(\delta _j-\sigma =1\). Then, the solution u of (3) belongs to \(\Lambda ^{2,\sigma }_{\varvec{\delta }}(\Omega )\) and depends continuously on the input data. This result remains true when replacing \(\Lambda \) by C.
Proof
The proof for the regularity in \(\Lambda \)-spaces can be deduced from [18, Theorem 1.4.5]. In order to show the regularity result in the weighted C-spaces we basically follow the ideas used in [28, Lemma 3.13]. First, introduce the numbers \(\nu _j\in \mathbb Z\), \(j\in \mathcal C\), such that \(\nu _j< \delta _j-\sigma < \nu _j+1\). Then, we split the solution into
with smooth cut-off functions \(\eta _j\) satisfying \(\eta _j\equiv 1\) in \(\Omega _{R/2}^j\) and \({{\,\mathrm{supp}\,}}{\eta _j}\subset \overline{\Omega }_R^j\) for all \(j\in \mathcal C\), and polynomials \(p_j\) of order not greater than \(1-\nu _j\). The key idea is to reuse the regularity results shown in the weighted \(\Lambda \)-spaces for the function w. A direct application of these results to u would not be possible as the homogeneous weights \(r_j(x)^{\delta -k-\sigma +|\alpha |}\), \(j\in \mathcal C\), could, depending on the choice of \(\delta \) and \(\sigma \), tend to infinity as \(r_j\rightarrow 0\). As a remedy, the polynomials \(p_j\) have to be chosen in such a way that \(w(c_j)=0\) if \(\nu _j=1\) and \(w(c_j)=D^\alpha w(c_j)=0\) for all \(|\alpha |=1\) if \(\nu _j=0\).
Once regularity for w in a weighted \(\Lambda \)-space is shown, the desired result follows from certain relations between C- and \(\Lambda \)-spaces. First, we observe that w solves the boundary value problem
Our aim is to show that w belongs to \(\Lambda ^{2,\sigma }_{\varvec{\delta }}(\Omega )\) which would follow under the assumption \(F\in \Lambda ^{0,\sigma }_{\varvec{\delta }}(\Omega )\) and \(G\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Gamma )\). To achieve this, we have to construct the polynomials \(p_j\) appropriately and therefore, we define the projection
for \(j\in \mathcal C\) and \(k\in \mathbb N_0\). In a similar way we construct a projection for functions defined on the boundary by means of \(q_k^\partial (v,\varvec{c}_j):=\gamma _0 q_k(\tilde{v};\varvec{c}_j)\), where \(\tilde{v}\) is an arbitrary extension of v and \(\gamma _0\) is the trace operator. The polynomial \(q_k^\partial (v;\varvec{c}_j)\) is independent of the extension \(\tilde{v}\), and hence, there holds \(\gamma _0 q_k(u; \varvec{c}_j) = q_k^\partial (z;\varvec{c}_j)\) as \(u\equiv z\) on \(\Gamma \). In the following we use the choice \(p_j := q_{1-\nu _j}(u;\varvec{c}_j)\). That F belongs to \(\Lambda ^{0,\sigma }_{\varvec{\delta }}(\Omega )\) is obvious, as f is assumed to be contained in \(C^{0,\sigma }_{\varvec{\delta }}(\Omega )\) and this space is equivalent to \(\Lambda ^{0,\sigma }_{\varvec{\delta }}(\Omega )\) if \(\varvec{\delta }\ge \sigma \), see the arguments before Lemma 6.7.1 in [22]. Moreover, the cut-off functions \(\eta _j\) are constant in the neighborhood of the corners and thus, the products \(\nabla \eta _j\cdot \nabla p_j\) and \(\Delta \eta _j p_j\) belong trivially to that space. Consequently, we get
With the definition (8) and the imposed Dirichlet boundary conditions, taking into account \(p_j|_\Gamma = q_{1-\nu _j}^\partial (z; \varvec{c}_j)\), \(j\in \mathcal C\), we deduce
for \(\delta _j - \sigma < 1+\nu _j\) and sufficiently small \(\varepsilon >0\), and \(\tilde{z}\) is a suitable extension of z, see (7). The second and third step follow from the equivalence of \(C^{k,\sigma }_{0}(\Gamma _R^j)\) and \(C^{k,\sigma }(\Gamma _R^j)\) stated in [22, Lemma 6.7.2] and an embedding theorem for weighted Hölder spaces, see [22, Lemma 6.7.1]. The embedding used in the second to the last step is trivial.
The property \(G\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Gamma )\) follows from [22, Theorem 6.7.6] which provides the a priori estimate
The regularity result proved in [18, Theorem 1.4.5(2)] then guarantees \(w\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Omega )\), and with the triangle inequality, the trivial estimate \(\Vert v\Vert _{C^{2,\sigma }_{\varvec{\delta }}(\Omega )} \le c\, \Vert v\Vert _{\Lambda ^{2,\sigma }_{\varvec{\delta }}(\Omega )}\) for \(v\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Omega )\) and (10) we infer
An a priori estimate for the weighted \(\Lambda \)-norm of w can be concluded from [18, Theorem 1.4.5(1)], which leads to
where the second step follows from the estimates (9) and (11). The \(L^1(\Omega )\) norm of u can be bounded by the \(V^{2,2}_{\varvec{\beta }}(\Omega )\)-norm with weights \(\beta _j = \max \{0,\delta _j-\sigma -1\}+\varepsilon \), \(j\in \mathcal C\), and \(\varepsilon >0\) sufficiently small. Using the norm equivalence from [26, Theorem 5.6] or [22, Lemma 6.2.12] we arrive at
With Lemma 3 and embeddings of W- into C-spaces, see e. g. [28, Lemma 2.39], we deduce
The desired a priori estimate follows after insertion of (13) and (14) into (12).\(\square \)
The regularity results in weighted Hölder spaces allow us to extend the assertion of Lemma 3 to \(L^\infty \) based norms. This is a simple conclusion from the definition of the spaces V and \(\Lambda \) as well as W and C.
Corollary 1
Assume that \(\varvec{\delta }\in (\sigma ,2+\sigma )^d\) satisfies the assumptions of Lemma 4. Let \(\varvec{\gamma }\in (0,2)^d\) be a weight vector defined by \(\gamma _j:=\delta _j-\sigma \) for \(j\in \mathcal C\).
- (i)
If \(f\in \Lambda ^{0,\sigma }_{\varvec{\delta }}(\Omega )\) and \(z\in \Lambda ^{2,\sigma }_{\varvec{\delta }}(\Gamma )\), the solution u of (3) belongs to \(V^{2,\infty }_{\varvec{\gamma }}(\Omega )\).
- (ii)
If \(f\in C^{0,\sigma }_{\varvec{\delta }}(\Omega )\) and \(z\in C^{2,\sigma }_{\varvec{\delta }}(\Gamma )\), the solution u of (3) belongs to \(W^{2,\infty }_{\varvec{\gamma }}(\Omega )\).
3 Error estimates for normal derivatives
In this section we consider a finite element discretization for the weak form of the boundary value problem (3) which reads
Therefore, let \(\{\mathcal T_h\}_{h>0}\) be a quasi-uniform family of shape-regular triangulations of \(\Omega \), which are feasible in the sense of [10, Section 5]. The parameter h denotes the maximal diameter of all elements from \(\mathcal T_h\) and is always assumed to be sufficiently small. The trial and test spaces are defined by
Moreover, the traces of function from \(V_h\) belong to the space
The finite-element approximation \(u_h\in V_h\) of u is defined by
where \(z_h\in V_h^\partial \) is some appropriate interpolation or projection of z. In the following, \(z_h\) will be the \(L^2(\Gamma )\)-projection of z onto \(V_h^\partial \), this is, \(z_h:=Q_h(z)\). Moreover, we denote by
the nodal interpolant. Here, \(x_i\in \overline{\Omega }\), \(i=1,\ldots ,N\), denote the nodes of \(\mathcal T_h\) and \(\{\varphi _i\}_{i=1}^N\) the nodal basis of \(V_h\). Moreover, we will use a slightly modified interpolant defined by
where \(\tilde{S}_h:V_h^\partial \rightarrow V_h\) is the zero extension which vanishes in the interior nodes of \(\mathcal T_h\). For functions \(u\in C(\overline{\Omega })\) with \(u|_\Gamma = z\), the interpolant fulfills the essential property \([\tilde{I}_h u]|_\Gamma \equiv Q_h z\) that is needed for instance in the proof of a Céa-Lemma. As the local interpolation error estimates will frequently depend on the distance to the corners, we introduce the notation
We start our investigations with an interpolation error estimate for the boundary datum z.
Lemma 5
Let be given weight vectors \(\varvec{\alpha }\in [0,1/2)^d\) and \(\varvec{\gamma }\in [0,3/2)^d\). Then, the interpolation error estimates
are valid, provided that z possesses the regularity demanded by the right-hand side.
Proof
The first estimate can be deduced from [32, Lemma 3.2.4]. There, the desired error estimate in the \(L^2(\Gamma )\)- and \(H^1(\Gamma )\)-norm is proved. The estimate in \(H^{1/2}(\Gamma )\) follows from an interpolation argument.
To show the second estimate we will reuse existing interpolation error estimates exploiting regularity in weighted V-spaces. To this end, we split up the function z by means of \(z=z_0+\eta _j\, p_j\) with \(z_0\in V^{3/2,2}_{\varvec{\alpha }}(\Gamma )\), certain constants \(p_j\in \mathbb R\), \(j\in \mathcal C\), and smooth cut-off functions \(\eta _j=\eta _j(|x-\varvec{c}_j|)\) satisfying
Note that the nodal interpolant preserves the functions \(\eta _j\, p_j\) near the corners provided that \(h>0\) is sufficiently small. Hence, it suffices to prove an estimate for \(z_0\). In order to derive local interpolation error estimates we denote by \(\hat{E}:=(0,1)\) the reference interval, and by \(F_E:\hat{E}\rightarrow E\) the affine reference transformation. Moreover, we write \(\hat{v}(\hat{x}) = v(F_E(\hat{x}))\) for all \(\hat{x}\in \hat{E}\). The norms of the weighted Sobolev spaces on the reference element, \(V^{k-1/2,2}_{\alpha }(\hat{E})\), are defined analogous to the global norms introduced in Sect. 2.2 but the weight function is defined by \(\hat{r}(\hat{x}):=|\hat{x}|\). For elements \(E\in \mathcal E_h\) touching the corner \(\varvec{c}_j\), \(j\in \mathcal C\), there holds the property \(r_j(F_E(\hat{x})) \sim h\,\hat{r}(\hat{x})\).
For all elements \(E\in \mathcal E_h\) with \(r_{j,E}=0\) for some \(j\in \mathcal C\), we obtain the estimate
which follows from the arguments used in the proof of [4, Lemma 4.5]. In case of \(E\subset \Omega _{R/2}^j\) for some \(j\in \mathcal C\) and \(r_{j,E} > 0\) we deduce
where the argument used in the last step can also be found in [4, Lemma 4.5]. Far away from the corners, i. e., \(r_{j,E} > 1/4\) for all \(j\in \mathcal C\), we can use a standard estimate to get \(\Vert z_0-I_h z_0\Vert _{L^2(E)} \le c\, h^{3/2}\,|z_0|_{H^{3/2}(E)}\). Next, we sum up over all elements \(E\in \mathcal E_h\) (note that for Sobolev-Slobodeckij-norms there holds \(\sum _{E\in \mathcal E_h}|u|_{H^{3/2}(E)}^2\le C\,|u|_{H^{3/2}(\Gamma )}^2\), but not the reverse estimate) and together with a standard estimate for the error terms \(p_j\,\eta _j - I_h(p_j\,\eta _j)\) we obtain
The last step is a consequence of the norm equivalence stated in [26, Ch. 4, Theorem 5.7].
The estimate in the \(H^{1/2}(\Gamma )\)-norm follows from an interpolation argument between estimates in \(L^2(\Gamma )\) and \(H^1(\Gamma )\). Note that \(H^{1/2}(\Gamma )\) is equivalent to the interpolation space \([L^2(\Gamma ),H^1(\Gamma )]_{1/2,2}\), see [23, Theorem B.11]. The space \(H^1(\Gamma )\) can be defined in the usual way via local Lipschitz continuous parametrizations of \(\Gamma \). To show an estimate in \(H^1(\Gamma )\) we derive local estimates first. For all elements \(E\in \mathcal E_h\) with \(r_{j,E} = 0\) we obtain
where the second step is an application of the embedding \(V^{3/2,2}_{\alpha _j}(\hat{E})\hookrightarrow H^1(\hat{E})\), which is valid for \(\alpha _j < 1/2\). Otherwise, if \(E\subset \Omega _{R/2}^j\) and \(r_{j,E} > 0\), we obtain with similar arguments
In the far interior, i. e., for \(E\in \mathcal E_h\) with \(r_{j,E}> 1/4\) for all \(j\in \mathcal C\), we can use a standard estimate exploiting \(H^{3/2}\)-regularity. Summation over all elements \(E\in \mathcal E_h\) and an interpolation argument lead to the desired estimate for \(z_0-I_h z_0\) in the \(H^{1/2}(\Gamma )\)-norm. With the splitting \(z = z_0+\sum _{j\in \mathcal C} p_j\, \eta _j\) we get an estimate for \(z-I_h z\) when using the arguments from (18).\(\square \)
Using the ideas of [7] we can derive error estimates for the approximate solutions \(u_h\) in the norms \(H^1(\Omega )\) and \(L^2(\Omega )\) . However, in this reference \(H^2(\Gamma _j)\)-regularity (\(j\in \mathcal C\)) for the Dirichlet datum z is assumed. As we deal with optimal Dirichlet control problems, the boundary datum for the state is the control function which might be less regular. Thus, we repeat the proof assuming less regularity for z in some weighted Sobolev space.
Lemma 6
Assume that \(u\in W^{2,2}_{\varvec{\alpha }}(\Omega )\) and \(z\in W^{3/2,2}_{\varvec{\alpha }}(\Gamma )\) with a weight vector \(\varvec{\alpha }\in [0,1/2)^d\). Moreover, let \(z_h:= Q_h z\). Then, the solution \(u_h\in V_h\) of (15) satisfies the error estimates
with some sufficiently small \(\varepsilon (\Omega )\in (0,1/2]\) depending on the opening angles of the corners of \(\Omega \). For convex domains, the choice \(\varepsilon (\Omega )=1/2\) is possible.
Proof
First, we derive the error estimate in \(H^1(\Omega )\). We apply the error equation \((\nabla (u-u_h), \nabla (\tilde{I}_h u - u_h))_{L^2(\Omega )^2} = 0\) and obtain
With a standard interpolation error estimate exploiting weighted regularity, see e. g [28, Lemma 3.31], we get
Note that the zero extension satisfies \(\Vert \nabla \tilde{S}_h \phi _h\Vert _{L^2(\Omega )} \le c\, h^{-1/2}\,\Vert \phi _h\Vert _{L^2(\Gamma )}\), see e. g. [21, Lemma 3.3]. Thus, together with Lemma 5 we obtain an estimate for the second term
In order to derive an estimate in the \(L^2(\Omega )\)-norm we use a duality argument. Let \(w\in H^1_0(\Omega )\) be the weak solution of \(-\Delta w = u-u_h\) in \(\Omega \). With partial integration, the orthogonality of the \(L^2(\Gamma )\)-projection \(Q_h\), the estimate in the \(H^1(\Omega )\)-norm and Lemma 5, we obtain for sufficiently small \(\varepsilon (\Omega )\in (0,1/2]\)
The assertion follows from the a priori estimate
\(\square \)
The aim in the remainder of this section is to derive error estimates for the variational normal derivative of the approximate solution \(u_h\). Motivated by Green’s identity this is defined by
Note that both the left- and right-hand side are zero for test functions from \(V_{0h}\). Hence, in order to compute \(\partial _n^h u_h\), it suffices to test the equation (20) with the nodal basis functions that belong to the boundary nodes.
We start our considerations with an existence and stability result.
Lemma 7
For arbitrary input data \(f\in L^2(\Omega )\), \(z_h\in V_h^\partial \), the variational normal derivative \(\partial _n^h u_h\in V_h^\partial \) defined by (20) exists, is unique, and satisfies the estimate
Proof
In the following \(S_h:V_h^\partial \rightarrow V_h\) is the discrete harmonic extension which satisfies the estimate \(\Vert S_h v_h\Vert _{H^1(\Omega )} \le c\, \Vert v_h\Vert _{H^{1/2}(\Gamma )}\), see [21, Lemma 3.2]. Together with the discrete stability of functions from \(V_h^\partial \) in \(H^{1/2}(\Gamma )\) as well as (15) we obtain
The a priori estimate \(\Vert u_h\Vert _{H^1(\Omega )} \le c\,(\Vert f\Vert _{L^2(\Omega )} + \Vert z_h\Vert _{H^{1/2}(\Gamma )})\) implies the assertion.
\(\square \)
Next, we show an error estimate for the variational normal derivative, for which we exploit the \(W^{2,2}_{\varvec{\alpha }}(\Omega )\)-regularity of the solution. The result of the following theorem is sharp for non-convex domains \(\Omega \), and also for convex domains when the solution is not more regular than \(H^2(\Omega )\) (this happens e. g. when the right-hand side belongs to \(L^2(\Omega )\), but not to \(L^p(\Omega )\) with \(p>2\)). Later, we prove an estimate which promises a higher convergence rate for convex domains, provided that the solution belongs to \(W^{2,\infty }_{\varvec{\beta }}(\Omega )\).
Theorem 1
Let \(\Omega \) be an arbitrary polygonal domain. Moreover, let \(z_h = Q_h z\). Under the assumptions \(u\in W^{2,2}_{\varvec{\alpha }}(\Omega )\) and \(z\in W^{3/2,2}_{\varvec{\alpha }}(\Gamma )\) with \(\varvec{\alpha }\in [0,1/2)^d\), there holds the error estimate
Proof
Using the triangle inequality we split up the norm into an error term for the \(L^2(\Gamma )\)-projection onto \(V_h^\partial \), and a fully discrete term, this is
With a standard duality argument we obtain for the first term
Next, we show a best-approximation error estimate in the \(L^2(\Gamma )\)-norm. To this end, we use the splitting splitting \(u=u_0+p_j\,\eta _j\), see e. g. [26, Theorem 5.6(2)], with a function \(u_0\in V^{2,2}_{\varvec{\alpha }}(\Omega )\), certain constants \(p_j\), \(j\in \mathcal C\), and smooth cut-off functions \(\eta _j=\eta _j(|x-\varvec{c}_j|)\) which are equal to one near \(\varvec{c}_j\) and have support in \(\overline{\Omega }_R^j\). A similar argument has been already used in the proof of Lemma 4. For functions belonging to \(V^{2,2}_{\varvec{\alpha }}(\Omega )\) the estimate
can be found in the proof of Theorem 9 in [29] for some Clément-type interpolation operator \(C_h:L^1(\Gamma )\rightarrow V_h^\partial \). Note that \(\partial _n (p_j\,\eta _j)\) and its interpolant vanish and thus, we easily deduce an estimate for the function \(u\in W^{2,2}_{\varvec{\alpha }}(\Omega )\). Moreover, due to norm equivalences of V- and W-spaces [26, Theorem 5.6(2)], we obtain \(\Vert u_0\Vert _{V^{2,2}_{\varvec{\alpha }}(\Omega )} + \sum _{j\in \mathcal C} |u(\varvec{c}_j)| \sim \Vert u\Vert _{W^{2,2}_{\varvec{\alpha }}(\Omega )}\), which leads together with the previous estimate and (22) to
With the discrete stability used already in (21), the definition of \(\partial _n^h\) from (20), orthogonality of the \(L^2(\Gamma )\)-projection \(Q_h\) and Greens identity, we deduce
where \(S_h:V_h^\partial \rightarrow V_h\) is the discrete harmonic extension. This operator satisfies the estimate \(\Vert \nabla S_h\varphi _h\Vert _{L^2(\Omega )} \le c\, \Vert \varphi _h\Vert _{H^{1/2}(\Gamma )}\). Together with the \(H^1(\Omega )\)-error estimate from Lemma 6 applied to \(\Vert \nabla (u-u_h)\Vert _{L^2(\Omega )}\) we conclude the assertion.\(\square \)
As already mentioned before the previous theorem, we expect a convergence rate higher than one for convex domains, provided that the input data are more regular. The proof of sharp convergence rates in this case is more complicated and we start with some notation required in the following. As in [29] we introduce a dyadic decomposition towards the boundary of \(\Omega \), namely
where \(\rho (x):={{\,\mathrm{dist}\,}}(x,\Gamma )\). We set \(d_J:=2^{-J}\) for \(J=0,\ldots ,I\) and use modifications for the interior domain by \(d_{-1}:={{\,\mathrm{diam}\,}}(\Omega )\), and the outermost domain by \(d_{I+1}:=0\). Note that this forms a complete decomposition of \(\Omega \), i. e.,
In the following we will frequently exploit the following two properties that can be directly concluded from the definition:
The termination index I is chosen such that \(d_I = c_I h\) with some mesh-independent constant \(c_I>1\) specified later. This implies that \(I\sim |\ln h|\). Moreover, we introduce the patches with the adjacent subsets given by
Note that the patches satisfy the properties (26) as well due to \(d_{J+1} \sim d_J\) for \(J=-1,\ldots ,I-1\).
We start the proof of the desired finite element error estimate with some local error estimates for the nodal interpolant defined in (17).
Lemma 8
Assume that \(\Omega \) is convex and \(u\in W^{2,\infty }_{\varvec{\beta }}(\Omega )\), \(z\in W^{2,2}_{\varvec{\gamma }}(\Gamma )\) with \(\varvec{\beta }\in [0,2)^d\), \(\varvec{\gamma }\in [0,3/2)^d\). Then, there holds the estimate
with \(s=1\) if \(\overline{\varvec{\beta }}=1/2\), and \(s=0\) if \(\overline{\varvec{\beta }}\ne 1/2\).
Proof
Throughout the proof we will hide the constant \(c_I\) in the generic constant c as it is not needed for the terms considered here. For elements \(T\in \mathcal T_h\) touching a corner, i. e., \(r_{j,T}=0\) for some \(j\in \mathcal C\), we directly deduce the estimate
which follows from the estimate from [28, Corollary 3.33] and the property \(d_I \sim h\). On that part of \(\Omega _J\) excluding the elements touching a corner we obtain for \(J=-1,\ldots ,I\) with a standard estimate
where \(S_h:=\cup \{T\in \mathcal T_h:r_{j,T} = 0,\ j\in \mathcal C\}\). It remains to bound the term on the right-hand side of (28). Therefore, we bound \(\Vert \nabla ^2 u\Vert _{L^2(\Omega _{\tilde{J}}\setminus S_h)}\) for \(\tilde{J}=\max \{-1,J-1\},\ldots ,\min \{J+1,I\}\) by some weighted \(W^{2,\infty }(\Omega )\)-norm of u. This is done by an application of the Hölder inequality on a further dyadic decomposition of \(\Omega _{\tilde{J}}\) with respect to the corners. A similar technique is used e. g. in [5, 32] where error estimates in \(L^2(\Gamma )\) for the Neumann problem in three-dimensional polyhedral domains are derived. Therein, the domain is decomposed twice into dyadic subsets to resolve both edge and corner singularities. Following these ideas we introduce
and define the subdomains
for \(J=0,\ldots ,I\), \(K=0,\ldots ,J-1\) and \(j\in \mathcal C\). Here \(\tilde{\Gamma }_{j}\) stands for the straight line which coincides with the boundary edge \(\Gamma _j\). Each domain \(\Omega _{J,K}^{j,+}\) is a parallelogram bounded by that parallels to \(\tilde{\Gamma }_j\) having distance \(d_{J+1}\) and \(d_{J}\) from \(\tilde{\Gamma }_j\), and by that parallels to \(\tilde{\Gamma }_{j-1}\) having distance \(d_{J,K}\) and \(d_{J,K+1}\) from \(\tilde{\Gamma }_{j-1}\). In a similar way we define the subdomains \(\Omega _{J,K}^{j,-}\) by simply changing the roles of \(\tilde{\Gamma }_j\) and \(\tilde{\Gamma }_{j-1}\) in the definition (29). Note that \(\Omega _{J,0}^j := \Omega _{J,0}^{j,+} = \Omega _{J,0}^{j,-}\). These subdomains are illustrated in Fig. 1.
Definition of the domains \(\Omega _{J,K}^{j,\pm }\)
By construction we have the property
Moreover, we will exploit the property
for all \(J=0,\ldots ,I\), \(K=0,\ldots ,J-1\) and \(j\in \mathcal C\), which follows directly from the definition of the sets \(\Omega _{J,K}^{\pm ,j}\). This allows us to locally trade the quantities \(d_{J,K}\) by the weights \(r_j(x)\) contained in the weighted Sobolev spaces. The union of the domains introduced in (29) leads to a covering of our initial decomposition (25) near a ball of radius 1 around the corner \(\varvec{c}_j\), \(j\in \mathcal C\), i. e.,
In order to bound the term on the right-hand side of (28), we apply the Hölder inequality on each subset \(\Omega _{\tilde{J},K}^{j,\pm }\) using the property (30), and insert appropriate weights taking (31) into account. This implies
where the last step follows from the limit value of the geometric series and the property \(\tilde{J} \le I \sim |\ln h|\), i. e.,
With the Hölder inequality we obtain a similar estimate on the set \(\Omega _{\tilde{J}}\setminus \cup _{j\in \mathcal C} \Omega _R^j\), this is,
Combining the previous estimates and summing up over the indices \(\tilde{J}=\max \{-1,J-1\},\ldots ,\min \{J+1,I\}\) finally yields together with (27)
In case of \(J=I\), we still have to discuss the boundary terms to obtain an estimate for \(\tilde{I}_h\). This follows from
and the estimate derived in Lemma 5.\(\square \)
As an intermediate result we prove a weighted \(L^2(\Omega )\)-error estimate. The weight function we use is defined by
Note that such a weight function has been discussed already in Sect. 2.1. The regularizer in the present situation is the width of the outermost subset \(\Omega _I\). Here, the relation between the weight function \(\sigma \) and the dyadic decomposition (25) becomes clear, as the definition directly implies
Lemma 9
Assume that \(\Omega \subset \mathbb R^2\) is a convex polygonal domain. Let \(u\in W^{2,\infty }_{\varvec{\beta }}(\Omega )\) and \(z\in W^{2,2}_{\varvec{\gamma }}(\Gamma )\) with \(\varvec{\beta }\in [0,2)^d\) and \(\varvec{\gamma }\in [0,3/2)^d\). Moreover, let \(z_h:=Q_h z\). Then the solutions of (15) fulfill the error estimate
provided that \(c_I\) is sufficiently large.
Proof
We follow the arguments of the Nitsche trick using the slightly modified dual problem
with \(\psi = \sigma ^{-2}\,(u-u_h) / \Vert \sigma ^{-2}\,(u-u_h)\Vert _{L^2(\Omega )}\). Note that \(\Vert \psi \Vert _{L^2(\Omega )} = 1\). With partial integration and the Galerkin orthogonality we conclude
First, we consider the second term on the right-hand side of (38). With the orthogonality of the \(L^2(\Gamma )\)-projection we obtain
where the second step follows from Lemma 5 and the estimate \(\Vert \partial _n w\Vert _{H^{1/2}(\Gamma )} \le c\, \Vert w\Vert _{H^2(\Omega )} \le c\, \Vert \sigma ^{-2}\,\psi \Vert _{L^2(\Gamma )} \le c\, h^{-2}\) which is a consequence of a trace theorem, an a priori estimate, and \(\sigma (x) \ge d_I \sim h\) for all \(x\in \Omega \).
Next, we discuss the first term on the right-hand side of (38). A subset-wise application of the Cauchy-Schwarz inequality with respect to the dyadic decomposition (25) yields
Moreover, with the local finite-element error estimates from [11] we obtain
for all \(J=-1,\ldots ,I\). Note that this estimate would not hold for \(I_h\) as the boundary traces of \(u_h\) and the used interpolant must coincide.
Next, we insert (41) into (40) and discuss the resulting terms separately. First, consider the product of the interpolation terms. For the interpolation error of the dual solution we apply a standard estimate and Lemma 2 in case of \(J=-1,\ldots ,I-2\). As we can locally trade \(\sigma \) by \(d_J\), see (35), we obtain
In case of \(J=I-1,I\) we use a global a priori estimate to arrive at
where the last step follows from the property \(\sigma (x) \ge d_I\) for \(x\in \Omega \). Together with the interpolation error estimates from Lemma 8 and the discrete Cauchy-Schwarz inequality we obtain
The last step follows from the limit value of the geometric series. Analogous to (33) this can be calculated by means of
with \(t=2\min \{-1/2,-\overline{\varvec{\beta }}\}\). Moreover, we exploited the property \(d_I\sim h\) and the estimates from Lemma 1 taking into account \(\Vert \sigma ^{-1}\,\psi \Vert \le c\,d_I^{-1}\le c\, h^{-1}\). Note that the constant \(c_I\) vanishes in c as it is not needed here.
Next, we discuss the product of the pollution term for the primal problem from (41) and the interpolation error for the dual problem. With similar arguments as in (44) we get
and in the last step we applied Lemma 1 and \(\Vert \sigma ^{-1}\,\psi \Vert _{L^2(\Omega )} \le c\, d_I^{-1} = c\, c_I^{-1}\, h^{-1}\). Finally, insertion of (44) and (46) into (40) and the resulting estimate together with (39) into (38) leads to
The last term on the right-hand side can be neglected when \(c_I\) is chosen sufficiently large such that \(c c_I^{-1} \le 1/2\). This implies the assertion.\(\square \)
Now, we are in the position to show an improved convergence rate for the variational normal derivative in case of convex domains.
Theorem 2
Let \(z_h:=Q_h z\). Assume that \(\Omega \) is a convex polygonal domain. Let \(u\in H^2(\Omega )\cap W^{2,\infty }_{\varvec{\beta }}(\Omega )\) and \(z\in W^{2,2}_{\varvec{\gamma }}(\Gamma )\) with \(\varvec{\beta }\in [0,1)^d\), \(\varvec{\gamma }\in [0,3/2)^d\). Moreover, it is assumed that \(\partial _n u\) is continuous in the corners of \(\Omega \). Then, there holds the error estimate
with \(s=1\) if \(\overline{\varvec{\beta }}=1/2\) and \(s=0\) otherwise.
Proof
The beginning of the proof is analogous to the proof of Theorem 1. First, we derive an interpolation error estimates for some interpolant of \(\partial _n u\) in the \(L^2(\Gamma )\)-norm. Therefore, we use the a Clément-type interpolant \(C_h:C(\Gamma )\rightarrow V_h^\partial \) with a slight modification in the nodes located in a corner of \(\Omega \). In the following \(\{x_i\}_{i=1}^{N_{\text {bd}}}\) are the nodes of \(\mathcal E_h\), and \(\{\varphi _i\}_{i=1}^{N_{\text {bd}}}\) are the nodal basis functions of \(\mathcal E_h\). Each basis function is the boundary trace of a nodal basis function of \(V_h\) (the 2D “hat functions”). The precise definition of \(C_h\) is given by
where \(\sigma _i :=\cup \{E\in \mathcal E_h:x_i\in \overline{E}\}\) if \(x_i\not \in \{\varvec{c}_j,\ j\in \mathcal C\}\). For the nodes \(x_i\) located in the vertices of \(\Omega \) we simply set \(\sigma _i:=\emptyset \). For some \(E\in \mathcal E_h\) we denote by T the corresponding triangle from \(\mathcal T_h\), this is, \(E\subset \bar{T}\), and by \(F_T:\hat{T}\rightarrow T\) the affine mappings from the reference triangle \(\hat{T}:=\text {conv}\{(0,0),\, (1,0),\,(0,1)\}\) to the world element T. Moreover, we will use the notation \(\hat{v}(\hat{x}) := v(F_T(\hat{x}))\). In addition, we introduce the patches \(S_E:=\cup \{\sigma _i:x_i\in \bar{E}\}\) and \(D_E:=\cup \{T\in \mathcal T_h:\bar{T}\cap \bar{E}\ne \emptyset \}\), as well as the corresponding reference patches \(S_{\hat{E}} := F_T^{-1}(S_E)\) and \(D_{\hat{E}} := F_T^{-1}(D_E)\).
First, we easily see that the interpolant satisfies the stability estimate
for an arbitrary function \(v\in L^\infty (E)\). For elements \(E\in \mathcal E_h\) touching the corner \(\varvec{c}_j\), \(j\in \mathcal C\), we insert an arbitrary first-order polynomial p and infer with the triangle inequality and the stability estimate for \(C_h\)
We proceed with the embedding \(W^{2,2+\varepsilon }(D_{\hat{E}}) \hookrightarrow W^{1,\infty }(D_{\hat{E}})\), the Bramble-Hilbert Lemma, as well as the embedding \(W^{0,\infty }_{\beta _j}(D_{\hat{E}}) \hookrightarrow L^{2+\varepsilon }(D_{\hat{E}})\), which holds for all \(\beta _j < 1\), provided that \(\varepsilon >0\) is sufficiently small. The weighted Sobolev spaces in the reference setting are defined analogous to the spaces defined in Sect. 2.2 with the exception that the weight function is defined by \(\hat{r}:=|\hat{x}|\). When assuming w.l.o.g that \(F_T(0) = \varvec{c}_j\) we obtain the property \(\hat{r}(\hat{x}) \sim r_j(F_T(\hat{x}))\, h^{-1}\). A transformation of variables then yields
For elements \(E\in \mathcal E_h\) away from the corners we apply similar arguments, but use instead the stability estimate \(\Vert C_h v\Vert _{L^2(E)} \le c\, \Vert v\Vert _{L^2(S_E)}\), to arrive at
Summation over all boundary elements \(E\in \mathcal E_h\) with \(E\not \subset S_h:=\cup \{E\in \mathcal E_h:r_{j,E} = 0,\ j\in \mathcal C\}\) yields
where the last step is an application of the estimate (34) with \(J=I\) taking into account \(d_I\sim h\). Together with the estimates (22) and (48) we deduce
The fully discrete part \(Q_h(\partial _nu) - \partial _n^h u_h\) is treated as in (23) with the only difference that we use instead of \(S_h\varphi _h\) the extension \(I_h S\varphi _h\) (note that the choice of the discrete extension in (23) is arbitrary). Here, \(S:H^{1/2}(\Gamma )\rightarrow H^1(\Omega )\) denotes the harmonic extension, i. e., \([S\varphi ]|_\Gamma \equiv \varphi \) and \((\nabla S\varphi ,\nabla v)_{L^2(\Omega )^2} = 0\) for all \(v\in H_0^1(\Omega )\). We have to derive an upper bound for \((\nabla (u-u_h),\nabla I_h (S \varphi _h))_{L^2(\Omega )^2}\) which must depend linearly on \(\Vert \varphi \Vert _{H^{1/2}(\Gamma )}\). Introducing \(S\varphi _h\) as intermediate function yields
The latter term is the simpler one. With integration by parts and the trace theorem for normal derivatives from [25, Theorem 1.3.2] we obtain
In the last step we used the fact that the \(L^2(\Gamma )\)-projection is stable in \(H^{1/2}(\Gamma )\), see [31], and fulfills thus a best-approximation property in the \(H^{1/2}(\Gamma )\)-norm. The desired estimate then follows from Lemma 5.
The first term on the right-hand side of (49) has the structure of the term (40) from the proof of Lemma 9. The only difference is that the dual solution w used in that lemma, has to be replaced by the function \(S\varphi _h\). To this end, we first confirm the estimate
provided that \(\varepsilon \in (0,1/2)\). With the same arguments as in (44) and (46), taking into account (42) with w and \(\psi \) replaced by \(S\varphi _h\) and 0 for \(J=-1,\ldots ,I-2\) and (51) for \(J=I-1,I\), as well as the interpolation error estimates from Lemma 8 and the property \(d_I\sim h\), we obtain
The last term in the parentheses on the right-hand side is discussed in Lemma 9 already and we can bound this term by the first two ones.
Insertion of the previous estimate, (50) and (49) into (23) and canceling out the terms \(\Vert \varphi _h\Vert _{H^{1/2}(\Gamma )}\) leads to the desired estimate for the term \(\Vert Q_h(\partial _n u) - \partial _n^h u_h\Vert _{H^{-1/2}(\Gamma )}\).\(\square \)
Remark 1
The best possible convergence rate of 3 / 2 is achieved when \(z\in H^2(\Gamma )\) and \(u\in W^{2,\infty }_{\varvec{\beta }}(\Omega )\) with \(\beta _j < 1/2\) for all \(j\in \mathcal C\). In general, the latter assumption is only satisfied when the opening angles of the corners of \(\Omega \) satisfy \(\omega _j < 2\pi /3\), \(j\in \mathcal C\), and when f is sufficiently smooth. As an example, assuming f to be Hölder continuous would be sufficient, compare Corollary 1. Otherwise, for angles larger than \(2\pi /3\) we find a relation between the convergence rate and the exponent of the dominating singularity \(\bar{\lambda }= \pi /\omega _{\text {max}}\) by choosing \(\bar{\beta }= 2-\bar{\lambda }+ \varepsilon \) if \(\omega _{\text {max}}\in (2\pi /3,\pi )\) and \(\bar{\alpha }= 1-\bar{\lambda }+\varepsilon \) if \(\omega _{\text {max}}\in (\pi ,2\pi )\) for arbitrary but sufficiently small \(\varepsilon > 0\). Under the assumption that f and z are sufficiently smooth we then infer
4 Dirichlet control problems
This section is devoted to the numerical approximation of the optimal control problem
subject to the constraints
Here, \(f, u_d\in L^2(\Omega )\) are given functions and \(\nu >0\) is a regularization parameter. The operator \(N:H^{1/2}(\Gamma )\rightarrow H^{-1/2}(\Gamma )\) is a Steklov–Poincaré operator which is used to realize an \(H^{1/2}(\Gamma )\)-seminorm.
We introduce the linear operators \(S:H^{1/2}(\Gamma )\rightarrow H^1(\Omega )\) and \(P:H^{1}(\Omega )^*\rightarrow H_0^1(\Omega )\) defined by
We can express the operator N by means of \(Nz:=\partial _n (Sz)\). Note that the regularization term is equivalent to the square of the \(H^{1/2}(\Gamma )\)-seminorm of z.
Necessary optimality conditions, that are also sufficient due to the convexity of this optimization problem, can be found in [27]. Therein, it is shown that the pair \((u,z)\in H^1(\Omega )\times H^{1/2}(\Gamma )\) is the unique global minimizer of (52)–(53) if and only if an adjoint state \(p\in H^1(\Omega )\) exists such that the optimality system
is fulfilled. One can reformulate the optimality system using the operators S and P introduced above. Taking also into account the relation \(S^* u=\partial _n Pu \) leads to a compact form of the optimality system
Eliminating u leads to the variational problem
with
where \(u_f:=Pf\). The existence of a unique solution z of (55) follows from the Lax-Milgram Lemma. Note that the operator T is coercive due to
It remains to discuss the regularity of the optimal solution and the corresponding state and adjoint state. These results will be needed for sharp discretization error estimates.
Lemma 10
Assume that \(f,u_d\in L^2(\Omega )\). Let \(\varvec{\alpha }\in [0,1)^d\) be a weight vector satisfying \(1-\lambda _j< \alpha _j\), \(j\in \mathcal C\). Then, the solution of (54) possesses the regularity
Moreover, if \(u_d\in C^{0,\sigma }(\overline{\Omega })\) for some \(\sigma \in (0,1)\), there holds
with \(\varvec{\beta }\in [0,2)^d\), \(\varvec{\gamma }\in [0,3/2)^d\) satisfying \(2-\lambda _j < \beta _j\) and \(3/2-\lambda _j<\gamma _j\) for \(j\in \mathcal C\).
Proof
In order to transfer the regularity of the adjoint state to the state, we introduce the auxiliary function \(u_0\) solving the boundary value problem
Note that the function \(u_0\) can be determined uniquely as the optimal state satisfies \(\int _\Omega u = \int _\Omega u_d\), see e. g. [16, Section 3.2.3]. With (54) it is easy to confirm that the state can be decomposed by means of \(Sz = u_0 - \frac{1}{\nu }\,p\).
The assertion then follows from bootstrapping arguments. Standard regularity results, and in particular [13, Theorem 4.4.3.7], immediately imply
for arbitrary \(q\in [1,\infty )\) satisfying \(2/q > 2-\lambda _j\) for all \(j\in \mathcal C\). The regularity results collected in (56) then directly follow from Lemma 3.
From embedding theorems we moreover conclude that \(u, p\in C^{0,\sigma '}(\overline{\Omega })\) for some \(\sigma ' \in (0,\min \{1,\bar{\lambda }\})\), and with Corollary 1 we directly infer (57). The assertion \(z\in W^{2,2}_{\varvec{\gamma }}(\Gamma )\) follows from the \(W^{2,\infty }_{\varvec{\beta }}(\Gamma )\)-regularity due to the Hölder inequality.\(\square \)
In order to discretize the optimality system we replace S and P by the finite element solution operators \(S_h:V_h^\partial \rightarrow V_h\) and \(P_h:H^1(\Omega )^*\rightarrow V_h^\partial \) defined by
Instead of \(S^*\) we use its discrete version \(S_h^*:=\partial _n^h P_h\) which is the adjoint operator to \(S_h\). Then, we seek a state \(u_h\in V_h\) and a control \(z_h\in V_h^\partial \) as solution of the finite-dimensional optimization problem
subject to
In order to define an appropriate discrete Steklov–Poincaré operator we use the variational normal derivative introduced in (20), and define \(N_h:V_h^\partial \rightarrow V_h^\partial \) by \(N_h z_h := \partial _n^h (S_h z_h)\). Note that by this definition, the functional \(\left<N_h\cdot , \cdot \right>\) induces a mesh-independent \(H^{1/2}(\Gamma )\)-seminorm for functions in \(V_h^\partial \). Analogous to the continuous case we can derive the discrete optimality system
with an adjoint state \(p_h\in V_{0h}\). This system can be rewritten by means of
with \(T_h = S_h^*S_h + \nu \, N_h\), \(g_h:=S_h^*(u_d - u_{f,h})\) and \(u_{f,h}:=P_h f\).
The remainder of this section is devoted to the proof of error estimates for the finite-element approximation \((u_h,z_h,p_h)\). To this end, we introduce an auxiliary function \(\tilde{z}_h\in V_h^\partial \) solving the variational formulation
The Lax-Milgram-Lemma guarantees the existence and uniqueness of the solution \(\tilde{z}_h\) of (62). Moreover, by the Céa-Lemma and the interpolation error estimates from Lemma 5 we obtain the following intermediate result:
Lemma 11
Let \(z\in H^{1/2}(\Gamma )\) be the optimal control solving (52)–(53). The approximate solutions \(\tilde{z}_h\) of (62) satisfy the estimate
The weights \(\varvec{\alpha }\) and \(\varvec{\gamma }\) are chosen as in Lemma 10.
It remains to derive an estimate for the error between the continuous and the discrete control z and \(z_h\), respectively. In the following Lemma we present a general estimate. The idea of the proof is taken from [27].
Lemma 12
The solutions z and \(z_h\) of (55) and (61), respectively, satisfy the general error estimate
with \(u_h(Q_h z) = S_h(Q_h z) + u_{f,h}\) and \(p_h(u) = P_h(u-u_d)\).
Proof
First, we confirm that the bilinear form \(\left<T_h \cdot , \cdot \right>\) is \(V_h^\partial \)-elliptic and continuous, this is, for all \(v_h,w_h\in V_h^\partial \) there holds
with some constant \(\gamma >0\) independent of h. This follows directly from the mapping properties of \(N_h\), \(S_h\) and \(S_h^*\) as well as Lemma 7. In the following we write \(w_h:= z_h - \tilde{z}_h\). With the ellipticity, the equations (61) and (62) and Young’s inequality, we obtain
Insertion of the definitions of g and \(g_h\) yields
Rearrangement of the remaining terms and the definitions of T and \(T_h\) lead to
Next, we insert (64) and (65) into (63), apply the triangle inequality, and use the abbreviations
as well as their discrete counterparts
Insertion of (64) and (65) into (63), and exploiting the stability estimates
that can be concluded from Lemma 7, as well as the stability of \(Q_h\) in \(H^{1/2}(\Gamma )\) [31], leads to the estimate
With the triangle inequality \(\Vert z-z_h\Vert _{H^{1/2}(\Gamma )} \le \Vert z-\tilde{z}_h\Vert _{H^{1/2}(\Gamma )} + \Vert w_h\Vert _{H^{1/2}(\Gamma )}\) we conclude the assertion.\(\square \)
This general estimate and the estimates presented in Lemma 6, Theorems 1, 2 and Lemma 11 lead to the main result of this section.
Theorem 3
Let \(\Omega \subset \mathbb R^2\) be an arbitrary polygonal domain and assume that \(f, u_d\in L^2(\Omega )\). Let (u, z, p) be the solution of (54), and \((u_h,z_h,p_h)\) the corresponding finite element approximation solving (60). Then, the error estimate
is valid for arbitrary \(\varepsilon >0\).
Furthermore, if \(\Omega \) is convex and \(u_d\in C^{0,\sigma }(\overline{\Omega })\) for some \(\sigma \in (0,1)\), there holds the estimate
Note that \(\bar{\lambda }:= \pi /\max _{j\in \mathcal C} \omega _j\).
The constant c depends linearly on the functions z, Sz, Pf and p, more precisely,
The weights are defined by \(\alpha _j:=\max \{0,1-\lambda _j+\varepsilon \}\), \(\beta _j:=\max \{0,2-\lambda _j+\varepsilon \}\) and \(\gamma _j:=\max \{0,3/2-\lambda _j+\varepsilon \}\) for all \(j\in \mathcal C\).
Proof
The terms involving the approximation of the normal derivative require an application of Theorem 1 or Theorem 2. In the latter theorem continuity of the normal derivatives in the corners is assumed. This is trivially fulfilled for \(\partial _n p\) as \(p=0\) on \(\Gamma \) and due to the optimality condition \(Nz = -\frac{1}{\nu }\partial _n p\) this property is transferred to \(\partial _n (Sz) = Nz\).\(\square \)
As a simple conclusion we also obtain an error estimate for the state variable in the energy norm.
Corollary 2
Assume that \(f,\,u_d\in L^2(\Omega )\). Let \(u\in H^1(\Omega )\) and \(u_h\in V_h\) be the optimal states of (52)–(53) and (58)–(59), respectively. Then, the error estimate
holds for arbitrary but sufficiently small \(\varepsilon >0\). The constant \(c>0\) is the same as in the previous theorem.
Proof
With the triangle inequality we get
Note that \(S_h Q_h z + u_{f,h}\) is the finite element approximation of \(u:=Sz + u_f\). Thus, we infer with Lemma 6
Moreover, with stability properties of \(S_h\) and \(Q_h\) we get
and with (66) we conclude the assertion.\(\square \)
5 Numerical experiments
In order to confirm the theoretically predicted convergence results we present some numerical experiments measuring the convergence rates. Thus, we computed the problem (52)–(53) in the domains
with input data \(\nu = 1\), \(f\equiv 0\) and \(u_d(x_1,x_2) = x_1+x_2\).
We start with a structured grid consisting of 2, 3 or 6 triangles, respectively, and compute the discrete solutions solving (60) on a sequence of meshes obtained by bisection of each element so that new nodes of the grid are inserted at the midpoints of the longest edge of each element. The solution was computed by a GMRES method applied to the system (61) and in each iteration the linearized state and adjoint equation have to be solved. This was done by the parallel direct solver MUMPS which allows to reuse the factorization of the stiffness matrix. The implementation is written in C++ and the tests were performed on a Intel-Core-i7-4770 (\(4\times 3400\) MHz) machine with 32 GB RAM.
As an explicit representation of the exact solution is not available for the given input data we measured the error by comparison with the solution on a very fine mesh with maximal element diameter \(h_{ref}=2^{-10}\). In Tables 1, 2 and 3 we report the error of the state in \(H^1(\Omega )\), and the control in the \(L^2(\Gamma )\)-norm and the \(H^{1/2}(\Gamma )\)-seminorm, respectively. The latter norm is realized by the discrete harmonic extension \(S_h\), this is,
The convergence rates measured for the domain \(\Omega _{90}\) are the same as in the experiments from [27]. These results confirm the rates predicted in Theorem 3 and Corollary 2. Note that the largest opening angle is \(\bar{\omega }=\pi /2\) and thus, \(\bar{\lambda }= 2\). Our theory moreover claims that the convergence rate for the discrete control is reduced when the largest opening angle exceeds the limiting case \(2\pi /3\). This is the case for the domain \(\Omega _{135}\), where we have \(\bar{\omega }= 3\pi /4\) and \(\bar{\lambda }= 4/3\). The rate 4 / 3 for the control in the \(H^{1/2}(\Gamma )\)-norm, which is predicted in Theorem 3, is the rate we also observe numerically. The convergence rate for the discrete states is still 1 as proved in Corollary 2. The fact that our error estimates are also valid and sharp for non-convex domains is confirmed by the experiment for the domain \(\Omega _{270}\). Here, the rate \(\bar{\lambda }=2/3\) is almost observed numerically for the discrete states and controls in \(H^1(\Omega )\) and \(H^{1/2}(\Gamma )\), respectively. Note that the convergence rates in the experiments are always slightly better than predicted which is due to the approximate computation of the error by comparison with a reference solution on a fine grid.
Moreover, we have to notice that we have not proved error estimates for the control in \(L^2(\Gamma )\), but the experiments confirm in all cases that this convergence rate is higher by 1 / 2 compared to the rate obtained in the \(H^{1/2}(\Gamma )\)-norm. In case of Dirichlet control problems with \(L^2(\Gamma )\)-regularization the convergence rate for the control in the \(L^2(\Gamma )\)-norm is \(\min \{1,\bar{\lambda }-1/2-\varepsilon \}\) (up to logarithmic factors), see [3], or \(\min \{1,\bar{\lambda }/2\}-\varepsilon \) in case of state-constraints, see [20]. Obviously, the rate is higher by 1 when \(H^{1/2}(\Gamma )\)-regularization is used which is due to the fact that the solutions are more regular. However, in order to prove estimates in \(L^2(\Gamma )\) for the energy regularization approach, one has to establish a Nitsche trick for the non-conforming approximation (61) of (55). This will be subject of future research.
References
Agoshkov, V.I., Lebedev, V.I.: Poincaré–Steklov operators and methods of partition of the domain in variational problems. Comput. Process. Syst. 2, 173–227 (1985)
Apel, Th, Mateos, M., Pfefferer, J., Rösch, A.: On the regularity of the solutions of Dirichlet optimal control problems in polygonal domains. SIAM J. Control Optim. 53(6), 3620–3641 (2015)
Apel, Th, Mateos, M., Pfefferer, J., Rösch, A.: Error estimates for Dirichlet control problems in polygonal domains: Quasi-uniform meshes. Math. Control Relat. Fields 8(1), 217–245 (2018)
Apel, Th, Pfefferer, J., Rösch, A.: Finite element error estimates for Neumann boundary control problems on graded meshes. Comput. Optim. Appl. 52(1), 3–28 (2012)
Apel, Th, Pfefferer, J., Winkler, M.: Error estimates for the postprocessing approach applied to neumann boundary control problems in polyhedral domains. IMA J. Numer. Anal. 38(4), 1984–2025 (2018)
Apel, Th, Steinbach, O., Winkler, M.: Error estimates for Neumann boundary control problems with energy regularization. J. Numer. Math. 24(4), 207–233 (2016)
Bartels, S., Carstensen, C., Dolzmann, G.: Inhomogeneous Dirichlet conditions in a priori and a posteriori finite element error analysis. Numer. Math. 99(1), 1–24 (2004)
Casas, E., Raymond, J.-P.: Error estimates for the numerical approximation of Dirichlet boundary control of semilinear elliptic equations. SIAM J. Control Optim. 45(5), 1586–1611 (2006)
Chowdhury, S., Gudi, T., Nandakumaran, A.K.: Error bounds for a Dirichlet boundary control problem based on energy spaces. Math. Comp. 86(305), 1103–1126 (2017)
Ciarlet, P.G.: Basic error estimates for elliptic problems. Finite Element Methods. Handbook of Numerical Analysis, vol. 2, pp. 17–352. Elsevier, North-Holland (1991)
Demlow, A., Guzmán, J., Schatz, A.H.: Local energy estimates for the finite element method on sharply varying grids. Math. Comp. 80(273), 1–9 (2011)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. In: Totaro, N.S., Waldschmidt, A.V. (eds.) Grundlehren der Mathematischen Wissenschaften, vol. 224, 2nd edn. Springer, Berlin (1998)
Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985)
Gunzburger, M. D., Hou, L. S., Svobodny, Th. P.: Analysis and finite element approximation of optimal control problems for the stationary Navier-Stokes equations with Dirichlet controls. ESAIM: Math. Model. Numer. Anal., 25(6):711–748, 1991
Horger, T., Melenk, M., Wohlmuth, B.I.: On optimal L2- and surface flux convergence in FEM. Comput. Vis. Sci. 16(5), 231–246 (2015)
John, L. J.: Optimal boundary control in energy spaces preconditioning and applications. PhD thesis, TU Graz, 2014
John, L.J., Swierczynski, P., Wohlmuth, B.I.: Energy corrected FEM for optimal Dirichlet boundary control problems. Numer. Math. 139(4), 913–938 (2018)
Kozlov, V.A., Maz’ya, V.G., Rossmann, J.: Spectral Problems Associated with Corner Singularities of Solutions to Elliptic Equations. Mathematical Surveys and Monographs, vol. 85. American Mathematical Society, Providence (2001)
Lions, J.L.: Optimal Control of Systems Governed by Partial Differential Equations. Grundlehren der mathematischen Wissenschaften. Springer, New York (1971)
Mateos, M., Neitzel, I.: Dirichlet control of elliptic state constrained problems. Comput. Optim. Appl. 63(3), 825–853 (2016)
May, S., Rannacher, R., Vexler, B.: Error analysis for a finite element approximation of elliptic Dirichlet boundary control problems. SIAM J. Control Optim. 51(3), 2585–2611 (2013)
Maz’ya, V.G., Rossmann, J.: Elliptic Equations in Polyhedral Domains. AMS, Providence (2010)
McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000)
Melenk, M., Wohlmuth, B.I.: Quasi-optimal approximation of surface based lagrange multipliers in finite element methods. SIAM J. Numer. Anal. 50(4), 2064–2087 (2012)
Mikhailov, S.E.: Traces, extensions, co-normal derivatives and solution regularity of elliptic systems with smooth and non-smooth coefficients. J. Math. Anal. Appl. 378, 324–342 (2012)
Nazarov, S.A., Plamenevskij, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. De Gruyter, Berlin (1994)
Of, G., Phan, T.X., Steinbach, O.: An energy space finite element approach for elliptic Dirichlet boundary control problems. Numer. Math. 129(4), 723–748 (2015)
Pfefferer, J.: Numerical analysis for elliptic Neumann boundary control problems on polygonal domains. PhD thesis, Universität der Bundeswehr München, 2014
Pfefferer, J., Winkler, M.: Finite element error estimates for normal derivatives on boundary concentrated meshes. SIAM J. Numer. Anal. 57(5), 2043–2073 (2019)
Quarteroni, A., Valli, A.: Domain Decomposition Methods for Partial Differential Equations. Numerical Mathematics and Scientific Computing. Clarendon Press, Oxford (1999)
Steinbach, O.: On the stability of the \(L_2\) projection in fractional Sobolev spaces. Numer. Math. 88(2), 367–379 (2001)
Winkler, M.: Finite element error analysis for neumann boundary control problems on polygonal and polyhedral domains. PhD thesis, Universität der Bundeswehr München, 2015
Wloka, J., Thomas, C.B., Thomas, M.J.: Partial Differential Equations. Cambridge University Press, Cambridge (1987)
Xu, J., Zhang, S.: Preconditioning the Poincaré-Steklov operator by using Green’s function. Math. Comp. 66(217), 125–138 (1997)
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The author acknowledges the fruitful discussions with Johannes Pfefferer, Hannes Meinlschmidt and Marco Zank during the preparation of the manuscript.
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Winkler, M. Error estimates for variational normal derivatives and Dirichlet control problems with energy regularization. Numer. Math. 144, 413–445 (2020). https://doi.org/10.1007/s00211-019-01091-1
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DOI: https://doi.org/10.1007/s00211-019-01091-1