Identifying conductivity in electrical impedance tomography with total variation regularization

Abstract

In this paper we investigate the problem of identifying the conductivity in electrical impedance tomography from one boundary measurement. A variational method with total variation regularization is here proposed to tackle this problem. We discretize the PDE as well as the conductivity with piecewise linear, continuous finite elements. We prove the stability and convergence of this technique. For the numerical solution we propose a projected Armijo algorithm. Finally, a numerical experiment is presented to illustrate our theoretical results.

1 Introduction

Let $\Omega$ be an open bounded connected domain in $\mathbb {R}^d, d\in \{2,3\}$ with polygonal boundary $\partial \Omega$ and $f\in {H^1(\Omega )}^* := H^{-1}(\Omega )$ be given. We consider the following elliptic boundary value problem

$$\begin{aligned} -\nabla \cdot \big (q \nabla \Phi \big )&= f \text{ in } \Omega , \end{aligned}$$

(1.1)

$$\begin{aligned} q \nabla \Phi \cdot {n}&= j^\dag \text{ on } \partial \Omega \text{ and } \end{aligned}$$

(1.2)

$$\begin{aligned} \Phi&= g^\dag \text{ on } \partial \Omega , \end{aligned}$$

(1.3)

where ${n}$ is the unit outward normal on $\partial \Omega$.

The system (1.1)–(1.3) is overdetermined, i.e. if the Neumann and Dirichlet boundary conditions $j^\dag \in H^{-1/2}(\partial \Omega ) := {H^{1/2}(\partial \Omega )}^*, ~g^\dag \in H^{1/2}(\partial \Omega )$ and the conductivity

$$\begin{aligned} q\in \mathcal {Q} := \left\{ q \in L^{\infty }(\Omega ) ~\big |~ \underline{q} \le q(x) \le \overline{q} \text{ a.e. } \text{ in } \Omega \right\} \end{aligned}$$

(1.4)

are given, then there may be no $\Phi$ satisfying this system. Here $\underline{q}$ and $\overline{q}$ are some given positive constants.

In this paper we assume that the system is consistent and our aim is to identify the conductivity $q^\dag \in \mathcal {Q}$ and the electric potential $\Phi ^\dag \in H^1(\Omega )$ in the system (1.1)–(1.3) from current and voltage, i.e. Neumann and Dirichlet measurements at the boundary $\left( j_\delta ,g_\delta \right) \in H^{-1/2}(\partial \Omega ) \times H^{1/2}(\partial \Omega )$ of the exact $\big (j^\dag ,g^\dag \big )$ satisfying

$$\begin{aligned} \big \Vert j_\delta -j^\dag \big \Vert _{H^{-1/2}(\partial \Omega )} + \big \Vert g_\delta -g^\dag \big \Vert _{H^{1/2}(\partial \Omega )} \le \delta \text{ with } \delta >0. \end{aligned}$$

Note that using the $H^{-1/2}(\partial \Omega ) \times H^{1/2}(\partial \Omega )$ topology for the data is natural from the point of view of solution theory for elliptic PDEs but unrealistic with regard to practical measurements. We will comment in this issue in Remark 2.2 below.

For the purpose of conductivity identification—a problem which is very well known in literature and practice as electrical impedance tomography EIT, see below for some references—we simultaneously consider the Neumann problem

$$\begin{aligned} -\nabla \cdot (q\nabla u) = f \text{ in } \Omega \text{ and } q\nabla u\cdot {n} = j_\delta \text{ on } \partial \Omega \end{aligned}$$

(1.5)

and the Dirichlet problem

$$\begin{aligned} -\nabla \cdot (q\nabla v) = f \text{ in } \Omega \text{ and } v = g_\delta \text{ on } \partial \Omega \end{aligned}$$

(1.6)

and respectively denote by $\mathcal {N}_q j_\delta$, $\mathcal {D}_q g_\delta$ the unique weak solutions of the problems (1.5), (1.6), which depend nonlinearly on q, where $\mathcal {N}_q j_\delta$ is normalized with vanishing mean on the boundary. We adopt the variational approach of Kohn and Vogelius in [30,31,32] to the identification problem. In fact, for estimating the conductivity q from the observation $\left( j_\delta ,g_\delta \right)$ of the exact data $\big (j^\dag ,g^\dag \big )$, we use the functional

$$\begin{aligned} \mathcal {J}_\delta (q) := \int _\Omega q\nabla \left( \mathcal {N}_q j_\delta - \mathcal {D}_q g_\delta \right) \cdot \nabla \left( \mathcal {N}_q j_\delta - \mathcal {D}_q g_\delta \right) dx. \end{aligned}$$

For simplicity of exposition we restrict ourselves to the case of just one Neumann–Dirichlet pair, while the approach described here can be easily extended to multiple measurements $\left( j_\delta ^i,g_\delta ^i \right) _{i=1,\ldots ,I}$, see also Example 5.3 below. It is well-known that such a finite number of boundary data in general only allows to identify conductivities taking finitely many different values in the domain $\Omega$, see, e.g., [2].

Indeed, we are interested in estimating such piecewise constant conductivities and therefore use total variation regularization, i.e. we consider the minimization problem

$$\begin{aligned} \min _{q\in \mathcal {Q}_{ad}} \mathcal {J}_\delta (q) + \rho \int _\Omega \left| \nabla q\right| , \end{aligned}$$

(1.7)

where $\mathcal {Q}_{ad} := \mathcal {Q} \cap BV(\Omega )$ is the admissible set of the sought conductivities, $BV(\Omega )$ is the space of all functions with bounded total variation (see §2.1 for its definition) and $\rho >0$ is the regularization parameter, and consider a minimizer $q_{\rho ,\delta }$ of (1.7) as reconstruction.

For each $q\in \mathcal {Q}$ let $\mathcal {N}^h_qj_\delta$ and $\mathcal {D}^h_q g_\delta$ be corresponding approximations of $\mathcal {N}_qj_\delta$ and $\mathcal {D}_qg_\delta$ in the finite dimensional space $\mathcal {V}^h_1$ of piecewise linear, continuous finite elements and $q^h_{\rho ,\delta }$ denote a minimizer of the discrete regularized problem corresponding to (1.7), i.e. of the following minimization problem

$$\begin{aligned} \min _{q \in \mathcal {Q}^h_{ad}} \int _\Omega q\nabla \left( \mathcal {N}^h_q j_\delta - \mathcal {D}^h_q g_\delta \right) \cdot \nabla \left( \mathcal {N}^h_q j_\delta - \mathcal {D}^h_q g_\delta \right) dx + \rho \int _\Omega \sqrt{\left| \nabla q\right| ^2+\epsilon ^h} \end{aligned}$$

(1.8)

with $\mathcal {Q}^h_{ad} := \mathcal {Q}_{ad} \cap \mathcal {V}^h_1$ and $\epsilon ^h$ being a positive functional of the mesh size h satisfying $\lim _{h\rightarrow 0} \epsilon ^h =0$.

In Sect. 4 we will show the stability of approximations for fixed positive $\rho$. Furthermore as $h,\delta \rightarrow 0$ and with an appropriate a priori regularization parameter choice $\rho =\rho (h,\delta )$, there exists a subsequence of $\big (q^h_{\rho ,\delta }\big )$ converging in the $L^1(\Omega )$-norm to a total variation-minimizing solution $q^\dag$ defined by

$$\begin{aligned} q^\dag \in \arg \min _{\big \{q\in \mathcal {Q}_{ad} ~|~ \mathcal {N}_qj ^\dag = \mathcal {D}_q g^\dag \big \}} \int _\Omega | \nabla q|. \end{aligned}$$

In particular, if $q^\dag$ is uniquely defined, then this convergence holds for the whole sequence $\big (q^h_{\rho ,\delta }\big )$. The corresponding state sequences $\Big (\mathcal {N}^h_{q^h_{\rho ,\delta }} j_\delta \Big )$ and $\Big (\mathcal {D}^h_{q^h_{\rho ,\delta }} g_\delta \Big )$ converge in the $H^1(\Omega )$-norm to $\Phi ^\dag = \Phi ^\dag (q^\dag ,j^\dag ,g^\dag )$ solving the system (1.1)–(1.3). Finally, for the numerical solution of the discrete regularized problem (1.8), in Sect. 5 we employ a projected Armijo algorithm. Numerical results show the efficiency of the proposed method and illustrate our theoretical findings.

We conclude this introduction with a selection of references from the vast literature on EIT, which has evolved to a highly relevant imaging and diagnostics tool in industrial and medical applications and has attracted great attention of many scientists in the last few decades.

To this end, for any fixed $q\in \mathcal {Q}$ we define the Neumann-to-Dirichlet map $\Lambda _q: H^{-1/2}(\partial \Omega ) \rightarrow H^{1/2}(\partial \Omega )$, by

$$\begin{aligned} j\mapsto \Lambda _qj := \gamma \mathcal {N}_qj, \end{aligned}$$

where $\gamma : H^1(\Omega ) \rightarrow H^{1/2}(\partial \Omega )$ is the Dirichlet trace operator. Calderón in 1980 posed the question whether an unknown conductivity distribution inside a domain can be determined from an infinite number of boundary observations, i.e. from the Neumann-to-Dirichlet map $\Lambda _q$:

$$\begin{aligned} p, q \in \mathcal {Q} \subset L^\infty (\Omega ) \text{ with } \Lambda _p = \Lambda _q \quad \Rightarrow \quad p=q \ ? \end{aligned}$$

(1.9)

Calderón did not answer his question (1.9); however, in [15] he proved that the problem linearized at constant conductivities has a unique solution. In dimensions three and higher Sylvester and Uhlmann [41] proved the unique identifiability of a $C^\infty$-smooth conductivity. Päivärinta et al. [37] and Brown and Torres [12] established uniqueness in the inverse conductivity problem for $W^{3/2,p}$-smooth conductivities with $p=\infty$ and $p>2d$, respectively. In the two dimensional setting, Nachman [34] and Brown and Uhlmann [13] proved uniqueness results for conductivities which are in $W^{2,p}$ with $p>1$ and $W^{1,p}$ with $p>2$, respectively. Finally, in 2006 the question (1.9) has been answered to be positive by Astala and Päivärinta [3] in dimension two. For surveys on the subject, we refer the reader to [10, 17, 20, 33, 43] and the references therein.

Although there exists a large number of papers on the numerical solution of the inverse problems of EIT, among these also papers considering the Kohn–Vogelius functional (see, e.g., [28, 29]) and total variation regularization (see, e.g., [21, 36]), we have not yet found investigations on the discretization error in a combination of both functionals for the fully nonlinear setting, a fact which motivated the research presented in this paper.

Throughout the paper we use the standard notion of Sobolev spaces $H^1(\Omega )$, $H^1_0(\Omega )$, $W^{k,p}(\Omega )$, etc from, for example, [1]. If not stated otherwise we write $\int _\Omega \cdots$ instead of $\int _\Omega \cdots dx$.

2 Problem setting and preliminaries

2.1 Notations

Let us denote by

$$\begin{aligned} \gamma : H^1(\Omega ) \rightarrow H^{1/2}(\partial \Omega ) \end{aligned}$$

the continuous Dirichlet trace operator while

$$\begin{aligned} \gamma ^{-1} : H^{1/2}(\partial \Omega ) \rightarrow H^1(\Omega ) \end{aligned}$$

is the continuous right inverse operator of $\gamma$, i.e. $(\gamma \circ \gamma ^{-1}) g =g$ for all $g\in H^{1/2}(\partial \Omega )$. With $f\in H^{-1}(\Omega )$ (with a slight abuse of notation) in (1.1) being given, let us denote

$$\begin{aligned} c_f := (f,1), \end{aligned}$$

where the expression $(f,\varphi )$ denotes the value of the functional $f\in H^{-1}(\Omega )$ at $\varphi \in H^1(\Omega )$. We also denote

$$\begin{aligned} H^{-1/2}_{-c_f}(\partial \Omega ) := \left\{ j \in H^{-1/2}(\partial \Omega ) ~\big |~ \langle j,1 \rangle = -c_f\right\} , \end{aligned}$$

where the notation $\left\langle j,g\right\rangle$ stands for the value of the functional $j\in H^{-1/2}(\partial \Omega )$ at $g\in H^{1/2}(\partial \Omega )$. Similarly, we denote

$$\begin{aligned} H^{1/2}_\diamond (\partial \Omega ) := \left\{ g \in H^{1/2}(\partial \Omega ) ~\Big |~ \int _{\partial \Omega } g(s) =0\right\} \end{aligned}$$

while $H^1_\diamond (\Omega )$ is the closed subspace of $H^1(\Omega )$ consisting of all functions with zero mean on the boundary, i.e.

$$\begin{aligned} H^1_\diamond (\Omega ) := \left\{ u \in H^1(\Omega ) ~\Big |~ \int _{\partial \Omega } \gamma u =0\right\} . \end{aligned}$$

Let us denote by $C^\Omega _\diamond$ the positive constant appearing in the Poincaré–Friedrichs inequality (see, for example, [38])

$$\begin{aligned} C^\Omega _\diamond \int _\Omega \varphi ^2 \le \int _\Omega |\nabla \varphi |^2 \text{ for } \text{ all } \varphi \in H^1_\diamond (\Omega ). \end{aligned}$$

(2.1)

Then for all $q \in \mathcal {Q}$ defined by (1.4), the coercivity condition

$$\begin{aligned} \Vert \varphi \Vert ^2_{H^1(\Omega )} \le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond } \int _\Omega | \nabla \varphi |^2 \le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \int _\Omega q\nabla \varphi \cdot \nabla \varphi \end{aligned}$$

(2.2)

holds for all $\varphi \in H^1_\diamond (\Omega )$. Furthermore, since $H^1_0(\Omega ) := \left\{ u \in H^1(\Omega ) ~\big |~ \gamma u =0\right\} \subset H^1_\diamond (\Omega )$, the inequality (2.2) remains valid for all $\varphi \in H^1_0(\Omega )$.

Finally, for the sake of completeness we briefly introduce the space of functions with bounded total variation; for more details one may consult [4, 24]. A scalar function $q\in L^{1}(\Omega )$ is said to be of bounded total variation if

$$\begin{aligned} TV(q) {:=} \int _{\Omega }\left| \nabla q\right| :=\sup \left\{ \int _{\Omega } q\text {div}~ \Xi ~\big |~ \Xi \in C^1_c(\Omega )^d,~ \left| \Xi (x)\right| _{\infty } \le 1,~x\in \Omega \right\} <\infty . \end{aligned}$$

Here $\left| \cdot \right| _{\infty }$ denotes the $\ell _{\infty }$-norm on $\mathbb {R}^d$ defined by $\left| x\right| _{\infty }=\max \limits _{1\le i\le d}\left| x_i\right|$ and $C^1_c(\Omega )$ the space of continuously differentiable functions with compact support in $\Omega$. The space of all functions in $L^{1}(\Omega )$ with bounded total variation is denoted by

$$\begin{aligned} BV(\Omega )=\left\{ q\in L^{1}(\Omega ) ~\Big |~ \int _{\Omega }\left| \nabla q\right| <\infty \right\} \end{aligned}$$

which is a Banach space with the norm

$$\begin{aligned} \Vert q \Vert _{BV(\Omega )} := \Vert q \Vert _{L^1(\Omega )} + \int _{\Omega } |\nabla q|. \end{aligned}$$

Furthermore, if $\Omega$ is an open bounded set with Lipschitz boundary, then $W^{1,1}(\Omega )\varsubsetneq BV(\Omega )$.

2.2 Neumann operator, Dirichlet operator and Neumann-to-Dirichlet map

2.2.1 Neumann operator

We consider the following Neumann problem

$$\begin{aligned} -\nabla \cdot (q\nabla u) = f \text{ in } \Omega \text{ and } q\nabla u\cdot {n} = j \text{ on } \partial \Omega . \end{aligned}$$

(2.3)

By the coercivity condition (2.2) and the Riesz representation theorem, we conclude that for each $q\in \mathcal {Q}$ and $j\in H^{-1/2}_{-c_f}(\partial \Omega )$ there exists a unique weak solution u of the problem (2.3) in the sense that $u\in H^1_\diamond (\Omega )$ and satisfies the identity

$$\begin{aligned} \int _\Omega q \nabla u \cdot \nabla \varphi = \left\langle j,\gamma \varphi \right\rangle + (f,\varphi ) \end{aligned}$$

(2.4)

for all $\varphi \in H^1_\diamond (\Omega )$. By the imposed compatibility condition $\langle j,1 \rangle = -c_f$, i.e.

$$\begin{aligned} \left\langle j,1\right\rangle + (f,1) =0, \end{aligned}$$

(2.5)

and the fact that $H^1(\Omega )=H^1_\diamond (\Omega )+\text{ span }\{1\}$, equation (2.4) is satisfied for all $\varphi \in H^1(\Omega )$. Furthermore, this solution satisfies the following estimate

$$\begin{aligned} \left\| u\right\| _{H^1(\Omega )}&\le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \left( \left\| \gamma \right\| _{\mathcal {L}\big (H^1(\Omega ),H^{1/2}(\partial \Omega )\big )} \left\| j\right\| _{H^{-1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) \nonumber \\&\le C_{\mathcal {N}} \left( \left\| j\right\| _{H^{-1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) , \end{aligned}$$

(2.6)

where

$$\begin{aligned} C_{\mathcal {N}} := \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \max \left( 1, \left\| \gamma \right\| _{\mathcal {L}\big (H^1(\Omega ),H^{1/2}(\partial \Omega )\big )}\right) . \end{aligned}$$

Then for any fixed $j\in H^{-1/2}_{-c_f}(\partial \Omega )$ we can define the Neumann operator

$$\begin{aligned} \mathcal {N} : \mathcal {Q} \rightarrow H^1_\diamond (\Omega ) \text{ with } q\mapsto \mathcal {N}_qj \end{aligned}$$

which maps each $q \in \mathcal {Q}$ to the unique weak solution $\mathcal {N}_qj := u$ of the problem (2.3).

Remark 2.1

We note that the restriction $j\in H^{-1/2}_{-c_f}(\partial \Omega )$ instead of $j\in H^{-1/2}(\partial \Omega )$ preserves the compatibility condition (2.5) for the pure Neumann problem. In case this condition fails, the strong form of the problem (2.3) has no solution. This is the reason why we require $j\in H^{-1/2}_{-c_f}(\partial \Omega )$. However, its weak form, i.e. the variational equation (2.4), attains a unique solution independently of the compatibility condition. By working with the weak form only, all results in the present paper remain valid for $j\in H^{-1/2}(\partial \Omega )$.

2.2.2 Dirichlet operator

We now consider the following Dirichlet problem

$$\begin{aligned} -\nabla \cdot (q\nabla v) = f \text{ in } \Omega \text{ and } v = g \text{ on } \partial \Omega . \end{aligned}$$

(2.7)

For each $q\in \mathcal {Q}$ and $g\in H^{1/2}(\partial \Omega )$, by the coercivity condition (2.2), the problem (2.7) attains a unique weak solution v in the sense that $v\in H^1(\Omega )$, $\gamma v = g$ and satisfies the identity

$$\begin{aligned} \int _\Omega q \nabla v \cdot \nabla \psi = (f,\psi ) \end{aligned}$$

(2.8)

for all $\psi \in H^1_0(\Omega )$. We can rewrite

$$\begin{aligned} v =v_0 +G, \end{aligned}$$

(2.9)

where $G=\gamma ^{-1}g$ and $v_0 \in H^1_0(\Omega )$ is the unique solution to the following variational problem

$$\begin{aligned} \int _\Omega q \nabla v_0 \cdot \nabla \psi = (f,\psi ) - \int _\Omega q \nabla G \cdot \nabla \psi \end{aligned}$$

for all $\psi \in H^1_0(\Omega )$. Since

$$\begin{aligned} \left\| G\right\| _{H^1(\Omega )} \le \left\| \gamma ^{-1}\right\| _{\mathcal {L}\big (H^{1/2}(\partial \Omega ),H^1(\Omega )\big )} \left\| g\right\| _{H^{1/2}(\partial \Omega )}, \end{aligned}$$

we thus obtain the priori estimate

$$\begin{aligned} \left\| v\right\| _{H^1(\Omega )}&\le \left\| v_0\right\| _{H^1(\Omega )} + \left\| G\right\| _{H^1(\Omega )}\nonumber \\&\le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}}\Vert f\Vert _{H^{-1}(\Omega )} + \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \overline{q} \left\| G\right\| _{H^1(\Omega )} + \left\| G\right\| _{H^1(\Omega )}\nonumber \\&\le C_\mathcal {D} \left( \left\| g\right\| _{H^{1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) , \end{aligned}$$

(2.10)

where

$$\begin{aligned} C_\mathcal {D} := \max \left( \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}}, \left( \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \overline{q} + 1\right) \left\| \gamma ^{-1}\right\| _{\mathcal {L}\big (H^{1/2}(\partial \Omega ),H^1(\Omega )\big )}\right) . \end{aligned}$$

The Dirichlet operator is for any fixed $g \in H^{1/2}(\partial \Omega )$ defined as

$$\begin{aligned} \mathcal {D} : \mathcal {Q} \rightarrow H^1(\Omega ) \text{ with } q\mapsto \mathcal {D}_qg \end{aligned}$$

which maps each $q\in \mathcal {Q}$ to the unique weak solution $\mathcal {D}_qg := v$ of the problem (2.7).

2.2.3 Neumann-to-Dirichlet map

For any fixed $q\in \mathcal {Q}$ we can define the Neumann-to-Dirichlet map

$$\begin{aligned} \Lambda _q: H^{-1/2}_{-c_f}(\partial \Omega )&\rightarrow H^{1/2}_\diamond (\partial \Omega )\\ j&\mapsto \Lambda _qj := \gamma \mathcal {N}_qj. \end{aligned}$$

Since

$$\begin{aligned} \int _\Omega q \nabla \mathcal {N}_q j \cdot \nabla \psi = (f,\psi ) \end{aligned}$$

for all $\psi \in H^1_0(\Omega )$, in view of (2.8) we conclude that

$$\begin{aligned} \Lambda _qj = g \text{ if } \text{ and } \text{ only } \text{ if } \mathcal {N}_qj = \mathcal {D}_qg. \end{aligned}$$

2.3 Identification problem

The inverse problem is stated as follows.

$$\begin{aligned} { Given }\,f{\in } H^{-1}(\Omega ), \left( j^\dag , g^\dag \right) {\in } H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )\,{ with}\,\Lambda _{q^\dag } j^\dag {=} g^\dag ,\, { find}\, q^\dag {\in } \mathcal {Q}. \end{aligned}$$

In other words, the problem of interest is, given $f\in H^{-1}(\Omega )$, and a single Neumann–Dirichlet pair $\left( j^\dag , g^\dag \right) \in H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$, to find $q^\dag \in \mathcal {Q}$ and $\Phi ^\dag \in H^1_\diamond (\Omega )$ such that the system (1.1)–(1.3) is satisfied in the weak sense.

2.4 Total variation regularization

Assume that $\left( j_\delta ,g_\delta \right) \in H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ is the measured data of the exact boundary values $(j^\dag ,g^\dag )$ with

$$\begin{aligned} \big \Vert j_\delta -j^\dag \big \Vert _{H^{-1/2}(\partial \Omega )} + \big \Vert g_\delta -g^\dag \big \Vert _{H^{1/2}(\partial \Omega )} \le \delta \end{aligned}$$

(2.11)

for some measurement error $\delta >0$. Our problem is now to reconstruct the conductivity $q^\dag \in \mathcal {Q}$ from this perturbed data $\left( j_\delta ,g_\delta \right)$. For this purpose we consider the cost functional

$$\begin{aligned} \mathcal {J}_\delta (q) := \int _\Omega q\nabla \left( \mathcal {N}_qj_\delta - \mathcal {D}_qg_\delta \right) \cdot \nabla \left( \mathcal {N}_qj_\delta - \mathcal {D}_qg_\delta \right) , \end{aligned}$$

(2.12)

where $\mathcal {N}_qj_\delta$ and $\mathcal {D}_qg_\delta$ is the unique weak solutions of the problems (2.3) and (2.7), respectively, with j in (2.3) and g in (2.7) being replaced by $j_\delta$ and $g_\delta$. Furthermore, to estimate the possibly discontinuous conductivity, we here use the total variation regularization (cf., e.g., [14, 21, 22]), i.e. we consider the minimization problem

where

$$\begin{aligned} \mathcal {Q}_{ad} := \mathcal {Q} \cap BV(\Omega ) \end{aligned}$$

is the admissible set of the sought conductivities.

Remark 2.2

The noise model (2.11) is to some extent an idealized one, since in practice, measurement precision might be different for the current j and the voltage g, and, more importantly, it will first of all be given with respect to some $L^p$ norm (e.g., $p=2$ corresponding to normally and $p=\infty$ to uniformly distributed noise) rather than in $H^{-1/2}(\partial \Omega ) \times H^{1/2}(\partial \Omega )$. While the Neumann data part is unproblematic, by continuity of the embedding of $L^p(\partial \Omega )$ in $H^{-1/2}(\partial \Omega )$ for $p\ge 2\frac{d-1}{d}$, we can obtain an $H^{1/2}(\partial \Omega )$ version of the originally $L^p(\partial \Omega )$ Dirichlet data e.g. by Tikhonov regularization (cf. [22] and the references therein) as follows. For simplicity, we restrict ourselves to the Hilbert space case $p=2$ and assume that we have measurements $\tilde{g}_{\delta _g}\in L^2(\partial \Omega )$ such that

$$\begin{aligned} \Vert \tilde{g}_{\delta _g}-g^\dagger \Vert _{L^2(\partial \Omega )}\le \delta _g \end{aligned}$$

Tikhonov regularization applied to the embedding operator $K:H^{1/2}(\partial \Omega )\rightarrow L^2(\partial \Omega )$ amounts to finding a minimizer $g_\alpha ^{\delta _g}$ of

$$\begin{aligned} \min _{g\in H^{1/2}(\partial \Omega )} \Vert Kg-\tilde{g}_{\delta _g}\Vert _{L^2(\partial \Omega )}^2 +\alpha \Vert g\Vert _{H^{1/2}(\partial \Omega )}^2, \end{aligned}$$

where we use

$$\begin{aligned} \Vert g\Vert _{H^{1/2}(\partial \Omega )}:=\Vert \gamma ^{-1}g\Vert _{H^1(\Omega )} =\left( \int _\Omega (|\nabla \gamma ^{-1}g|^2+|\gamma ^{-1}g|^2)\, dx\right) ^{1/2} \end{aligned}$$

as a norm on $H^{1/2}(\partial \Omega )$. The first order optimality conditions for this quadratic minimization problem yield

$$\begin{aligned}&\int _{\partial \Omega } \phi (g_\alpha ^{\delta _g}-\tilde{g}_{\delta _g})\, ds +\alpha \int _\Omega (\nabla \gamma ^{-1} g_\alpha ^{\delta _g}\cdot \nabla \gamma ^{-1}\phi + \gamma ^{-1} g_\alpha ^{\delta _g}\gamma ^{-1}\phi )\, dx\\&\quad \qquad =0 \text{ for } \text{ all } \phi \in H^{1/2}(\partial \Omega ), \end{aligned}$$

which is equivalent to

$$\begin{aligned} \int _{\partial \Omega } \gamma \varphi (\gamma w-\tilde{g}_{\delta _g})\, ds +\alpha \int _\Omega (\nabla w\cdot \nabla \varphi + w\varphi )\, dx =0 \text{ for } \text{ all } \varphi \in H^1(\Omega ), \end{aligned}$$

for $w=\gamma ^{-1}g_\alpha ^{\delta _g}$, i.e. the weak form of the Robin problem

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta w+w &{}= 0 \text{ in } \Omega ,\\ \alpha \nabla w \cdot {n} + w&{}= \tilde{g}_{\delta _g} \text{ on } \partial \Omega . \end{array}\right. } \end{aligned}$$

(2.13)

Thus, according to well-known results from regularization theory (cf., e.g. [22]), the smoothed version $g_\delta :=g_\alpha ^{\delta _g}=\gamma w$ (where w weakly solves (2.13)) of $\tilde{g}_{\delta _g}$ converges to $g^\dagger$ as $\delta _g$ tends to zero, provided the regularization parameter $\alpha =\alpha (\delta _g,\tilde{g}_{\delta _g})$ is chosen appropriately. The latter can, e.g., be done by the discrepancy principle, where $\alpha$ is chosen such that

$$\begin{aligned} \Vert Kg_\alpha ^{\delta _g}-\tilde{g}_{\delta _g}\Vert _{L^2(\partial \Omega )}^2 =\int _{\partial \Omega } |g_\alpha ^{\delta _g}-\tilde{g}_{\delta _g}|^2\, dx\sim \delta _g^2. \end{aligned}$$

We also wish to mention the complete electrode model cf., e.g., [40], which fully takes into account the fact that current and voltage are typically not measured pointwise on the whole boundary, but via a set of finitely many electrodes with finite geometric extensions as well as contact impedances.

2.5 Auxiliary results

Now we summarize some useful properties of the Neumann and Dirichlet operators. The proof of the following result is based on standard arguments and therefore omitted.

Lemma 2.3

Let $(j,g)\in H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ be fixed.

• (i) The Neumann operator $\mathcal {N} : \mathcal {Q} \subset L^\infty (\Omega ) \rightarrow H^1_\diamond (\Omega )$ is continuously Fréchet differentiable on the set $\mathcal {Q}$. For each $q\in \mathcal {Q}$ the action of the Fréchet derivative in direction $\xi \in L^\infty (\Omega )$ denoted by $\eta _{\mathcal {N}} := \mathcal {N}'_qj(\xi ):=\mathcal {N}'(q)\xi$ is the unique weak solution in $H^1_\diamond (\Omega )$ to the Neumann problem

  $$\begin{aligned} -\nabla \cdot (q\nabla \eta _{\mathcal {N}}) = \nabla \cdot (\xi \nabla \mathcal {N}_qj) \text{ in } \Omega \text{ and } q\nabla \eta _{\mathcal {N}}\cdot {n} = -\xi \nabla \mathcal {N}_qj \cdot {n} \text{ on } \partial \Omega \end{aligned}$$

  in the sense that the identity

  $$\begin{aligned} \int _{\Omega }q \nabla \eta _{\mathcal {N}} \cdot \nabla \varphi = -\int _{\Omega } \xi \nabla \mathcal {N}_q j \cdot \nabla \varphi \end{aligned}$$

  (2.14)

  holds for all $\varphi \in H^1_\diamond (\Omega )$. Furthermore, the following estimate is fulfilled

  $$\begin{aligned} \Vert \eta _{\mathcal {N}} \Vert _{H^1(\Omega )}\le \frac{\left( 1+ C^\Omega _\diamond \right) C_{\mathcal {N}}}{C^\Omega _\diamond \underline{q}} \left( \left\| j\right\| _{H^{-1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) \Vert \xi \Vert _{L^\infty (\Omega )}. \end{aligned}$$

  (2.15)

• (ii) The Dirichlet operator $\mathcal {D} : \mathcal {Q} \subset L^\infty (\Omega ) \rightarrow H^1_\diamond (\Omega )$ is continuously Fréchet differentiable on the set $\mathcal {Q}$. For each $q\in \mathcal {Q}$ the action of the Fréchet derivative in direction $\xi \in L^\infty (\Omega )$ denoted by $\eta _{\mathcal {D}} := \mathcal {D}'_qg(\xi )=: \mathcal {D}'(q)\xi$ is the unique weak solution in $H^1_0(\Omega )$ to the Dirichlet problem

  $$\begin{aligned} -\nabla \cdot (q\nabla \eta _{\mathcal {D}}) = \nabla \cdot (\xi \nabla \mathcal {D}_qg) \text{ in } \Omega \text{ and } \eta _{\mathcal {D}} = 0 \text{ on } \partial \Omega \end{aligned}$$

  in the sense that it satisfies the equation

  $$\begin{aligned} \int _{\Omega }q \nabla \eta _{\mathcal {D}} \cdot \nabla \psi = -\int _{\Omega } \xi \nabla \mathcal {D}_qg \cdot \nabla \psi \end{aligned}$$

  for all $\psi \in H^1_0(\Omega )$. Furthermore, the following estimate is fulfilled

  $$\begin{aligned} \Vert \eta _{\mathcal {D}} \Vert _{H^1(\Omega )}\le \frac{\left( 1+ C^\Omega _\diamond \right) C_{\mathcal {D}}}{C^\Omega _\diamond \underline{q}} \left( \left\| g\right\| _{H^{1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) \Vert \xi \Vert _{L^\infty (\Omega )}. \end{aligned}$$

Lemma 2.4

If the sequence $\left( q_n\right) \subset \mathcal {Q}$ converges to q in the $L^1(\Omega )$-norm, then $q\in \mathcal {Q}$ and for any fixed $(j_\delta ,g_\delta )\in H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ the sequence $\left( \mathcal {N}_{q_n}j_\delta ,\mathcal {D}_{q_n}g_\delta \right)$ converges to $\left( \mathcal {N}_{q}j_\delta ,\mathcal {D}_{q}g_\delta \right)$ in the $H^1(\Omega )\times H^1(\Omega )$-norm. Furthermore, there holds

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}_\delta \left( q_n\right) = \mathcal {J}_\delta \big (q\big ), \end{aligned}$$

where the functional $\mathcal {J}_\delta$ is defined in (2.12).

Proof

Since $\left( q_n\right) \subset \mathcal {Q}$ converges to q in the $L^1(\Omega )$-norm, up to a subsequence we assume that it converges to q a.e. in $\Omega$, which implies that $q\in \mathcal {Q}$. For all $\varphi \in H^1_\diamond (\Omega )$ we infer from (2.4) that

$$\begin{aligned} \int _\Omega q_n \nabla \mathcal {N}_{q_n}j_\delta \cdot \nabla \varphi&= \left\langle j_\delta ,\gamma \varphi \right\rangle +(f,\varphi ) = \int _\Omega q \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \varphi \end{aligned}$$

and so that

$$\begin{aligned} \int _\Omega q_n \nabla \left( \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right) \cdot \nabla \varphi = \int _\Omega \left( q -q_n\right) \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \varphi . \end{aligned}$$

(2.16)

Taking $\varphi = \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta$, by (2.2), we get

$$\begin{aligned} \frac{C^\Omega _\diamond \underline{q}}{1+ C^\Omega _\diamond }\left\| \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right\| ^2_{H^1(\Omega )}&\le \int _\Omega q_n \nabla \left( \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right) \cdot \nabla \left( \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right) \\&= \int _\Omega \left( q -q_n\right) \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \left( \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right) \\&\le \left( \int _\Omega | q - q_n |^2 \left| \nabla \mathcal {N}_qj_\delta \right| ^2 \right) ^{1/2} \left( \int _\Omega \left| \nabla \left( \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right) \right| ^2\right) ^{1/2} \end{aligned}$$

and so that

$$\begin{aligned} \left\| \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} \le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \left( \int _\Omega | q - q_n |^2 \left| \nabla \mathcal {N}_qj_\delta \right| ^2 \right) ^{1/2}. \end{aligned}$$

Hence, by the Lebesgue dominated convergence theorem, we deduce from the last inequality that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \mathcal {N}_{q_n}j_\delta - \mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} = 0. \end{aligned}$$

(2.17)

Similarly to (2.16), we also get

$$\begin{aligned} \int _\Omega q_n \nabla \left( \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right) \cdot \nabla \psi = \int _\Omega \left( q -q_n\right) \nabla \mathcal {D}_{q}g_\delta \cdot \nabla \psi \end{aligned}$$

for all $\psi \in H^1_0(\Omega )$. Since $\gamma \mathcal {D}_{q_n}g_\delta =\gamma \mathcal {D}_qg_\delta =g_\delta$, taking $\psi =\mathcal {D}_{q_n}g_\delta -\mathcal {D}_{q}g_\delta \in H^1_0(\Omega )$ in the last equation, we also obtain the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right\| _{H^1(\Omega )} = 0. \end{aligned}$$

(2.18)

Next, we rewrite the functional $\mathcal {J}_\delta$ as follows

$$\begin{aligned} \mathcal {J}_\delta \left( q_n\right)&= \int _\Omega q_n\nabla \mathcal {N}_{q_n}j_\delta \cdot \nabla \mathcal {N}_{q_n}j_\delta - 2\int _\Omega q_n\nabla \mathcal {N}_{q_n}j_\delta \cdot \nabla \mathcal {D}_{q_n}g_\delta \nonumber \\&\quad + \int _\Omega q_n\nabla \mathcal {D}_{q_n}g_\delta \cdot \nabla \mathcal {D}_{q_n}g_\delta \nonumber \\&= \left\langle j_\delta ,\gamma \mathcal {N}_{q_n}j_\delta \right\rangle + \left( f, \mathcal {N}_{q_n}j_\delta \right) - 2\left( \left\langle j_\delta ,g_\delta \right\rangle + \left( f, \mathcal {D}_{q_n}g_\delta \right) \right) \nonumber \\&\quad + \int _\Omega q_n\nabla \mathcal {D}_{q_n}g_\delta \cdot \nabla \mathcal {D}_{q_n}g_\delta \end{aligned}$$

(2.19)

and, by (2.17)–(2.18), have that

$$\begin{aligned} \left\langle j_\delta ,\gamma \mathcal {N}_{q_n}j_\delta \right\rangle + \left( f, \mathcal {N}_{q_n}j_\delta - 2 \mathcal {D}_{q_n}g_\delta \right) \rightarrow \left\langle j_\delta ,\gamma \mathcal {N}_{q}j_\delta \right\rangle + \left( f, \mathcal {N}_{q}j_\delta - 2 \mathcal {D}_{q}g_\delta \right) \end{aligned}$$

(2.20)

as n tends to $\infty$. We now consider the difference

$$\begin{aligned}&\int _\Omega q_n\nabla \mathcal {D}_{q_n}g_\delta \cdot \nabla \mathcal {D}_{q_n}g_\delta - \int _\Omega q\nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \\&\quad = \int _\Omega q_n\nabla \left( \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right) \cdot \nabla \left( \mathcal {D}_{q_n}g_\delta + \mathcal {D}_{q}g_\delta \right) \\&\qquad - \int _\Omega \left( q - q_n\right) \nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \end{aligned}$$

and note that

$$\begin{aligned} \int _\Omega \left( q - q_n\right) \nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \rightarrow 0 \end{aligned}$$

as n goes to $\infty$, by the Lebesgue dominated convergence theorem. Furthermore, then applying the Cauchy–Schwarz inequality, we also get that

$$\begin{aligned}&\bigg | \int _\Omega q_n\nabla \left( \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right) \cdot \nabla \left( \mathcal {D}_{q_n}g_\delta + \mathcal {D}_{q}g_\delta \right) \bigg |\\&\quad \le \overline{q} \left( \int _\Omega \left| \nabla \left( \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right) \right| ^2\right) ^{1/2} \left( \int _\Omega \left| \nabla \left( \mathcal {D}_{q_n}g_\delta + \mathcal {D}_{q}g_\delta \right) \right| ^2\right) ^{1/2}\\&\quad \le \overline{q} \left\| \mathcal {D}_{q_n}g_\delta - \mathcal {D}_{q}g_\delta \right\| _{H^1(\Omega )} \left( \left\| \mathcal {D}_{q_n}g_\delta \right\| _{H^1(\Omega )} + \left\| \mathcal {D}_{q}g_\delta \right\| _{H^1(\Omega )}\right) \rightarrow 0 \end{aligned}$$

as n approaches $\infty$, here we used (2.10) and (2.18). We thus obtain that

$$\begin{aligned} \int _\Omega q_n\nabla \mathcal {D}_{q_n}g_\delta \cdot \nabla \mathcal {D}_{q_n}g_\delta \rightarrow \int _\Omega q\nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \end{aligned}$$

(2.21)

as n tends to $\infty$. Then we deduce from (2.19)–(2.21) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}_\delta \left( q_n\right)&= \left\langle j_\delta ,\gamma \mathcal {N}_{q}j_\delta \right\rangle + \left( f, \mathcal {N}_{q}j_\delta \right) -2 \left\langle j_\delta ,g_\delta \right\rangle -2 \left( f, \mathcal {D}_{q}g_\delta \right) \\&\quad + \int _\Omega q\nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \\&= \int _\Omega q\nabla \mathcal {N}_{q}j_\delta \cdot \nabla \mathcal {N}_{q}j_\delta - 2\int _\Omega q\nabla \mathcal {N}_{q}j_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \nonumber \\&\quad + \int _\Omega q\nabla \mathcal {D}_{q}g_\delta \cdot \nabla \mathcal {D}_{q}g_\delta \\&= \mathcal {J}_\delta \big (q\big ), \end{aligned}$$

which finishes the proof. $\square$

Lemma 2.5

([24]).

• (i) Let $\left( q_n\right)$ be a bounded sequence in the $BV(\Omega )$-norm. Then a subsequence which is denoted by the same symbol and an element $q\in BV(\Omega )$ exist such that $\left( q_n\right)$ converges to q in the $L^1(\Omega )$-norm.

• (ii) Let $\left( q_n\right)$ be a sequence in $BV(\Omega )$ converging to q in the $L^1(\Omega )$-norm. Then $q \in BV(\Omega )$ and

  $$\begin{aligned} \int _\Omega \left| \nabla q\right| \le \liminf _{n\rightarrow \infty }\int _\Omega |\nabla q_n|. \end{aligned}$$

  (2.22)

We mention that equality need not be achieved in (2.22). Here is a counterexample from [24]. Let $\Omega =(0,2\pi )$ and $q_n(x)=\frac{1}{n}\sin nx$ for $x\in \Omega$ and $n\in \mathbb {N}$. Then $\Vert q_n\Vert _{L^1(\Omega )} \rightarrow 0$ as $n\rightarrow \infty$, but $\int _\Omega |\nabla q_n| =4$ for each $n\in \mathbb {N}$.

Let us quote the following useful result on approximation of BV-functions by smooth functions.

Lemma 2.6

([5, 16]). Assume that $w\in BV(\Omega )$. Then for all $\alpha >0$ an element $w^\alpha \in C^\infty (\Omega )$ exists such that

$$\begin{aligned}&\int _\Omega |w-w^\alpha | \le \alpha \int _\Omega |\nabla w|,~ \int _\Omega |\nabla w^\alpha | \le (1+C\alpha )\int _\Omega |\nabla w| \text{ and } \int _\Omega |D^2 w^\alpha | \\&\quad \le C\alpha ^{-1}\int _\Omega |\nabla w|, \end{aligned}$$

where the positive constant C is independent of $\alpha$.

Now, we are in a position to prove the main result of this section

Theorem 2.7

The problem $\left( \mathcal {P}_{\rho ,\delta }\right)$ attains a solution $q_{\rho ,\delta }$, which is called the regularized solution of the identification problem.

Proof

Let $\left( q_n\right) \subset \mathcal {Q}_{ad}$ be a minimizing sequence of the problem $\left( \mathcal {P}_{\rho ,\delta }\right)$, i.e.

$$\begin{aligned} \lim _{n\rightarrow \infty }\left( \mathcal {J}_\delta \left( q_n\right) + \rho \int _\Omega \left| \nabla q_n\right| \right) = \inf _{q\in \mathcal {Q}_{ad}} \left( \mathcal {J}_\delta (q) + \rho \int _\Omega \left| \nabla q\right| \right) . \end{aligned}$$

(2.23)

Then, due to Lemma 2.5, a subsequence which is not relabelled and an element $q \in \mathcal {Q}_{ad}$ exist such that $\left( q_n\right)$ converges to q in the $L^1(\Omega )$-norm and

$$\begin{aligned} \int _\Omega |\nabla q| \le \liminf _{n\rightarrow \infty }\int _\Omega |\nabla q_n|. \end{aligned}$$

(2.24)

Using Lemma 2.4 and by (2.23)–(2.24), we obtain that

$$\begin{aligned} \mathcal {J}_\delta (q) + \rho \int _\Omega |\nabla q|&\le \lim _{n\rightarrow \infty }\mathcal {J}_\delta \left( q_n\right) + \liminf _{n\rightarrow \infty }\rho \int _\Omega |\nabla q_n| \\&= \liminf _{n\rightarrow \infty }\left( \mathcal {J}_\delta \left( q_n\right) + \rho \int _\Omega \left| \nabla q_n\right| \right) \\&= \inf _{q\in \mathcal {Q}_{ad}} \left( \mathcal {J}_\delta (q) + \rho \int _\Omega \left| \nabla q\right| \right) . \end{aligned}$$

This means that q is a solution of the problem $\left( \mathcal {P}_{\rho ,\delta }\right)$, which finishes the proof. $\square$

3 Finite element method for the identification problem

Let $\left( \mathcal {T}^h\right) _{0<h<1}$ be a family of regular and quasi-uniform triangulations of the domain $\overline{\Omega }$ with the mesh size h such that each vertex of the polygonal boundary $\partial \Omega$ is a node of $\mathcal {T}^h$. For the definition of the discretization space of the state functions let us denote

$$\begin{aligned} \mathcal {V}_1^h := \left\{ v^h\in C(\overline{\Omega }) ~\big |~{v^h}_{|T} \in \mathcal {P}_1(T), ~~\forall T\in \mathcal {T}^h\right\} \end{aligned}$$

and

$$\begin{aligned} \mathcal {V}_{1,\diamond }^h := \mathcal {V}_1^h \cap H^1_\diamond (\Omega ) \text{ and } \mathcal {V}_{1,0}^h := \mathcal {V}_1^h \cap H^1_0(\Omega ) \subset \mathcal {V}_{1,\diamond }^h, \end{aligned}$$

where $\mathcal {P}_1$ consists of all polynomial functions of degree less than or equal to 1.

To go further, we introduce the following modified Clément’s interpolation operator, see [19].

Lemma 3.1

An interpolation operator $\Pi ^h_\diamond : L^1(\Omega ) \rightarrow \mathcal {V}^h_{1,\diamond }$ exists such that

$$\begin{aligned} \Pi ^h_\diamond \varphi ^h = \varphi ^h \text{ for } \text{ all } \varphi ^h \in \mathcal {V}^h_{1,\diamond } \text{ and } \Pi ^h_\diamond \big (H^1_0(\Omega )\big ) \subset \mathcal {V}^h_{1,0} \subset \mathcal {V}^h_{1,\diamond }. \end{aligned}$$

Furthermore, it satisfies the properties

$$\begin{aligned} \lim _{h\rightarrow 0} \big \Vert \vartheta - \Pi ^h_\diamond \vartheta \big \Vert _{H^1(\Omega )} =0 \text{ for } \text{ all } \vartheta \in H^1_\diamond (\Omega ) \end{aligned}$$

(3.1)

and

$$\begin{aligned} \big \Vert \vartheta - \Pi ^h_\diamond \vartheta \big \Vert _{H^1(\Omega )} \le Ch \Vert \vartheta \Vert _{H^2(\Omega )} \text{ for } \text{ all } \vartheta \in H^1_\diamond (\Omega )\cap H^2(\Omega ) \end{aligned}$$

(3.2)

with the positive constant C being independent of h and $\vartheta$.

Proof

It is well known (see [19] and some generalizations [6, 7, 39]) that there is an interpolation operator

$$\begin{aligned} \Pi ^h: L^1(\Omega ) \rightarrow \mathcal {V}^h_1 \text{ with } \Pi ^h\varphi ^h = \varphi ^h \text{ for } \text{ all } \varphi ^h \in \mathcal {V}^h_{1} \text{ and } \Pi ^h \big (H^1_0(\Omega )\big ) \subset \mathcal {V}^h_{1,0} \end{aligned}$$

which satisfies the following properties

$$\begin{aligned} \lim _{h\rightarrow 0} \big \Vert \vartheta - \Pi ^h \vartheta \big \Vert _{H^1(\Omega )} =0 \text{ for } \text{ all } \vartheta \in H^1(\Omega ) \end{aligned}$$

(3.3)

and

$$\begin{aligned} \big \Vert \vartheta - \Pi ^h \vartheta \big \Vert _{H^1(\Omega )} \le Ch \Vert \vartheta \Vert _{H^2(\Omega )} \text{ for } \text{ all } \vartheta \in H^2(\Omega ). \end{aligned}$$

(3.4)

We then define for each $\vartheta \in L^1(\Omega )$

$$\begin{aligned} \Pi ^h_\diamond \vartheta := \Pi ^h\vartheta -\frac{1}{|\partial \Omega |}\int _{\partial \Omega } \gamma \Pi ^h\vartheta \in \mathcal {V}^h_{1,\diamond }. \end{aligned}$$

Then $\Pi ^h_\diamond \big (L^1(\Omega )\big ) \subset \mathcal {V}^h_{1,\diamond }$, $\Pi ^h_\diamond \varphi ^h = \varphi ^h$ for all $\varphi ^h \in \mathcal {V}^h_{1,\diamond }$ and $\Pi ^h_\diamond \big (H^1_0(\Omega )\big ) \subset \mathcal {V}^h_{1,0}$. Furthermore, since $\nabla \Pi ^h_\diamond \vartheta = \nabla \Pi ^h\vartheta$ for all $\vartheta \in L^1(\Omega )$, the properties (3.1), (3.2) are deduced from (3.3), (3.4), respectively. The proof is completed. $\square$

We remark that the operator $\Pi ^h$ in the above proof satisfies the estimate $\Vert \vartheta -\Pi ^h\vartheta \Vert _{H^k(\Omega )} \le Ch^{l-k}\Vert \vartheta \Vert _{H^l(\Omega )}$ for $0\le k\le l\le 2$ and $\vartheta \in H^l(\Omega )$ (see [19]) which implies that

$$\begin{aligned} \left\| \Pi ^h_\diamond \vartheta \right\| _{H^1(\Omega )} \le C\left\| \vartheta \right\| _{H^1(\Omega )} \quad \text{ for } \text{ all }\quad \vartheta \in H^1(\Omega ), \end{aligned}$$

(3.5)

an estimate that is required for the proof of part (ii) of the following proposition.

Similarly to the continuous case we have the following result.

Proposition 3.2

• (i) Let q be in $\mathcal {Q}$ and j be in $H^{-1/2}_{-c_f}(\partial \Omega )$. Then the variational equation

  $$\begin{aligned} \int _\Omega q\nabla u^h \cdot \nabla \varphi ^h = \left\langle j,\gamma \varphi ^h\right\rangle + \left( f,\varphi ^h\right) \text{ for } \text{ all } \varphi ^h\in \mathcal {V}_{1,\diamond }^h \end{aligned}$$

  (3.6)

  admits a unique solution $u^h\in \mathcal {V}_{1,\diamond }^h$. Furthermore, there holds

  $$\begin{aligned} \big \Vert u^h\big \Vert _{H^1(\Omega )}\le C_{\mathcal {N}} \left( \left\| j\right\| _{H^{-1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) . \end{aligned}$$

  (3.7)

• (ii) Let q be in $\mathcal {Q}$ and g be in $H^{1/2}_\diamond (\partial \Omega )$. Then the equation

  $$\begin{aligned} \int _\Omega q\nabla v^h \cdot \nabla \psi ^h = \left( f,\psi ^h\right) \text{ for } \text{ all } \psi ^h\in \mathcal {V}_{1,0}^h \end{aligned}$$

  (3.8)

  with $\gamma v^h = \gamma \big (\Pi ^h_\diamond (\gamma ^{-1}g)\big )$ has a unique solution $v^h\in \mathcal {V}_{1,\diamond }^h$. Furthermore, the stability estimate

  $$\begin{aligned} \big \Vert v^h\big \Vert _{H^1(\Omega )}\le \bar{C}_{\mathcal {D}} \left( \left\| g\right\| _{H^{1/2}(\partial \Omega )} + \Vert f\Vert _{H^{-1}(\Omega )}\right) \end{aligned}$$

  (3.9)

  is satisfied, where $\bar{C}_\mathcal {D} := \max \Bigg ( \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}}, \left( \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \overline{q} + 1\right) \left\| \Pi ^h_\diamond \right\| _{\mathcal {L}\big (H^1(\Omega ),H^1(\Omega )\big )} \left\| \gamma ^{-1}\right\| _{\mathcal {L}\big (H^{1/2}(\partial \Omega ),H^1(\Omega )\big )}\Bigg )$.

Let u and $u^h$ be solutions to (2.4) and (3.6), respectively. Due to the standard theory of the finite element method (see, for example, [11, 18]), the estimate

$$\begin{aligned} \Vert u-u^h\Vert _{H^1(\Omega )} \le Ch\Vert u\Vert _{H^2(\Omega )} \end{aligned}$$

(3.10)

holds in case $u\in H^2(\Omega )$, where the positive constant C is independent of h and u.

Assume that v and $v^h$ are the solutions to (2.8) and (3.8), where $v\in H^2(\Omega )$, we then have (see, for example, [11, Section 5.4]) that

$$\begin{aligned} \Vert v-v^h\Vert _{H^1(\Omega )} \le \inf _{\psi ^h\in \mathcal {V}_{1,0}^h}\Vert v-\gamma ^{-1}g-\psi ^h\Vert _{H^1(\Omega )} + 2\Vert \gamma ^{-1}g - \Pi ^h_\diamond (\gamma ^{-1}g)\Vert _{H^1(\Omega )}. \end{aligned}$$

Since $v\in H^2(\Omega )$, it follows that $g=\gamma v\in H^{3/2}(\Omega )$ and so $\gamma ^{-1}g \in H^2(\Omega )$. Due to the approximation property of the finite dimensional spaces $\mathcal {V}_{1,0}^h \subset H^1_0(\Omega )$ (which states that $\inf _{\psi ^h\in \mathcal {V}_{1,0}^h}\Vert \psi -\psi ^h\Vert _{H^1(\Omega )} \le C h\Vert \psi \Vert _{H^2(\Omega )}$ for each $\psi \in H^2(\Omega )\cap H^1_0(\Omega )$, where the constant C is independent of h and $\psi$) and (3.4), we deduce

$$\begin{aligned} \Vert v-v^h\Vert _{H^1(\Omega )} \le Ch\left( \Vert v\Vert _{H^2(\Omega )} + \Vert \gamma ^{-1} g\Vert _{H^2(\Omega )}\right) . \end{aligned}$$

(3.11)

We also mention that above we approximate the Dirichlet boundary condition g by $g^h := \gamma \big (\Pi ^h_\diamond (\gamma ^{-1}g)\big )$. There exist some different choices for the approximation $g^h$; for example, the $L^2$-projection of g on the set $\mathcal {S}^h_{\partial \Omega } := \{\gamma \varphi ^h ~|~ \varphi ^h \in \mathcal {V}^h_1\}$, or the Lagrange interpolation of g in $\mathcal {S}^h_{\partial \Omega }$ in case g being smooth enough (see [23] for more details).

Definition 3.3

• (i) For any fixed $j\in H^{-1/2}_{-c_f}(\partial \Omega )$ the operator $\mathcal {N}^h: \mathcal {Q} \rightarrow \mathcal {V}_{1,\diamond }^h$ mapping each $q \in \mathcal {Q}$ to the unique solution $u^h =: \mathcal {N}^h_qj$ of the variational equation (3.6) is called the discrete Neumann operator.

• (ii) For any fixed $g\in H^{1/2}_\diamond (\partial \Omega )$ the operator $\mathcal {D}^h: \mathcal {Q} \rightarrow \mathcal {V}_{1,\diamond }^h$ mapping each $q \in \mathcal {Q}$ to the unique solution $v^h =: \mathcal {D}^h_q g$ of the variational equation (3.8) is called the discrete Dirichlet operator.

Next, the discretization space for the sought conductivity is defined by

$$\begin{aligned} \mathcal {Q}^h_{ad} := \mathcal {Q}\cap \mathcal {V}^h_1 \subset \mathcal {Q} \cap BV(\Omega ) = \mathcal {Q}_{ad}. \end{aligned}$$

Then, using the discrete operators $\mathcal {N}^h$ and $\mathcal {D}^h$ in Definition 3.3, we introduce the discrete cost functional

$$\begin{aligned} \Upsilon ^h_{\rho ,\delta } (q):= \mathcal {J}_\delta ^h (q) + \rho \int _\Omega \sqrt{\left| \nabla q\right| ^2+\epsilon ^h}, \end{aligned}$$

(3.12)

where $q \in \mathcal {Q}^h_{ad}$, $\epsilon ^h$ is a positive function of the mesh size h satisfying $\lim _{h\rightarrow 0} \epsilon ^h =0$ and

$$\begin{aligned} \mathcal {J}_\delta ^h (q):= \int _\Omega q\nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \text{ with } q\in \mathcal {Q}. \end{aligned}$$

(3.13)

The positive function $\epsilon ^h$ above acts as a smoothing parameter for the total variation.

Theorem 3.4

For any fixed h, $\rho$ and $\delta$ the minimization problem

attains a solution $q^h_{\rho ,\delta }$, which is called the discrete regularized solution of the identication problem.

Proof

We first note that $\mathcal {Q}^h_{ad}$ is a compact subset of the finite dimensional space $\mathcal {V}^h_1$. Let $\left( q_n\right) \subset \mathcal {Q}^h_{ad}$ be a minimizing sequence of the problem $\left( \mathcal {P}^h_{\rho ,\delta }\right)$, i.e.

$$\begin{aligned} \lim _{n\rightarrow \infty }\Upsilon ^h_{\rho ,\delta } \left( q_n\right) = \inf _{q\in \mathcal {Q}^h_{ad}} \Upsilon ^h_{\rho ,\delta } (q). \end{aligned}$$

(3.14)

Then a subsequence of $\left( q_n\right)$ which is denoted by the same symbol and an element $q \in \mathcal {Q}^h_{ad}$ exist such that $\left( q_n\right)$ converges to q in the $H^1(\Omega )$-norm. We have that

(3.15)

On the other hand, similarly to Lemma 2.4, we can prove that the sequence $\left( \mathcal {N}^h_{q_n}j_\delta ,\mathcal {D}^h_{q_n}g_\delta \right)$ converges to $\left( \mathcal {N}^h_{q}j_\delta ,\mathcal {D}^h_{q}g_\delta \right)$ in the $H^1(\Omega )\times H^1(\Omega )$-norm as n goes to $\infty$ and then obtain

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}^h_\delta \left( q_n\right) = \mathcal {J}^h_\delta (q). \end{aligned}$$

(3.16)

Thus, it follows from (3.14)–(3.16) that

$$\begin{aligned} \Upsilon ^h_{\rho ,\delta } (q) = \lim _{n\rightarrow \infty }\Upsilon ^h_{\rho ,\delta } \left( q_n\right) = \inf _{q\in \mathcal {Q}^h_{ad}} \Upsilon ^h_{\rho ,\delta } (q), \end{aligned}$$

which finishes the proof. $\square$

4 Convergence

From now on C is a generic positive constant which is independent of the mesh size h of $\mathcal {T}^h$, the noise level $\delta$ and the regularization parameter $\rho$. The following result shows the stability of the finite element method for the regularized identification problem.

Theorem 4.1

Let $(h_n)_n$ be a sequence with $\lim _{n\rightarrow \infty }h_n = 0$ and $\left( j_{\delta _n}, g_{\delta _n}\right)$ be a sequence in $H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ converging to $\left( j_\delta , g_\delta \right)$ in the $H^{-1/2}(\partial \Omega ) \times H^{1/2}(\partial \Omega )$-norm. For a fixed regularization parameter $\rho >0$ let $q^{h_n}_{\rho ,\delta _n} \in \mathcal {Q}^{h_n}_{ad}$ be a minimizer of $\left( \mathcal {P}^{h_n}_{\rho ,\delta _n} \right)$ for each $n\in \mathbb {N}$. Then a subsequence of $\big (q^{h_n}_{\rho ,\delta _n} \big )$ not relabelled and an element $q_{\rho ,\delta } \in \mathcal {Q}_{ad}$ exist such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\big \Vert q^{h_n}_{\rho ,\delta _n} - q_{\rho ,\delta }\big \Vert _{L^1(\Omega )} = 0 \text{ and } \lim _{n\rightarrow \infty }\int _\Omega \big |\nabla q^{h_n}_{\rho ,\delta _n} \big | = \int _\Omega \left| \nabla q_{\rho ,\delta } \right| . \end{aligned}$$

Furthermore, $q_{\rho ,\delta }$ is a solution to $\left( \mathcal {P}_{\rho ,\delta } \right)$.

To prove the theorem, we need the auxiliary results, starting with the following estimates.

Lemma 4.2

Let $(j_1,g_1)$ and $(j_2,g_2)$ be arbitrary in $H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$. Then the estimates

$$\begin{aligned} \left\| \mathcal {N}^{h}_{q} j_1 - \mathcal {N}^{h}_{q} j_2 \right\| _{H^1(\Omega )} \le \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \left\| \gamma \right\| _{\mathcal {L}\big (H^1(\Omega ),H^{1/2}(\partial \Omega )\big )} \left\| j_1 -j_2 \right\| _{H^{-1/2}(\partial \Omega )} \end{aligned}$$

(4.1)

and

$$\begin{aligned}&\left\| \mathcal {D}^{h}_{q} g_1 - \mathcal {D}^{h}_{q} g_2 \right\| _{H^1(\Omega )} \nonumber \\&\quad \le \left( \frac{1+ C^\Omega _\diamond }{C^\Omega _\diamond \underline{q}} \overline{q} + 1\right) \left\| \Pi ^h_\diamond \right\| _{\mathcal {L}\big (H^1(\Omega ),H^1(\Omega )\big )}\left\| \gamma ^{-1}\right\| _{\mathcal {L}\big (H^{1/2}(\partial \Omega ),H^1(\Omega )\big )} \left\| g_1 -g_2 \right\| _{H^{1/2}(\partial \Omega )} \end{aligned}$$

(4.2)

hold for all $q\in \mathcal {Q}$ and $h>0$.

Proof

According to the definition of the discrete Neumann operator, we have for all $\varphi ^{h}\in \mathcal {V}_{1,\diamond }^{h}$ that

$$\begin{aligned} \int _\Omega q \nabla \mathcal {N}^{h}_{q} j_i \cdot \nabla \varphi ^{h} = \left\langle j_i,\gamma \varphi ^{h}\right\rangle + \left( f,\varphi ^h\right) \text{ with } i=1,2. \end{aligned}$$

Thus, $\Phi ^{h}_{\mathcal {N}} := \mathcal {N}^{h}_{q} j_1 - \mathcal {N}^{h}_{q} j_2$ is the unique solution to the variational problem

$$\begin{aligned} \int _\Omega q \nabla \Phi ^{h}_{\mathcal {N}} \cdot \nabla \varphi ^{h} = \left\langle j_1-j_2,\gamma \varphi ^{h}\right\rangle \end{aligned}$$

for all $\varphi ^{h}\in \mathcal {V}_{1,\diamond }^{h}$ and so that (4.1) follows. Similarly, we also obtain (4.2), which finishes the proof. $\square$

Lemma 4.3

Let $(h_n)_n$ be a sequence with $\lim _{n\rightarrow \infty }h_n = 0$ and $\left( j_{\delta _n}, g_{\delta _n}\right) \subset H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ be a sequence converging to $\left( j_\delta , g_\delta \right)$ in the $H^{-1/2}(\partial \Omega ) \times H^{1/2}(\partial \Omega )$-norm. Then for any fixed $q\in \mathcal {Q}$ the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n} (q) = \mathcal {J}_{\delta } (q) \end{aligned}$$

(4.3)

holds. Furthermore, if $(q_n)$ is a sequence in $\mathcal {Q}$ which converges to q in the $L^1(\Omega )$-norm, then the sequence $\left( \mathcal {N}^{h_n}_{q_n}j_{\delta _n},\mathcal {D}^{h_n}_{q_n}g_{\delta _n}\right)$ converges to $\left( \mathcal {N}_{q}j_\delta ,\mathcal {D}_{q}g_\delta \right)$ in the $H^1(\Omega )\times H^1(\Omega )$-norm and the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n} (q_n) = \mathcal {J}_{\delta } (q) \end{aligned}$$

(4.4)

also holds.

Proof

We get for any fixed $q\in \mathcal {Q}$ that

$$\begin{aligned} \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {D}^{h_n}_{q} g_{\delta _n} = \left( \mathcal {N}_{q} j_{\delta } - \mathcal {D}_{q} g_{\delta }\right) + \left( \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {N}_{q} j_{\delta } + \mathcal {D}_{q} g_{\delta } - \mathcal {D}^{h_n}_{q} g_{\delta _n} \right) . \end{aligned}$$

Thus, with $\Phi _n := \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {N}_{q} j_{\delta } + \mathcal {D}_{q} g_{\delta } - \mathcal {D}^{h_n}_{q} g_{\delta _n}$ we have

$$\begin{aligned} \mathcal {J}^{h_n}_{\delta _n} (q)&= \int _\Omega q\nabla \left( \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {D}^{h_n}_{q} g_{\delta _n}\right) \cdot \nabla \left( \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {D}^{h_n}_{q} g_{\delta _n}\right) \\&= \mathcal {J}_{\delta } (q) +\int _\Omega q\nabla \Phi _n \cdot \nabla \Phi _n +2 \int _\Omega q\nabla \left( \mathcal {N}_{q} j_{\delta } - \mathcal {D}_{q} g_{\delta }\right) \cdot \nabla \Phi _n. \end{aligned}$$

Applying Lemma 4.2, we infer that

$$\begin{aligned} \left\| \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {N}_{q} j_{\delta } \right\| _{H^1(\Omega )}&\le \left\| \mathcal {N}^{h_n}_{q} j_{\delta } - \mathcal {N}_{q} j_{\delta } \right\| _{H^1(\Omega )} + \left\| \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {N}^{h_n}_{q} j_{\delta } \right\| _{H^1(\Omega )} \\&\le \left\| \mathcal {N}^{h_n}_{q} j_{\delta } - \mathcal {N}_{q} j_{\delta } \right\| _{H^1(\Omega )} \\&\quad + C \left\| j_{\delta _n} - j_{\delta } \right\| _{H^{-1/2}(\partial \Omega )} \rightarrow 0 \text{ as } n\rightarrow \infty , \end{aligned}$$

where we used the limit

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \mathcal {N}^{h_n}_{q} j_{\delta } - \mathcal {N}_{q} j_{\delta } \right\| _{H^1(\Omega )} =0, \end{aligned}$$

due to the standard theory (see, for example, [11, 18]). Similarly, we also have

$$\begin{aligned} \left\| \mathcal {D}^{h_n}_{q} g_{\delta _n} - \mathcal {D}_{q} g_{\delta } \right\| _{H^1(\Omega )} \rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

We thus get that

$$\begin{aligned} \Vert \Phi _n\Vert _{H^1(\Omega )}&\le \left\| \mathcal {N}^{h_n}_{q} j_{\delta _n} - \mathcal {N}_{q} j_{\delta } \right\| _{H^1(\Omega )} + \left\| \mathcal {D}_{q} g_{\delta } - \mathcal {D}^{h_n}_{q} g_{\delta _n} \right\| _{H^1(\Omega )} \rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

Therefore, we obtain that

$$\begin{aligned}&\lim _{n\rightarrow \infty }\left| \int _\Omega q\nabla \Phi _n \cdot \nabla \Phi _n +2 \int _\Omega q\nabla \left( \mathcal {N}_{q} j_{\delta } - \mathcal {D}_{q} g_{\delta }\right) \cdot \nabla \Phi _n \right| \\&\quad \le C \lim _{n\rightarrow \infty }\left( \Vert \Phi _n\Vert ^2_{H^1(\Omega )} + \Vert \Phi _n\Vert _{H^1(\Omega )}\right) =0 \end{aligned}$$

and (4.3) then follows.

Next, for $q_n$ converging to q in $L^1(\Omega )$, hence, along a subsequence again denoted by $(q_n)_n$, pointwise almost everywhere, by (3.6) and (2.4), we have

$$\begin{aligned} \int _\Omega q_n \nabla \mathcal {N}^{h_n}_{q_n}j_{\delta _n} \cdot \nabla \varphi ^{h_n}&= \big \langle j_{\delta _n},\gamma \varphi ^{h_n}\big \rangle + \big (f, \varphi ^{h_n}\big )= \big \langle j_{\delta },\gamma \varphi ^{h_n}\big \rangle + \big (f, \varphi ^{h_n}\big ) \\&\quad + \big \langle j_{\delta _n} - j_\delta ,\gamma \varphi ^{h_n}\big \rangle \\&= \int _\Omega q \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \varphi ^{h_n} + \big \langle j_{\delta _n} - j_\delta ,\gamma \varphi ^{h_n}\big \rangle \end{aligned}$$

for all $\varphi ^{h_n}\in \mathcal {V}_{1,\diamond }^{h_n}$ which implies that

$$\begin{aligned} \int _\Omega q_n \nabla&\left( \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right) \cdot \nabla \varphi ^{h_n} \nonumber \\&= \int _\Omega q \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \varphi ^{h_n} - \int _\Omega q_n \nabla \Pi _\diamond ^{h_n} \mathcal {N}_{q}j_\delta \cdot \nabla \varphi ^{h_n} + \big \langle j_{\delta _n} - j_\delta ,\gamma \varphi ^{h_n}\big \rangle \nonumber \\&= \int _\Omega \big (q-q_n\big ) \nabla \mathcal {N}_{q}j_\delta \cdot \nabla \varphi ^{h_n} + \int _\Omega q_n \nabla \left( \mathcal {N}_{q}j_\delta - \Pi _\diamond ^{h_n} \mathcal {N}_{q}j_\delta \right) \cdot \nabla \varphi ^{h_n} \nonumber \\&\quad + \big \langle j_{\delta _n} - j_\delta ,\gamma \varphi ^{h_n}\big \rangle , \end{aligned}$$

(4.5)

where the operator $\Pi _\diamond ^{h_n}$ is defined according to Lemma 3.1. Taking $\varphi ^{h_n} = \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \in \mathcal {V}_{1,\diamond }^h$, by (2.2) and using the Cauchy–Schwarz inequality, we get

$$\begin{aligned} \frac{C^\Omega _\diamond \underline{q}}{1+ C^\Omega _\diamond }&\left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right\| ^2_{H^1(\Omega )}\\&\le \left( \int _\Omega | q - q_n |^2 \left| \nabla \mathcal {N}_qj_\delta \right| ^2 \right) ^{1/2} \left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} \\&~\quad + \overline{q} \left\| \mathcal {N}_{q}j_\delta - \Pi _\diamond ^{h_n} \mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} \left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )}\\&~\quad + \left\| \gamma \right\| _{\mathcal {L}\big (H^1(\Omega ),H^{1/2}(\partial \Omega )\big )} \left\| j_{\delta _n} {-} j_\delta \right\| _{H^{-1/2}(\partial \Omega )} \left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} {-} \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} \end{aligned}$$

and so that

$$\begin{aligned}&\frac{C^\Omega _\diamond \underline{q}}{1+ C^\Omega _\diamond }\Big \Vert \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \Big \Vert _{H^1(\Omega )} \le \left( \int _\Omega | q - q_n |^2 \left| \nabla \mathcal {N}_qj_\delta \right| ^2 \right) ^{1/2} \\&\quad +\, \overline{q} \left\| \mathcal {N}_{q}j_\delta - \Pi _\diamond ^{h_n} \mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} + \left\| \gamma \right\| _{\mathcal {L}\big (H^1(\Omega ),H^{1/2}(\partial \Omega )\big )} \left\| j_{\delta _n} \right. \\&\quad \left. - j_\delta \right\| _{H^{-1/2}(\partial \Omega )}\rightarrow 0 \text{ as } n\rightarrow \infty , \end{aligned}$$

by the Lebesgue dominated convergence theorem and (3.1). Thus, we infer from the triangle inequality that

$$\begin{aligned} \left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )}&\le \left\| \mathcal {N}^{h_n}_{q_n}j_{\delta _n} - \Pi _\diamond ^{h_n}\mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )} \\&\quad + \left\| \Pi _\diamond ^{h_n} \mathcal {N}_{q}j_\delta -\mathcal {N}_{q}j_\delta \right\| _{H^1(\Omega )}\rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

Similarly, using (2.8) and (3.8), for all $\psi ^{h_n}\in \mathcal {V}_{1,0}^{h_n}$ we arrive at

$$\begin{aligned}&\int _\Omega q_n \nabla \left( \mathcal {D}^{h_n}_{q_n}g_{\delta } - \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta }\right) \cdot \nabla \psi ^{h_n}= \int _\Omega \big (q-q_n\big ) \nabla \mathcal {D}_{q}g_{\delta } \cdot \nabla \psi ^{h_n} \nonumber \\&\quad + \int _\Omega q_n \nabla \left( \mathcal {D}_{q}g_{\delta } - \Pi _\diamond ^{h_n} \mathcal {D}_{q}g_{\delta } \right) \cdot \nabla \psi ^{h_n}. \end{aligned}$$

(4.6)

We have

$$\begin{aligned} \gamma \mathcal {D}^{h_n}_{q_n}g_{\delta }=\gamma \big (\Pi ^{h_n}_\diamond (\gamma ^{-1}g_\delta )\big ), \end{aligned}$$

(4.7)

by Proposition 3.2 (ii). On the other hand, in view of (2.9), we get $\mathcal {D}_qg_\delta =v_0 +\gamma ^{-1}g_\delta$ with $v_0\in H^1_0(\Omega )$, and therefore

$$\begin{aligned} \gamma \big (\Pi _\diamond ^{h_n} \mathcal {D}_{q}g_{\delta }\big )&= \gamma \big (\Pi _\diamond ^{h_n} (v_0 + \gamma ^{-1} g_{\delta })\big ) = \gamma \big (\Pi _\diamond ^{h_n} v_0\big ) + \gamma \big (\Pi _\diamond ^{h_n} (\gamma ^{-1} g_{\delta })\big ) \nonumber \\&= \gamma \big (\Pi _\diamond ^{h_n} (\gamma ^{-1} g_{\delta })\big ), \end{aligned}$$

(4.8)

since $\gamma \big (\Pi _\diamond ^{h_n} v_0\big )=0$. It follows from (4.7)–(4.8) that

$$\begin{aligned} \psi ^{h_n}_* : = \mathcal {D}^{h_n}_{q_n}g_{\delta } - \Pi _\diamond ^{h_n} \mathcal {D}_{q}g_{\delta } \in \mathcal {V}_{1,0}^{h_n}. \end{aligned}$$

Taking $\psi ^{h_n} := \psi ^{h_n}_*$ in the above equation (4.6), it is deduced that

$$\begin{aligned} \lim _{n\rightarrow \infty }\left\| \mathcal {D}^{h_n}_{q_n}g_{\delta } - \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta } \right\| _{H^1(\Omega )} = 0. \end{aligned}$$

Using Lemma 4.2, we therefore obtain that

$$\begin{aligned} \Big \Vert \mathcal {D}^{h_n}_{q_n}g_{\delta _n}&- \mathcal {D}_{q}g_{\delta } \Big \Vert _{H^1(\Omega )} \\&\le \left\| \mathcal {D}^{h_n}_{q_n}g_{\delta _n} - \mathcal {D}^{h_n}_{q_n}g_{\delta } \right\| _{H^1(\Omega )} + \left\| \mathcal {D}^{h_n}_{q_n}g_{\delta } - \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta } \right\| _{H^1(\Omega )}\\&\quad + \left\| \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta } - \mathcal {D}_{q}g_{\delta } \right\| _{H^1(\Omega )}\\&\le C\left\| g_{\delta _n} - g_\delta \right\| _{H^{1/2}(\partial \Omega )} + \left\| \mathcal {D}^{h_n}_{q_n}g_{\delta } - \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta } \right\| _{H^1(\Omega )}\\{}&\quad + \left\| \Pi _\diamond ^{h_n}\mathcal {D}_{q}g_{\delta } - \mathcal {D}_{q}g_{\delta } \right\| _{H^1(\Omega )}\\&\rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

Since $\big ( q_n\big )$ converges to q in the $L^1(\Omega )$-norm while the sequence $\left( \mathcal {N}^{h_n}_{q_n}j_{\delta _n},\mathcal {D}^{h_n}_{q_n}g_{\delta _n}\right)$ converges to $\left( \mathcal {N}_{q}j_\delta ,\mathcal {D}_{q}g_\delta \right)$ in the $H^1(\Omega )\times H^1(\Omega )$-norm, we conclude, similarly to the proof of Lemma 2.4 that

$$\begin{aligned} \lim _{n\rightarrow \infty }\mathcal {J}_{\delta _n}^{h_n} \big (q_n\big ) = \mathcal {J}_\delta (q), \end{aligned}$$

which finishes the proof. $\square$

Proof of Theorem 4.1

To simplify notation we write $q_n := q^{h_n}_{\rho ,\delta _n}$. Let $q \in \mathcal {Q}_{ad}$ be arbitrary. Using Lemma 2.6, for any fixed $\alpha \in (0,1)$ an element $q^\alpha \in C^\infty (\Omega )$ exists such that

$$\begin{aligned} \left\| q-q^\alpha \right\| _{L^1(\Omega )} \le \overline{C}\alpha \text{ and } \int _\Omega \left| \nabla q^\alpha \right| \le \overline{C}\alpha +\int _\Omega \left| \nabla q\right| , \end{aligned}$$

(4.9)

where the positive constant $\overline{C}$ is independent of $\alpha$. Setting

$$\begin{aligned} q^\alpha _P := \max \left( \underline{q}, \min \left( q^\alpha , \overline{q} \right) \right) \in W^{1,\infty }(\Omega )\cap \mathcal {Q} \subset \mathcal {Q}_{ad} \text{ and } q^\alpha _n := I^{h_n}_1 q^\alpha _P \in \mathcal {Q}^{h_n}_{ad}, \end{aligned}$$

where

$$\begin{aligned} I^h_1 : W^{1,p}(\Omega )\hookrightarrow C(\overline{\Omega }) \rightarrow \mathcal {V}^h_1 \text{ with } p>d \end{aligned}$$

is the usual nodal value interpolation operator. Since the sequence $\big (q^\alpha _n\big )$ converges to $q^\alpha _P$ in the $H^1(\Omega )$-norm as n tends to $\infty$ (see, for example, [11, 18]), we get the equation

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega \sqrt{\left| \nabla q^\alpha _n\right| ^2+\epsilon ^{h_n}} = \int _\Omega |\nabla q^\alpha _P|. \end{aligned}$$

(4.10)

Indeed, we have that

$$\begin{aligned}&\left| \int _\Omega \sqrt{\left| \nabla q^\alpha _n\right| ^2+\epsilon ^{h_n}} - \int _\Omega \left| \nabla q^\alpha _n\right| \right| \le \int _\Omega \frac{\epsilon ^{h_n}}{\sqrt{\left| \nabla q^\alpha _n\right| ^2+\epsilon ^{h_n}} + \left| \nabla q^\alpha _n\right| } \\&\quad \le |\Omega |\sqrt{\epsilon ^{h_n}} \ \rightarrow 0 \text{ as } n\rightarrow \infty \end{aligned}$$

and by the reverse triangle as well as the Cauchy–Schwarz inequality

$$\begin{aligned} \left| \int _\Omega \left| \nabla q^\alpha _n\right| - \int _\Omega |\nabla q^\alpha _P| \right|&\le \left\| \nabla q^\alpha _n - \nabla q^\alpha _P\right\| _{L^1(\Omega )} \le |\Omega |^{1/2} \left\| \nabla q^\alpha _n - \nabla q^\alpha _P\right\| _{L^2(\Omega )} \\&\rightarrow 0 \text{ as } n\rightarrow \infty \end{aligned}$$

so that (4.10) follows from the triangle inequality. By (4.9) and the fact that $q^\alpha _P$ is constant on $\{x\in \Omega ~|~ q^\alpha _P(x) \not = q^\alpha (x)\}$, we have that

$$\begin{aligned} \int _\Omega |\nabla q^\alpha _P|&= \int _{ \{x\in \Omega ~|~ q^\alpha _P(x) = q^\alpha (x)\}} |\nabla q^\alpha _P| \le \int _\Omega |\nabla q^\alpha | \le \overline{C}\alpha + \int _\Omega |\nabla q|. \end{aligned}$$

(4.11)

By the optimality of $q_n$, we get for all $n\in \mathbb {N}$ that

$$\begin{aligned} \mathcal {J}^{h_n}_{\delta _n} \left( q_n\right) + \rho \int _{\Omega } \sqrt{|\nabla q_n|^2 + \epsilon ^{h_n}} \le \mathcal {J}^{h_n}_{\delta _n} \left( q^\alpha _n\right) + \rho \int _{\Omega } \sqrt{|\nabla q^\alpha _n|^2 + \epsilon ^{h_n}}, \end{aligned}$$

(4.12)

where, by (3.7) and (3.9),

$$\begin{aligned} \mathcal {J}^{h_n}_{\delta _n} \left( q^\alpha _n\right) \le C \end{aligned}$$

holds for some C independent of n and $\alpha$. We then deduce from (4.10)–(4.12) that

$$\begin{aligned} \int _{\Omega } |\nabla q_n| \le \int _{\Omega } \sqrt{|\nabla q_n|^2 + \epsilon ^{h_n}}\le C(\rho ) \end{aligned}$$

for another constant $C(\rho )$ independent of n and $\alpha$, but depending on $\rho$, so the sequence $\left( q_n\right)$ is bounded in the $BV(\Omega )$-norm. Thus, by Lemma 2.5, a subsequence which is denoted by the same symbol and an element $\widehat{q}\in \mathcal {Q}_{ad}$ exist such that $\left( q_n\right)$ converges to $\widehat{q}$ in the $L^1(\Omega )$-norm and

$$\begin{aligned} \int _\Omega \left| \nabla \widehat{q}\right| \le \liminf _{n\rightarrow \infty }\int _\Omega \left| \nabla q_n\right| \le \liminf _{n\rightarrow \infty }\int _\Omega \sqrt{\left| \nabla q_n\right| ^2+\epsilon ^{h_n}}. \end{aligned}$$

(4.13)

Furthermore, due to Lemma 4.3 we get that

$$\begin{aligned} \mathcal {J}_\delta (\widehat{q}) = \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n}\left( q_n\right) \end{aligned}$$

(4.14)

and

$$\begin{aligned} \mathcal {J}_\delta \left( q^\alpha _P \right) = \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n}\left( q^\alpha _n\right) . \end{aligned}$$

(4.15)

Therefore, by (4.10)–(4.15), we have that

$$\begin{aligned} \mathcal {J}_\delta (\widehat{q}) +\rho \int _\Omega \left| \nabla \widehat{q}\right|&\le \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n}\left( q_n\right) + \liminf _{n\rightarrow \infty }\rho \int _\Omega \sqrt{\left| \nabla q_n\right| ^2+\epsilon ^{h_n}}, \text{ by } 4.14 \hbox { and } 4.13 \nonumber \\&=\liminf _{n\rightarrow \infty }\left( \mathcal {J}^{h_n}_{\delta _n}\left( q_n\right) + \rho \int _\Omega \sqrt{\left| \nabla q_n\right| ^2+\epsilon ^{h_n}}\right) \nonumber \\&\le \liminf _{n\rightarrow \infty }\left( \mathcal {J}^{h_n}_{\delta _n} \left( q^\alpha _n\right) + \rho \int _{\Omega } \sqrt{|\nabla q^\alpha _n|^2 + \epsilon ^{h_n}}\right) , \text{ by } 4.12 \nonumber \\{}&= \mathcal {J}_\delta \left( q^\alpha _P \right) + \rho \int _\Omega |\nabla q^\alpha _P|, \text{ by } 4.15 \hbox { and } 4.10 \nonumber \\&\le \mathcal {J}_\delta \left( q^\alpha _P \right) + \rho \int _\Omega |\nabla q | +\overline{C}\alpha \rho , \text{ by } 4.11. \end{aligned}$$

(4.16)

Now, by the definition of $q^\alpha _P$, we get $|q^\alpha _P -q| \le |q^\alpha - q|$ a.e. in $\Omega$ and therefore

$$\begin{aligned} \left\| q^\alpha _P - q\right\| _{L^1(\Omega )} \le \left\| q^\alpha - q\right\| _{L^1(\Omega )} \le \overline{C}\alpha . \end{aligned}$$

Sending $\alpha$ to zero in the last inequality and applying Lemma 2.4, we arrive at

$$\begin{aligned} \mathcal {J}_\delta (\widehat{q}) +\rho \int _\Omega \left| \nabla \widehat{q}\right| \le \mathcal {J}_\delta (q) +\rho \int _\Omega \left| \nabla q\right| , \end{aligned}$$

where $\widehat{q}\in \mathcal {Q}_{ad}$ and $q\in \mathcal {Q}_{ad}$ is arbitrary. This means that $\widehat{q}$ is a solution to $\left( \mathcal {P}_{\rho ,\delta } \right)$ and $\left( q_n\right)$ converges to $\widehat{q}$ in the $L^1(\Omega )$-norm.

Next, as above, from $\widehat{q}$ we can obtain $\widehat{q}^\alpha$, $\widehat{q}^\alpha _P$, $\widehat{q}^\alpha _n$ and note that $\big (\widehat{q}^\alpha _n\big )$ converges to $\widehat{q}^\alpha _P$ in the $H^1(\Omega )$-norm, so also in the $L^1(\Omega )$-norm, as n tends to $\infty$ while $\big (\widehat{q}^\alpha _P\big )$ converges to $\widehat{q}$ in the $L^1(\Omega )$-norm as $\alpha$ tends to 0. Then, by the optimality of $q_n$, we have that

$$\begin{aligned} \mathcal {J}^{h_n}_{\delta _n} \left( q_n\right) + \rho \int _{\Omega } \sqrt{|\nabla q_n|^2 + \epsilon ^{h_n}}&\le \mathcal {J}^{h_n}_{\delta _n} \left( \widehat{q}^\alpha _n\right) + \rho \int _{\Omega } \sqrt{|\nabla \widehat{q}^\alpha _n|^2 + \epsilon ^{h_n}}. \end{aligned}$$

(4.17)

By (4.14), we then obtain that

$$\begin{aligned} \rho \limsup _{n\rightarrow \infty }\int _\Omega \left| \nabla q_n\right|&= \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n}\left( q_n\right) + \rho \limsup _{n\rightarrow \infty }\int _\Omega \left| \nabla q_n\right| - \mathcal {J}_\delta (\widehat{q})\\&\le \limsup _{n\rightarrow \infty }\left( \mathcal {J}^{h_n}_{\delta _n}\left( q_n\right) + \rho \int _\Omega \sqrt{\left| \nabla q_n\right| ^2 + \epsilon ^{h_n}} \right) - \mathcal {J}_\delta (\widehat{q}) \\&\le \limsup _{n\rightarrow \infty }\left( \mathcal {J}^{h_n}_{\delta _n} \left( \widehat{q}^\alpha _n\right) + \rho \int _{\Omega } \sqrt{|\nabla \widehat{q}^\alpha _n|^2 + \epsilon ^{h_n}}\right) - \mathcal {J}_\delta (\widehat{q}), \text{ by } 4.17\\&= \mathcal {J}_\delta (\widehat{q}^\alpha _P) + \rho \int _\Omega |\nabla \widehat{q}^\alpha _P| - \mathcal {J}_\delta (\widehat{q}), \text{ by } \text{ Lemma } 4.3\\&\le \mathcal {J}_\delta (\widehat{q}^\alpha _P) + \rho \int _\Omega |\nabla \widehat{q}| +\overline{C}\alpha \rho - \mathcal {J}_\delta (\widehat{q}). \end{aligned}$$

Sending $\alpha$ to zero, we obtain from the last inequality that $\limsup _{n\rightarrow \infty }\int _\Omega \left| \nabla q_n\right| \le \int _\Omega |\nabla \widehat{q}|$. Combining this with (4.13), we conclude $\lim _{n\rightarrow \infty }\int _\Omega \left| \nabla q_n\right| = \int _\Omega |\nabla \widehat{q}|$, which finishes the proof. $\square$

Next we show convergence of the regularized finite element approximations to a solution of the identification problem. Before doing so, we introduce the notion of the total variation-minimizing solution.

Lemma 4.4

The problem

attains a solution, which is called the total variation-minimizing solution of the identification problem, where

$$\begin{aligned} \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right) :=\left\{ q\in \mathcal {Q}_{ad} ~\big |~ \Lambda _qj^\dag = g^\dag \right\} = \left\{ q\in \mathcal {Q}_{ad} ~\big |~ \mathcal {N}_qj^\dag = \mathcal {D}_qg^\dag \right\} . \end{aligned}$$

(4.18)

Proof

By our assumption on consistency of the exact boundary data, the set $\mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right)$ is non-empty. Let $\left( q_n\right) \subset \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right)$ be a minimizing sequence of the problem $\left( \mathcal {IP}\right)$, i.e.

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _{\Omega }\left| \nabla q_n\right| = \inf _{q\in \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right) } \int _{\Omega }\left| \nabla q\right| . \end{aligned}$$

(4.19)

Then due to Lemma 2.5, a subsequence which is denoted by the same symbol and an element $\tilde{q}\in \mathcal {Q}_{ad}$ exist such that $\left( q_n\right)$ converges to $\tilde{q}$ in the $L^1(\Omega )$-norm and

$$\begin{aligned} \int _\Omega \left| \nabla \tilde{q}\right| \le \lim _{n\rightarrow \infty }\int _\Omega |\nabla q_n|. \end{aligned}$$

(4.20)

On the other hand, by Lemma 2.4, we have that

$$\begin{aligned} \left( \mathcal {N}_{q_n}j^\dag ,\mathcal {D}_{q_n}g^\dag \right) \rightarrow \left( \mathcal {N}_{\tilde{q}}j^\dag ,\mathcal {D}_{\tilde{q}}g^\dag \right) \text{ in } \text{ the } H^1(\Omega )\times H^1(\Omega )\text{-norm }. \end{aligned}$$

By the definition of the set $\mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right)$, we get that $\mathcal {N}_{q_n}j^\dag =\mathcal {D}_{q_n}g^\dag$ which implies $\mathcal {N}_{\tilde{q}}j^\dag =\mathcal {D}_{\tilde{q}}g^\dag$. Combining this with (4.19) and (4.20), we conclude that

$$\begin{aligned} \int _{\Omega }\left| \nabla \tilde{q}\right| \le \inf _{q\in \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right) } \int _{\Omega }\left| \nabla q\right| , \end{aligned}$$

where $\tilde{q}\in \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right)$, which finishes the proof. $\square$

Remark 4.5

Note that due to the lack of strict convexity of the cost functional and the admissible set, a solution of $\left( \mathcal {IP}\right)$ may be nonunique.

Lemma 4.6

For any fixed $q\in \mathcal {Q}_{ad}$ an element $\widehat{q}^h \in \mathcal {Q}^h_{ad}$ exists such that

$$\begin{aligned} \big \Vert \widehat{q}^h-q\big \Vert _{L^1(\Omega )} \le Ch|\log h| \end{aligned}$$

(4.21)

and

$$\begin{aligned} \lim _{h\rightarrow 0}\int _\Omega \big |\nabla \widehat{q}^h\big | = \int _\Omega |\nabla q|. \end{aligned}$$

(4.22)

In case $q\in W^{1,p}(\Omega )\hookrightarrow C(\overline{\Omega })$ with $p>d$ the above element $\widehat{q}^h$ can be taken as $I^h_1q$.

Proof

According to Lemma 2.6, for any fixed $\alpha \in (0,1)$ an element $q^\alpha \in C^\infty (\Omega )$ exists such that

$$\begin{aligned} \left\| q{-}q^\alpha \right\| _{L^1(\Omega )} {\le } \overline{C}\alpha ,~ \int _\Omega \left| \nabla q^\alpha \right| \le \overline{C}\alpha {+}\int _\Omega \left| \nabla q\right| \text{ and } \int _\Omega |D^2 q^\alpha | \le \overline{C}\alpha ^{-1}\int _\Omega |\nabla q|, \end{aligned}$$

where the positive constant $\overline{C}$ is independent of $\alpha$. Setting

$$\begin{aligned} q^\alpha _P := \max \left( \underline{q}, \min \left( q^\alpha , \overline{q} \right) \right) \in W^{1,\infty }(\Omega )\cap \mathcal {Q} \subset \mathcal {Q}_{ad} \text{ and } \widehat{q}^h := I^h_1 q^\alpha _P \in \mathcal {Q}^h_{ad}, \end{aligned}$$

we then have

$$\begin{aligned} \big | \widehat{q}^h(x) - q(x)\big | = \big | I^h_1q^\alpha (x) - q(x)\big | \text{ a.e. } \text{ in } \Omega _1 := \{x\in \Omega ~|~ \underline{q} \le q^\alpha \le \overline{q}\} \end{aligned}$$

and

$$\begin{aligned} \big | \widehat{q}^h(x) - q(x)\big | \le \left| q^\alpha (x) - q(x)\right| \text{ a.e. } \text{ in } \Omega \setminus \Omega _1. \end{aligned}$$

We thus have, using for example [11, Theorem 4.4.20], with an another positive constant C independent of $\alpha$ that

$$\begin{aligned} \big \Vert \widehat{q}^h-q\big \Vert _{L^1(\Omega )}&\le \big \Vert I^h_1 q^\alpha -q\big \Vert _{L^1(\Omega _1)} +\left\| q^\alpha -q\right\| _{L^1(\Omega \setminus \Omega _1)}\nonumber \\&\le \big \Vert I^h_1 q^\alpha -q^\alpha \big \Vert _{L^1(\Omega )} + \left\| q-q^\alpha \right\| _{L^1(\Omega _1)}+\left\| q^\alpha -q\right\| _{L^1(\Omega \setminus \Omega _1)} \\&\le Ch \int _\Omega \left| \nabla q^\alpha \right| + \left\| q-q^\alpha \right\| _{L^1(\Omega )}\\&\le Ch \left( \overline{C}\alpha + \int _\Omega \left| \nabla q\right| \right) +\overline{C}\alpha \le C(h +\alpha )\\&\le Ch|\log h| \end{aligned}$$

for $\alpha \sim h|\log h|$. To establish the limit (4.22) we first note that

$$\begin{aligned} \int _\Omega \big | \nabla I^h_1 q^\alpha _P\big | \le \int _\Omega \big | \nabla I^h_1 q^\alpha \big |. \end{aligned}$$

(4.23)

Indeed, we rewrite

$$\begin{aligned} \int _\Omega \big | \nabla I^h_1 q^\alpha _P\big | {=} \sum _{T\in \mathcal {T}^h_1}\int _T \big | \nabla I^h_1 q^\alpha _P\big | {+} \sum _{T\in \mathcal {T}^h_2}\int _T \big | \nabla I^h_1 q^\alpha _P\big | + \sum _{T\in \mathcal {T}^h \setminus \big (\mathcal {T}^h_1 \cup \mathcal {T}^h_2\big )}\int _T \big | \nabla I^h_1 q^\alpha _P\big |, \end{aligned}$$

(4.24)

where $\mathcal {T}^h_1$ includes all triangles $T\in \mathcal {T}^h$ with its vertices $x_1, \ldots , x_d,x_{d+1}$ at which

$$\begin{aligned} \text{ either } q^\alpha (x_1), \ldots , q^\alpha (x_{d+1}) < \underline{q} \text{ or } q^\alpha (x_1), \ldots , q^\alpha (x_{d+1}) > \overline{q} \end{aligned}$$

while $\mathcal {T}^h_2$ consists all triangles $T\in \mathcal {T}^h$ with its vertices $x_1, \ldots , x_d,x_{d+1}$ at which

$$\begin{aligned} q^\alpha (x_1), \ldots , q^\alpha (x_{d+1}) \in [ \underline{q}, \overline{q}]. \end{aligned}$$

We then have that

$$\begin{aligned} \sum _{T\in \mathcal {T}^h_1}\int _T \big | \nabla I^h_1 q^\alpha _P\big | =0 \text{ and } \sum _{T\in \mathcal {T}^h_2}\int _T \big | \nabla I^h_1 q^\alpha _P\big | = \sum _{T\in \mathcal {T}^h_2}\int _T \big | \nabla I^h_1 q^\alpha \big |. \end{aligned}$$

(4.25)

Now let $T\in \mathcal {T}^h \setminus \big (\mathcal {T}^h_1 \cup \mathcal {T}^h_2\big )$ be arbitrary. In Cartesian coordinate system Oxz with $x\in \mathbb {R}^d$ we consider plane surfaces $z=I^h_1 q^\alpha _P(x)$ and $z=I^h_1 q^\alpha (x)$ with $x\in T$ and denote by ${m}_P$ and ${m}$ the constant unit normals on these surfaces in the upward z direction, respectively. By the definition of the projection $q^\alpha _P$, we get $0< \widehat{(Oz,{m}_P)} \le \widehat{(Oz,{m})} < \pi /2$ and so that $0< \cos \widehat{(Oz,{m})} \le \cos \widehat{(Oz,{m}_P)} <1.$ Since

$$\begin{aligned} \cos \widehat{(Oz,{m})} = \frac{1}{\sqrt{\big | \nabla I^h_1 q^\alpha \big |^2 + 1}} \text{ and } \cos \widehat{(Oz,{m}_P)} = \frac{1}{\sqrt{\big | \nabla I^h_1 q^\alpha _P\big |^2 + 1}}, \end{aligned}$$

it follows that $\big | \nabla I^h_1 q^\alpha (x)\big | \ge \big | \nabla I^h_1 q^\alpha _P(x)\big |$ for all $x\in T$. We thus have that

$$\begin{aligned} \sum _{T\in \mathcal {T}^h \setminus \big (\mathcal {T}^h_1 \cup \mathcal {T}^h_2\big )}\int _T \big | \nabla I^h_1 q^\alpha _P\big | \le \sum _{T\in \mathcal {T}^h \setminus \big (\mathcal {T}^h_1 \cup \mathcal {T}^h_2\big )}\int _T \big | \nabla I^h_1 q^\alpha \big |. \end{aligned}$$

(4.26)

The inequality (4.23) is then directly deduced from (4.24)–(4.26). We therefore have with a constant C independent of $\alpha$ that

$$\begin{aligned} \int _\Omega \big | \nabla \widehat{q}^h\big | - \int _\Omega \big | \nabla q\big |&= \int _\Omega \big | \nabla I^h_1 q^\alpha _P\big | - \int _\Omega \big | \nabla q\big | \le \int _\Omega \big | \nabla I^h_1 q^\alpha \big | - \int _\Omega \big | \nabla q\big | \\&\le \int _\Omega \big | \nabla \big ( I^h_1 q^\alpha -q^\alpha \big ) \big | + \int _\Omega \big | \nabla q^\alpha \big |- \int _\Omega \big | \nabla q\big | \\&\le Ch \int _\Omega \big | D^2 q^\alpha \big | + \overline{C}\alpha \\&\le C\overline{C}h\alpha ^{-1} \int _\Omega |\nabla q| + \overline{C}\alpha \\&\le C\left( |\log h|^{-1} + h|\log h|\right) \rightarrow 0 \text{ as } h\rightarrow 0 \text{ and } \text{ for } \alpha \sim h|\log h|. \end{aligned}$$

Combining this with (4.21) and Lemma 2.5, we obtain that

$$\begin{aligned} \int _\Omega | \nabla q| \le \liminf _{h\rightarrow 0} \int _\Omega \big | \nabla \widehat{q}^h\big | \le \limsup _{h\rightarrow 0} \int _\Omega \big | \nabla \widehat{q}^h\big |\le \int _\Omega | \nabla q|, \end{aligned}$$

which finishes the proof. $\square$

Lemma 4.7

Let $(q,j,g) \in \mathcal {Q}_{ad} \times H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ be arbitrary. Then the convergence

$$\begin{aligned} \widehat{\varrho }^h_{q} \left( j,g\right) := \left\| \mathcal {N}^h_{\widehat{q}^h}j - \mathcal {N}_q j \right\| _{H^1(\Omega )} + \left\| \mathcal {D}^h_{\widehat{q}^h}g - \mathcal {D}_q g \right\| _{H^1(\Omega )} \rightarrow 0 \text{ as } h\rightarrow 0 \end{aligned}$$

holds, where $\widehat{q}^h$ is generated from q according to Lemma 4.6.

Proof

The assertion follows directly from Lemmas 4.3 and 4.6. $\square$

Additional smoothness assumptions enable an error estimate of $\widehat{\varrho }^h_{q} \left( j,g\right)$.

Lemma 4.8

Let $(q,j,g) \in \mathcal {Q}_{ad} \times H^{-1/2}_{-c_f}(\partial \Omega ) \times H^{1/2}_\diamond (\partial \Omega )$ be arbitrary. Assume that $\mathcal {N}_{ q}j, \mathcal {D}_{ q}g \in H^2(\Omega )$. Then

$$\begin{aligned} \widehat{\varrho }^h_{q} \left( j,g\right) \le C_r \big ( h|\log h|\big )^r \text{ with } {\left\{ \begin{array}{ll} r<1/2&{} \text{ if } d=2 \text{ and } \\ r=1/3&{} \text{ if } d=3. \end{array}\right. } \end{aligned}$$

(4.27)

Proof

Due to Lemma 3.1, since $\mathcal {N}_{ q}j \in H^2(\Omega )$, we get that

$$\begin{aligned} \left\| \mathcal {N}_{q}j - \Pi _\diamond ^h \mathcal {N}_{q}j \right\| _{H^1(\Omega )} \le Ch. \end{aligned}$$

(4.28)

Furthermore, it follows from Lemma 4.6 that

$$\begin{aligned} \big \Vert q - \widehat{q}^h\big \Vert _{L^p(\Omega )}&=\left( \int _\Omega |q-\widehat{q}^h|\, |q-\widehat{q}^h|^{p-1}\right) ^{1/p} \le \left( (2\overline{q})^{p-1}\, Ch|\log h| \right) ^{1/p} \nonumber \\&\le C\big (h|\log h|\big )^{1/p} \end{aligned}$$

(4.29)

for $p\in [1,\infty )$. Like in (4.5), using (3.6) and (2.4), we infer that

$$\begin{aligned} \int _\Omega \widehat{q}^h \nabla \mathcal {N}^h_{\widehat{q}^h}j \cdot \nabla \varphi ^h = \big \langle j,\gamma \varphi ^h\big \rangle + \left( f,\varphi ^h\right) = \int _\Omega q \nabla \mathcal {N}_{q}j \cdot \nabla \varphi ^h \end{aligned}$$

for all $\varphi ^h\in \mathcal {V}_{1,\diamond }^h$ and obtain that

$$\begin{aligned} \int _\Omega \widehat{q}^h \nabla \left( \mathcal {N}^h_{\widehat{q}^h}j - \Pi _\diamond ^h \mathcal {N}_{q}j\right) \cdot \nabla \varphi ^h&= \int _\Omega \big ( q - \widehat{q}^h\big ) \nabla \mathcal {N}_{q}j \cdot \nabla \varphi ^h\nonumber \\&\quad + \int _\Omega \widehat{q}^h \nabla \left( \mathcal {N}_{q}j - \Pi _\diamond ^h \mathcal {N}_{q}j\right) \cdot \nabla \varphi ^h. \end{aligned}$$

(4.30)

Since $H^2(\Omega )$ is embedded in $W^{1,s}(\Omega )$ with

$$\begin{aligned} s{\left\{ \begin{array}{ll} <\infty &{} \text{ if } d=2\\ =6&{} \text{ if } d=3 \end{array}\right. } \end{aligned}$$

(4.31)

(see, for example, [1, Theorem 5.4]), it follows from Cauchy–Schwarz and Hölder’s inequality that

$$\begin{aligned} \int _\Omega \big ( q - \widehat{q}^h\big ) \nabla \mathcal {N}_{q}j \cdot \nabla \varphi ^h&\le \left( \int _\Omega \big ( q - \widehat{q}^h\big )^2 |\nabla \mathcal {N}_{q}j|^2 \right) ^{1/2} \left( \int _\Omega \big |\nabla \varphi ^h\big |^2 \right) ^{1/2}\\&\le \big \Vert q - \widehat{q}^h\big \Vert _{L^{2s/(s-2)}(\Omega )} \big \Vert \nabla \mathcal {N}_{q}j\big \Vert _{L^s(\Omega )} \big \Vert \varphi ^h\big \Vert _{H^1(\Omega )}\\&\le C \big \Vert q - \widehat{q}^h\big \Vert _{L^{2s/(s-2)}(\Omega )} \big \Vert \varphi ^h\big \Vert _{H^1(\Omega )}. \end{aligned}$$

Then taking $\varphi ^h = \mathcal {N}^h_{\widehat{q}^h}j - \Pi _\diamond ^h \mathcal {N}_{q}j \in \mathcal {V}_{1,\diamond }^h$ and using (2.2), we infer from (4.30) that

$$\begin{aligned} \left\| \mathcal {N}^h_{\widehat{q}^h}j - \Pi _\diamond ^h \mathcal {N}_{q}j\right\| _{H^1(\Omega )}&\le C \left( \big \Vert q - \widehat{q}^h\big \Vert _{L^{2s/(s-2)}(\Omega )} + \left\| \mathcal {N}_{q}j - \Pi _\diamond ^h \mathcal {N}_{q}j \right\| _{H^1(\Omega )} \right) \\&\le C\big (h|\log h|\big )^{(s-2)/(2s)} + Ch \ \le C\big (h|\log h|\big )^{(s-2)/(2s)}, \end{aligned}$$

by (4.28)–(4.29). Thus, applying the triangle inequality and (4.28) again, we infer that

$$\begin{aligned} \left\| \mathcal {N}_{q}j - \mathcal {N}^h_{\widehat{q}^h}j \right\| _{H^1(\Omega )}&\le \left\| \mathcal {N}_{q}j - \Pi _\diamond ^h \mathcal {N}_{q}j \right\| _{H^1(\Omega )} + \left\| \Pi _\diamond ^h \mathcal {N}_{q}j - \mathcal {N}^h_{\widehat{q}^h}j\right\| _{H^1(\Omega )} \\&\le C\big (h|\log h|\big )^{(s-2)/(2s)}. \end{aligned}$$

Similarly, we also get $\left\| \mathcal {D}_{q} g - \mathcal {D}^h_{\widehat{q}^h} g \right\| _{H^1(\Omega )} \le C\big (h|\log h|\big )^{(s-2)/(2s)}$ and so that

$$\begin{aligned} \widehat{\varrho }^h_{q} \left( j,g\right) \le C\big (h|\log h|\big )^{(s-2)/(2s)} \end{aligned}$$

for s as in (4.31), which yields the assertion. $\square$

With an appropriate a priori choice of the regularization parameter we get convergence under conditions similar to those stated, e.g., in [35] in the Hilbert space setting.

Theorem 4.9

Let $\left( h_n\right) _n$, $\left( \delta _n\right) _n$ and $\left( \rho _n\right) _n$ be any positive sequences such that

$$\begin{aligned} \rho _n\rightarrow 0, ~\frac{\delta _n}{\sqrt{\rho _n}} \rightarrow 0 \text{ and } \frac{\widehat{\varrho }^{h_n}_{q} \left( j^\dag ,g^\dag \right) }{\sqrt{\rho _n}} \rightarrow 0 \text{ as } n\rightarrow \infty , \end{aligned}$$

(4.32)

where q is any solution to $\mathcal {N}_q j^\dag = \mathcal {D}_q g^\dag$. Moreover, assume that $\big (j_{\delta _n}, g_{\delta _n}\big )$ is a sequence satisfying

$$\begin{aligned} \big \Vert j_{\delta _n} - j^\dag \big \Vert _{H^{-1/2}(\partial \Omega )} + \big \Vert g_{\delta _n} - g^\dag \big \Vert _{H^{1/2}(\partial \Omega )} \le \delta _n \end{aligned}$$

and that $q_n := q_{\rho _n, \delta _n}^{h_n}$ is an arbitrary minimizer of $\left( \mathcal {P}_{\rho _n,\delta _n}^{h_n} \right)$ for each $n\in \mathbb {N}$. Then a subsequence of $(q_n)$ which is not relabelled and a solution $q^\dag$ to $\left( \mathcal {IP}\right)$ exist such that

$$\begin{aligned} \lim _{n\rightarrow \infty }\big \Vert q_n - q^\dag \big \Vert _{L^1(\Omega )} = 0 \text{ and } \lim _{n\rightarrow \infty }\int _\Omega |\nabla q_n | = \int _\Omega \big | \nabla q^\dag \big |. \end{aligned}$$

(4.33)

Furthermore, $\left( \mathcal {N}^{h_n}_{q_n}j_{\delta _n}\right)$ and $\left( \mathcal {D}^{h_n}_{q_n}g_{\delta _n}\right)$ converge to the unique weak solution $\Phi ^\dag = \Phi ^\dag (q^\dag ,j^\dag ,g^\dag )$ of the boundary value problem (1.1)–(1.3) in the $H^1(\Omega )$-norm. If $q^\dag$ is unique, then convergence (4.33) holds for the whole sequence.

Uniform $L^\infty$ boundedness of $(q_n)$ together with interpolation implies that convergence actually takes place in any $L^p$ space with $p\in [1,\infty ]$.

Remark 4.10

In case $\mathcal {N}_{ q}j^\dag , \mathcal {D}_{ q}g^\dag \in H^2(\Omega )$ Lemma 4.8 shows that $\widehat{\varrho }^h_{q} \left( j^\dag ,g^\dag \right) \le C\big (h|\log h|\big )^r$ with r as in (4.27). Therefore, in view of (4.32), convergence is obtained if the sequence $(\rho _n)$ is chosen such that

$$\begin{aligned} \rho _n\rightarrow 0, ~\frac{\delta _n}{\sqrt{\rho _n}} \rightarrow 0 \text{ and } \frac{\big (h_n|\log h_n|\big )^r}{\sqrt{\rho _n}} \rightarrow 0 \text{ as } n\rightarrow \infty . \end{aligned}$$

By regularity theory for elliptic boundary value problems (see, for example, [26, 42]), if $j^\dag \in H^{1/2}(\Omega )$, $g^\dag \in H^{3/2}(\Omega )$, $q\in C^{0,1}(\Omega )$, $f\in L^2(\Omega )$ and either $\partial \Omega$ is $C^{1,1}$-smooth or the domain $\Omega$ is convex, then $\mathcal {N}_{ q}j^\dag , \mathcal {D}_{ q}g^\dag \in H^2(\Omega )$.

Proof of Theorem 4.9

We have from the optimality of $q_n$ that

$$\begin{aligned} \mathcal {J}^{h_n}_{\delta _n} \left( q_n\right) + \rho _n \int _\Omega \sqrt{| \nabla q_n |^2 + \epsilon ^{h_n}}&\le \mathcal {J}^{h_n}_{\delta _n} \big ({\widehat{q}}^{h_n}\big ) + \rho _n \int _\Omega \sqrt{\big | \nabla {\widehat{q}}^{h_n} \big |^2 + \epsilon ^{h_n}}, \end{aligned}$$

(4.34)

where ${\widehat{q}}^{h_n}$ is generated from q according to Lemma 4.6, and

$$\begin{aligned}&\mathcal {J}^{h_n}_{\delta _n} \big ({\widehat{q}}^{h_n}\big ) = \int _\Omega {\widehat{q}}^{h_n} \nabla \left( \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}}j_{\delta _n}- \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}}g_{\delta _n}\right) \cdot \nabla \left( \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}}j_{\delta _n}- \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}}g_{\delta _n}\right) \\&\quad \le \overline{q} \left\| \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}}j_{\delta _n}- \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}}g_{\delta _n}\right\| ^2_{H^1(\Omega )}\\&\quad = \overline{q}\left\| \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j_{\delta _n} - \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j^\dag + \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j^\dag - \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}} g^\dag - \mathcal {N}_q j^\dag \right. \\&\qquad \left. +\, \mathcal {D}_q g^\dag + \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}} g^\dag - \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}}g_{\delta _n} \right\| ^2_{H^1(\Omega )}\\&\quad \le 4\overline{q}\Bigl ( \left\| \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j_{\delta _n} - \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j^\dag \right\| ^2_{H^1(\Omega )} + \left\| \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}} g^\dag - \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}}g_{\delta _n} \right\| ^2_{H^1(\Omega )}\\&~\qquad + \left\| \mathcal {N}^{h_n}_{{\widehat{q}}^{h_n}} j^\dag - \mathcal {N}_{q} j^\dag \right\| ^2_{H^1(\Omega )} + \left\| \mathcal {D}^{h_n}_{{\widehat{q}}^{h_n}} g^\dag -\mathcal {D}_{q} g^\dag \right\| ^2_{H^1(\Omega )}\Bigr )\\{}&\quad \le C \left( \left\| j_{\delta _n} - j^\dag \right\| ^2_{H^{-1/2}(\partial \Omega )} + \left\| g_{\delta _n} - g^\dag \right\| ^2_{H^{-1/2}(\partial \Omega )}\right) + C \widehat{\varrho }^{h_n}_{q} \left( j^\dag ,g^\dag \right) ^2\\&\quad \le C\left( \delta ^2_n + \widehat{\varrho }^{h_n}_{q} \left( j^\dag ,g^\dag \right) ^2\right) , \end{aligned}$$

where we have used Lemma 4.2 and the fact $\mathcal {N}_{q} j^\dag = \mathcal {D}_{q} g^\dag$. Moreover, by Lemma 4.6, we have that

$$\begin{aligned} \lim _{n\rightarrow \infty }\int _\Omega \sqrt{\big |\nabla {\widehat{q}}^{h_n} \big |^2 + \epsilon ^{h_n}} = \lim _{n\rightarrow \infty }\int _\Omega \big |\nabla {\widehat{q}}^{h_n} \big | = \int _\Omega |\nabla q |. \end{aligned}$$

(4.35)

We therefore conclude from (4.34) and (4.32) that

$$\begin{aligned} \lim _{n\rightarrow \infty }\frac{\mathcal {J}^{h_n}_{\delta _n} \left( \widehat{q}^{h_n}\right) }{\rho _n} =0, \quad \lim _{n\rightarrow \infty }\mathcal {J}^{h_n}_{\delta _n} \left( q_n\right) =0 \end{aligned}$$

(4.36)

and

$$\begin{aligned} \limsup _{n\rightarrow \infty }\int _\Omega | \nabla q_n | \le \limsup _{n\rightarrow \infty }\int _\Omega \sqrt{\big | \nabla q_n \big |^2 + \epsilon ^{h_n}} \le \limsup _{n\rightarrow \infty }\int _\Omega \sqrt{\big |\nabla {\widehat{q}}^{h_n} \big |^2 + \epsilon ^{h_n}} = \int _\Omega |\nabla q |. \end{aligned}$$

(4.37)

Thus, $\left( q_n\right)$ is bounded in the $BV(\Omega )$-norm. A subsequence which is denoted by the same symbol and an element $q^\dag \in \mathcal {Q}_{ad}$ exist such that $\left( q_n\right)$ converges to $q^\dag$ in the $L^1(\Omega )$-norm and

$$\begin{aligned} \int _\Omega \big |\nabla q^\dag \big | \le \liminf _{n\rightarrow \infty }\int _\Omega |\nabla q_n |. \end{aligned}$$

(4.38)

Using Lemma 4.2 again, we infer that

$$\begin{aligned} \Big \Vert \mathcal {N}^{h_n}_{q_n} j^\dag&- \mathcal {D}^{h_n}_{q_n} g^\dag \Big \Vert ^2_{H^1(\Omega )} \\&\le 3\left( \left\| \mathcal {N}^{h_n}_{q_n} j^\dag - \mathcal {N}^{h_n}_{q_n}j_{\delta _n} \right\| ^2_{H^1(\Omega )} + \left\| \mathcal {D}^{h_n}_{q_n} g^\dag - \mathcal {D}^{h_n}_{q_n}g_{\delta _n} \right\| ^2_{H^1(\Omega )}\right. \\&\quad \left. +\left\| \mathcal {N}^{h_n}_{q_n} j_{\delta _n} - \mathcal {D}^{h_n}_{q_n} g_{\delta _n} \right\| ^2_{H^1(\Omega )} \right) \\&\le C\delta ^2_n +3\left\| \mathcal {N}^{h_n}_{q_n} j_{\delta _n} - \mathcal {D}^{h_n}_{q_n} g_{\delta _n} \right\| ^2_{H^1(\Omega )} \\&\le C \left( \delta ^2_n + \mathcal {J}^{h_n}_{\delta _n} \left( q_n\right) \right) . \end{aligned}$$

Thus, using Lemma 4.3, we obtain from the last inequality and (4.36) that

$$\begin{aligned} \left\| \mathcal {N}_{q^\dag } j^\dag - \mathcal {D}_{q^\dag } g^\dag \right\| ^2_{H^1(\Omega )} = \lim _{n\rightarrow \infty }\left\| \mathcal {N}^{h_n}_{q_n} j^\dag - \mathcal {D}^{h_n}_{q_n} g^\dag \right\| ^2_{H^1(\Omega )} =0 \end{aligned}$$

and so that

$$\begin{aligned} \mathcal {N}_{q^\dag } j^\dag = \mathcal {D}_{q^\dag } g^\dag , \text{ i.e. } q^\dag \in \mathcal {I}_{\mathcal {Q}_{ad}} \left( j^\dag ,g^\dag \right) . \end{aligned}$$

(4.39)

Furthermore, it follows from (4.37)–(4.38) that

$$\begin{aligned} \int _\Omega \big |\nabla q^\dag \big | \le \liminf _{n\rightarrow \infty }\int _\Omega |\nabla q_n| \le \limsup _{n\rightarrow \infty }\int _\Omega | \nabla q_n | \le \int _\Omega |\nabla q | \end{aligned}$$

for any solution q to $\mathcal {N}_q j^\dag = \mathcal {D}_q g^\dag$, hence, in view of (4.39), $q^\dag$ is a total variation minimizing solution of the identification problem, i.e. a solution to $\left( \mathcal {IP}\right)$. Moreover, by setting $q=q^\dagger$, we get

$$\begin{aligned} \int _\Omega \big |\nabla q^\dag \big | = \lim _{n\rightarrow \infty }\int _\Omega |\nabla q_n |. \end{aligned}$$

Finally, Lemma 4.3 shows that the sequence $\left( \mathcal {N}^{h_n}_{q_n}j_{\delta _n}, \mathcal {D}^{h_n}_{q_n}g_{\delta _n}\right)$ converges in the $H^1(\Omega ) \times H^1(\Omega )$-norm to $\left( \mathcal {N}_{q^\dag } j^\dag , \mathcal {D}_{q^\dag } g^\dag \right)$, where $\Phi ^\dag := \mathcal {N}_{q^\dag } j^\dag = \mathcal {D}_{q^\dag } g^\dag$ is the unique weak solution of the elliptic system (1.1)–(1.3), which finishes the proof. $\square$

5 Projected Armijo algorithm and numerical test

In this section we present the projected Armijo algorithm (see [27, Chapter 5]) for numerically solving the minimization problem $\left( \mathcal {P}^h_{\rho ,\delta }\right)$. We note that many other efficient solution methods are available, see for example [8].

5.1 Projected Armijo algorithm

5.1.1 Differentiability of the cost functional

Similarly to Lemma 2.3 one also sees that the discrete Neumann and Dirichlet operators $\mathcal {N}^h$, $\mathcal {D}^h$ are Fréchet differentiable on the set $\mathcal {Q}$. For given $j_\delta \in H^{-1/2}(\partial \Omega )$ and each $q \in \mathcal {Q}$ the Fréchet derivative ${\mathcal {N}^h}'(q)\xi =:{\mathcal {N}^h_q}'j_\delta (\xi )$ in the direction $\xi \in L^\infty (\Omega )$ is an element of $\mathcal {V}^h_{1,\diamond }$ and satisfies the equation

$$\begin{aligned} \int _\Omega q\nabla {\mathcal {N}^h_q}'j_\delta (\xi ) \cdot \nabla \varphi ^h&= -\int _{\Omega } \xi \nabla \mathcal {N}^h_q j_\delta \cdot \nabla \varphi ^h \end{aligned}$$

(5.1)

for all $\varphi ^h \in \mathcal {V}^h_{1,\diamond }$. Likewise, for fixed $g_\delta \in H^{1/2}(\partial \Omega )$ and each $q \in \mathcal {Q}$ the Fréchet derivative ${\mathcal {D}^h}'(q)\xi =:{\mathcal {D}^h_q}'g_\delta (\xi )$ in the direction $\xi \in L^\infty (\Omega )$ is an element of $\mathcal {V}_{1,0}^h$ and satisfies the equation

$$\begin{aligned} \int _\Omega q\nabla {\mathcal {D}^h_q}'g_\delta (\xi ) \cdot \nabla \psi ^h&= -\int _{\Omega } \xi \nabla \mathcal {D}^h_q g_\delta \cdot \nabla \psi ^h \end{aligned}$$

(5.2)

for all $\psi ^h \in \mathcal {V}_{1,0}^h$.

The functional $\mathcal {J}_\delta ^h$ is therefore Fréchet differentiable on the set $\mathcal {Q}$. For each $q\in \mathcal {Q}$ the action of the Fréchet derivative in the direction $\xi \in L^\infty (\Omega )$ is given by

$$\begin{aligned} {\mathcal {J}_\delta ^h}'(q) (\xi )&= \int _\Omega \xi \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \\&~\quad + 2\int _\Omega q \nabla \left( {\mathcal {N}^h_q}'j_\delta (\xi )- {\mathcal {D}^h_q}' g_\delta (\xi )\right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \\&= \int _\Omega \xi \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \\&~\quad + 2\int _\Omega q \nabla {\mathcal {N}^h_q}'j_\delta (\xi ) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) -2\int _\Omega q \nabla \mathcal {N}^h_qj_\delta \cdot \nabla {\mathcal {D}^h_q}' g_\delta (\xi ) \\&~\quad + 2\int _\Omega q \nabla \mathcal {D}^h_qg_\delta \cdot \nabla {\mathcal {D}^h_q}' g_\delta (\xi ). \end{aligned}$$

Since $\mathcal {N}^h_qj_\delta , \mathcal {D}^h_q g_\delta \in \mathcal {V}^h_{1,\diamond }$ and ${\mathcal {D}^h_q}' g_\delta (\xi ) \in \mathcal {V}^h_{1,0} \subset \mathcal {V}^h_{1,\diamond }$, it follows from (5.1), (3.6) and (3.8) that

$$\begin{aligned} \int _\Omega&q \nabla {\mathcal {N}^h_q}'j_\delta (\xi ) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) -\int _\Omega q \nabla \mathcal {N}^h_qj_\delta \cdot \nabla {\mathcal {D}^h_q}' g_\delta (\xi ) \\&\quad + \int _\Omega q \nabla \mathcal {D}^h_qg_\delta \cdot \nabla {\mathcal {D}^h_q}' g_\delta (\xi )\\&= - \int _\Omega \xi \nabla \mathcal {N}^h_q j_\delta \cdot \nabla \left( \mathcal {N}^h_qj_\delta -\mathcal {D}^h_qg_\delta \right) -\left\langle j_\delta , \gamma {\mathcal {D}^h_q}' g_\delta (\xi )\right\rangle \\&\quad - \left( f, {\mathcal {D}^h_q}' g_\delta (\xi )\right) + \left( f, {\mathcal {D}^h_q}' g_\delta (\xi )\right) \\&= - \int _\Omega \xi \nabla \mathcal {N}^h_q j_\delta \cdot \nabla \left( \mathcal {N}^h_qj_\delta -\mathcal {D}^h_qg_\delta \right) \end{aligned}$$

and so that

$$\begin{aligned} {\mathcal {J}_\delta ^h}'(q) (\xi )&= \int _\Omega \xi \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta - \mathcal {D}^h_qg_\delta \right) \\&\quad -2\int _\Omega \xi \nabla \mathcal {N}^h_q j_\delta \cdot \nabla \left( \mathcal {N}^h_qj_\delta -\mathcal {D}^h_qg_\delta \right) \\&=\int _\Omega \xi \left( \nabla \mathcal {D}^h_qg_\delta \cdot \nabla \mathcal {D}^h_qg_\delta - \nabla \mathcal {N}^h_qj_\delta \cdot \nabla \mathcal {N}^h_qj_\delta \right) . \end{aligned}$$

Therefore, the derivative of the cost functional $\Upsilon ^h_{\rho ,\delta }$ of $\left( \mathcal {P}^h_{\rho ,\delta }\right)$ at $q\in \mathcal {Q}^h_{ad}$ in the direction $\xi \in \mathcal {V}^h_1$ is given by

$$\begin{aligned} {\Upsilon ^h_{\rho ,\delta }}'(q) (\xi ) = \int _\Omega \xi \left( \nabla \mathcal {D}^h_qg_\delta \cdot \nabla \mathcal {D}^h_qg_\delta - \nabla \mathcal {N}^h_qj_\delta \cdot \nabla \mathcal {N}^h_qj_\delta \right) + \rho \int _\Omega \frac{\nabla q \cdot \nabla \xi }{\sqrt{|\nabla q|^2 +\epsilon ^h}}. \end{aligned}$$

(5.3)

Let $\{N_j~:~j=1,\ldots ,M^h\}$ be the set of nodes of the triangulation $\mathcal {T}^h$, then $\mathcal {V}^h_1$ is a finite dimensional vector space with dimension $M^h$. Let $\{\phi _1, \ldots ,\phi _{M^h}\}$ be the basis of $\mathcal {V}^h_1$ consisting hat functions, i.e. $\phi _i(N_j) =\delta _{ij}$ for all $1\le i,j\le M^h$, where $\delta _{ij}$ is the Kronecker symbol. Each functional $u\in \mathcal {V}^h_1$ then can be identified with a vector $(u_1,\ldots ,u_{M^h})\in \mathbb {R}^{M^h}$ consisting of the nodal values of u, i.e.

$$\begin{aligned} u=\sum _{j=1}^{M^h} u_j \phi _j \text{ with } u_j=u(N_j). \end{aligned}$$

In $\mathcal {V}^h_1$ we use the Euclidean inner product $\langle \cdot ,\cdot \rangle _E$. For each $u=(u_1,\ldots ,u_{M^h})$ and $v=(v_1,\ldots ,v_{M^h})$, we have $\langle u,v\rangle _E =\sum _{j=1}^{M^h} u_jv_j$. Let us denote the gradient of $\Upsilon ^h_{\rho ,\delta }$ at $q\in \mathcal {Q}^h_{ad}$ by $\nabla \Upsilon ^h_{\rho ,\delta }(q) = (\Upsilon _1,\ldots ,\Upsilon _{M^h})$. We then have from (5.3) with $\xi =(\xi _1,\ldots ,\xi _{M^h}) \in \mathcal {V}^h_1$ that

$$\begin{aligned} \sum _{j=1}^{M^h} \xi _j \int _\Omega \left( \phi _j \left( \nabla \mathcal {D}^h_{q}g_\delta \cdot \nabla \mathcal {D}^h_{q}g_\delta - \nabla \mathcal {N}^h_{q}j_\delta \cdot \nabla \mathcal {N}^h_{q}j_\delta \right) + \frac{\rho \nabla q \cdot \nabla \phi _j}{\sqrt{|\nabla q|^2 +\epsilon ^h}}\right) {=}\sum _{j=1}^{M^h} \xi _j \Upsilon _j \end{aligned}$$

which yields

$$\begin{aligned} \Upsilon _j = \int _\Omega \phi _j \left( \nabla \mathcal {D}^h_{q}g_\delta \cdot \nabla \mathcal {D}^h_{q}g_\delta - \nabla \mathcal {N}^h_{q}j_\delta \cdot \nabla \mathcal {N}^h_{q}j_\delta \right) + \rho \int _\Omega \frac{\nabla q \cdot \nabla \phi _j}{\sqrt{|\nabla q|^2 +\epsilon ^h}} \end{aligned}$$

(5.4)

for all $j=1,\ldots ,M^h$.

5.1.2 Algorithm

The projected Armijo algorithm is then read as: given a step size control $\beta \in (0,1)$, an initial approximation $q^h_0 \in \mathcal {Q}^h_{ad}$, a smoothing parameter $\epsilon ^h$, number of iteration N and setting $k=0$.

• 1. Compute $\mathcal {N}^h_{q^h_k} j_\delta$ and $\mathcal {D}^h_{q^h_k} g_\delta$ from the variational equations

  $$\begin{aligned} \int _\Omega q^h_k \nabla \mathcal {N}^h_{q^h_k} j_\delta \cdot \nabla \varphi ^h = \big \langle j_\delta ,\gamma \varphi ^h\big \rangle + \left( f,\varphi ^h\right) \text{ for } \text{ all } \varphi ^h\in \mathcal {V}_{1,\diamond }^h \end{aligned}$$

  (5.5)

  and

  $$\begin{aligned} \int _\Omega q^h_k \nabla \mathcal {D}^h_{q^h_k} g_\delta \cdot \nabla \psi ^h = \left( f,\psi ^h\right) \text{ for } \text{ all } \psi ^h\in \mathcal {V}_{1,0}^h, \end{aligned}$$

  (5.6)

  respectively, as well as $\Upsilon _{\rho ,\delta }^h (q^h_k)$ according to (3.12), (3.13).

• 2. Compute the gradient $\nabla \Upsilon ^h_{\rho ,\delta }(q^h_k)$ with the $j^{\mathrm{th}}$-component given by

  $$\begin{aligned} \Upsilon _j = \int _\Omega \phi _j \left( \nabla \mathcal {D}^h_{q^h_k}g_\delta \cdot \nabla \mathcal {D}^h_{q^h_k}g_\delta - \nabla \mathcal {N}^h_{q^h_k}j_\delta \cdot \nabla \mathcal {N}^h_{q^h_k}j_\delta \right) + \rho \int _\Omega \frac{\nabla q^h_k \cdot \nabla \phi _j}{\sqrt{|\nabla q^h_k |^2 +\epsilon ^h}}, \end{aligned}$$

  due to (5.4).

• 3. Set $G^h_k :=\sum _{j=1}^{M^h} \Upsilon _j\phi _j$.

  • (a) Compute

    $$\begin{aligned} \tilde{q}^h_k := \max \left( \underline{q}, \min \left( q^h_k - \beta G^h_k, \overline{q} \right) \right) , \end{aligned}$$

    $\mathcal {N}^h_{\tilde{q}^h_k} j_\delta$, $\mathcal {D}^h_{\tilde{q}^h_k} g_\delta$, according to (5.5), (5.6), $\Upsilon _{\rho ,\delta }^h (\tilde{q}^h_k)$, according to (3.12), (3.13), and

    $L:= \Upsilon _{\rho ,\delta }^h (\tilde{q}^h_k) - \Upsilon _{\rho ,\delta }^h (q^h_k) + \tau \beta \big \Vert \tilde{q}^h_k - q^h_k \big \Vert ^2_{L^2(\Omega )}$ with $\tau =10^{-4}$.

  • (b) If $L\le 0$

          go to the next step (c) below

    else

          set $\beta := \frac{\beta }{2}$ and then go back (a)

  • (c) Update $q^h_k = \tilde{q}^h_k$, set $k=k+1$.

• 4. Compute

  $$\begin{aligned} \text{ Tolerance }:= \big \Vert \nabla \Upsilon ^h_{\rho ,\delta }(q^h_k) \big \Vert _{L^2(\Omega )} -\tau _1 -\tau _2\big \Vert \nabla \Upsilon ^h_{\rho ,\delta }(q^h_0) \big \Vert _{L^2(\Omega )} \end{aligned}$$

  (5.7)

  with $\tau _1 := 10^{-3}h^{1/2}$ and $\tau _2 := 10^{-2}h^{1/2}$. If $\text{ Tolerance } \le 0$ or $k>N$, then stop; otherwise go back Step 1.

5.2 Numerical tests

We now illustrate the theoretical result with numerical examples. For this purpose we consider the the boundary value problem

$$\begin{aligned} -\nabla \cdot \big (q^\dag \nabla \Phi \big )&= f \text{ in } \Omega , \end{aligned}$$

(5.8)

$$\begin{aligned} q^\dag \nabla \Phi \cdot {n}&= j^\dag \text{ on } \partial \Omega \text{ and } \end{aligned}$$

(5.9)

$$\begin{aligned} \Phi&= g^\dag \text{ on } \partial \Omega \end{aligned}$$

(5.10)

with $\Omega = \{ x = (x_1,x_2) \in \mathbb {R}^2 ~|~ -1< x_1, x_2 < 1\}$. The special constants $\underline{q}$ and $\overline{q}$ in the definition of the set $\mathcal {Q}$ according to (1.4) are respectively chosen as 0.05 and 10.

We assume that the known source f is discontinuous and given by

$$\begin{aligned} f= \frac{3}{2}\chi _D - \frac{1}{2} \chi _{\Omega \setminus D}, \end{aligned}$$

where $\chi _D$ is the characteristic function of $D:= \{ (x_1, x_2) \in \Omega ~\big |~ |x_1| \le 1/2 \text{ and } |x_2| \le 1/2 \}$. Note that $(f,1)=0$, so that $c_f =0$. The sought conductivity $q^\dag$ in the equation (5.8)–(5.9) is assumed to be discontinuous and given by

$$\begin{aligned} q^\dag = 3 \chi _{\Omega _{1}} + 2 \chi _{\Omega _{2}} + \chi _{\Omega \setminus (\Omega _1 \cup \Omega _2)}, \end{aligned}$$

where

$$\begin{aligned}&\Omega _1 := \left\{ (x_1, x_2) \in \Omega ~\big |~ 9\Big (x_1+1/2\Big )^2 + 16\Big (x_2-1/2\Big )^2 \le 1\right\} \text{ and } \\&\Omega _2 := \left\{ (x_1, x_2) \in \Omega ~\big |~ \Big (x_1-1/2\Big )^2 + \Big (x_2+1/2\Big )^2 \le 1/16\right\} . \end{aligned}$$

For the discretization we divide the interval $(-1,1)$ into $\ell$ equal segments and so that the domain $\Omega = (-1,1)^2$ is divided into $2\ell ^2$ triangles, where the diameter of each triangle is $h_{\ell } = \frac{\sqrt{8}}{\ell }$. In the minimization problem $\left( \mathcal {P}_{\rho ,\delta }^{h} \right)$ we take $h=h_\ell$ and $\rho = \rho _\ell = 0.01 \sqrt{h_\ell }$. We use the projected Armijo algorithm which is described in Sect. 5.1 for computing the numerical solution of the problem $\left( \mathcal {P}_{\rho _\ell ,\delta _\ell }^{h_\ell } \right)$. The step size control is chosen with $\beta =0.75$ while the smoothing parameter $\epsilon ^{h_\ell } = \rho _\ell$. The initial approximation is the constant function defined by $q^{h_\ell }_0 = 1.5$.

Example 5.1

In this example the Neumann boundary condition $j^\dag \in H^{-1/2}_{-c_f}(\partial \Omega )$ in the Eq. (5.9) is chosen to be the piecewise constant function defined by

$$\begin{aligned} \begin{aligned} j^\dag&= \chi _{(0,1]\times \{-1\}} - \chi _{[-1,0]\times \{1\}} + 2\chi _{(0,1]\times \{1\}} -2\chi _{[-1,0]\times \{-1\}}\\&~\quad +3\chi _{\{-1\}\times (-1,0]} - 3\chi _{\{1\}\times (0,1)} + 4\chi _{\{1\}\times (-1,0]} - 4\chi _{\{-1\}\times (0,1)} \end{aligned} \end{aligned}$$

(5.11)

so that $\left\langle j^\dag , 1\right\rangle =0$. The Dirichlet boundary condition $g^\dag \in H^{1/2}_\diamond (\partial \Omega )$ in the Eq. (5.10) is then defined as $g^\dag = \gamma \mathcal {N}_{q^\dag } j^\dag ,$ where $\mathcal {N}_{q^\dag } j^\dag$ is the unique weak solution to the Neumann problem (5.8)–(5.9). For the numerical solution of the pure Neumann problem (5.8)–(5.9) we use the penalty technique, see e.g. [9, 25] for more details. Furthermore, to avoid a so-called inverse crime, we generate the data on a finer grid than those used in the computations. To do so, we first solve the Neumann problem (5.8)–(5.9) on the very fine grid $\ell = 128$, and then handle $(j^\dag ,g^\dag )$ on this grid for our computational process below.

We assume that noisy observations are available in the form

$$\begin{aligned} \left( j_{\delta _{\ell }}, g_{\delta _{\ell }} \right) = \left( j^\dag + \theta _\ell \cdot R_{j^\dag }, g^\dag + \theta _\ell \cdot R_{g^\dag }\right) \quad \text{ for } \text{ some } \quad \theta _\ell >0 \quad \text{ depending } \text{ on } \quad \ell , \end{aligned}$$

(5.12)

where $R_{j^\dag }$ and $R_{g^\dag }$ are $\partial M^{h_\ell }\times 1$-matrices of random numbers on the interval $(-1,1)$ which are generated by the MATLAB function “rand” and $\partial M^{h_\ell }$ is the number of boundary nodes of the triangulation $\mathcal {T}^{h_\ell }$. The measurement error is then computed as $\delta _\ell = \big \Vert j_{\delta _\ell } -j^\dag \big \Vert _{L^2(\partial \Omega )} + \big \Vert g_{\delta _\ell } -g^\dag \big \Vert _{L^2(\partial \Omega )}.$ To satisfy the condition $\delta _\ell \cdot \rho ^{-1/2}_\ell \rightarrow 0$ as $\ell \rightarrow \infty$ in Theorem 4.9 we below take $\theta _\ell = h_\ell \sqrt{\rho _\ell }$. In doing so, we reversely mimic the situation of a given sequence of noise levels $\delta _\ell$ tending to zero and of choosing the discretization level as well as the regularization parameter in dependence of the noise level.

Our computational process will be started with the coarsest level $\ell =4$. In each iteration k we compute Tolerance defined by (5.7). Then the iteration is stopped if $\text{ Tolerance } \le 0$ or the number of iterations reaches the maximum iteration count of 1000. After obtaining the numerical solution of the first iteration process with respect to the coarsest level $\ell =4$, we use its interpolation on the next finer mesh $\ell =8$ as an initial approximation $q^{h_\ell }_0$ for the algorithm on this finer mesh, and so on for $\ell =16,32,64$.

Let $q_\ell$ denote the conductivity obtained at the final iterate of the algorithm corresponding to the refinement level $\ell$. Furthermore, let $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell }$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell }$ denote the computed numerical solution to the Neumann problem

$$\begin{aligned} -\nabla \cdot (q_\ell \nabla u) = f \text{ in } \Omega \text{ and } q_\ell \nabla u\cdot {n} = j_{\delta _\ell } \text{ on } \partial \Omega \end{aligned}$$

and the Dirichlet problem

$$\begin{aligned} -\nabla \cdot (q_\ell \nabla v) = f \text{ in } \Omega \text{ and } v = g_\ell \text{ on } \partial \Omega , \end{aligned}$$

respectively. The notations $\mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q^\dag } g^\dag$ of the exact numerical solutions are to be understood similarly. We use the following abbreviations for the errors

$$\begin{aligned}&L^2_q = \big \Vert q_\ell - q^\dag \big \Vert _{L^2(\Omega )}, \quad L^2_\mathcal {N} = \big \Vert \mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag \big \Vert _{L^2(\Omega )}\\&\quad \text{ and } \quad L^2_\mathcal {D} = \big \Vert \mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag \big \Vert _{L^2(\Omega )}. \end{aligned}$$

The numerical results are summarized in Tables 1 and 2, where we present the refinement level $\ell$, the mesh size $h_\ell$ of the triangulation, the regularization parameter $\rho _\ell$, the measurement noise $\delta _\ell$, the number of iterations, the value of Tolerance, the errors $L^2_q$, $L^2_{\mathcal {N}}$, $L^2_{\mathcal {D}}$, and their experimental order of convergence (EOC) defined by

$$\begin{aligned} \text{ EOC }_\Xi := \dfrac{\ln \Xi (h_1) - \ln \Xi (h_2)}{\ln h_1 - \ln h_2} \end{aligned}$$

with $\Xi (h)$ being an error functional with respect to the mesh size h. The convergence history given in Tables 1 and 2 shows that the projected Armijo algorithm performs well for our identification problem.

All figures presented hereafter correspond to the finest level $\ell = 64$. Figure 1 from left to right shows the interpolation $I_1^{h_{\ell }} q^\dag$, the numerical solution $q_\ell$ computed by the algorithm at the 953th iteration, and the differences $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag$.

Table 1 Refinement level $\ell$, mesh size $h_\ell$ of the triangulation, regularization parameter $\rho _\ell$, measurement noise $\delta _\ell$, number of iterates and value of tolerance

Table 2 Errors $L^2_q$, $L^2_{\mathcal {N}}$, $L^2_{\mathcal {D}}$, and their EOC between finest and coarsest level

Fig. 1

Interpolation $I_1^{h_{\ell }} q^\dag$, computed numerical solution $q_\ell$ of the algorithm at the $953{\mathrm{th}}$ iteration, and the differences $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag$, for $\ell =64$, $\delta _\ell =6.7743e-3$

We observe a decrease of all errors as the noise level gets smaller, as expected from our convergence result, however, with respect to different norms. In particular, in our computations we use an $L^2$ noise level, as realistic in applications.

Example 5.2

In this example we consider noisy observations in the form

$$\begin{aligned} \left( j_{\delta _{\ell }}, g_{\delta _{\ell }} \right) = \left( j^\dag + \theta \cdot R_{j^\dag }, g^\dag + \theta \cdot R_{g^\dag }\right) , \end{aligned}$$

where $j^\dag$ is defined by (5.11). This is different from (5.12), since here $\theta >0$ is independent of $\ell$.

Using the computational process which was described as in Example 5.1 starting with $\ell =4$, in Table 3 we perform the numerical results for the finest grid $\ell =64$ and with different values of $\theta$.

Table 3 Numerical results for the finest grid $\ell =64$ and with different values of $\theta$

In Fig. 2 from left to right we show the computed numerical solution $q_\ell$ of the algorithm at the final iteration, and the differences $q_\ell -I^{h_\ell }_1q^\dag$, $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag$ for $\ell =64$ and $\theta =0.005$, i.e. $\delta _\ell =0.0167$. Finally, Fig. 3 performs the analog differences, but with $\theta =0.1$, i.e. $\delta _\ell =0.3308$.

Fig. 2

Computed numerical solution $q_\ell$ of the algorithm at the $991{\mathrm{th}}$ iteration, and the differences $q_\ell -I^{h_\ell }_1q^\dag$, $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag$ for $\ell =64$ and $\theta =0.005$, i.e. $\delta _\ell =0.0167$

Fig. 3

Computed numerical solution $q_\ell$ of the algorithm at the $1000{\mathrm{th}}$ iteration, and the differences $q_\ell -I^{h_\ell }_1q^\dag$, $\mathcal {N}^{h_\ell }_{q_\ell } j_{\delta _\ell } - \mathcal {N}^{h_\ell }_{q^\dag } j^\dag$ and $\mathcal {D}^{h_\ell }_{q_\ell } g_{\delta _\ell } - \mathcal {D}^{h_\ell }_{q^\dag } g^\dag$ for $\ell =64$ and $\theta =0.1$, i.e. $\delta _\ell =0.3308$

Example 5.3

In this example we assume that multiple measurements are available, say $\left( j_\delta ^i,g_\delta ^i \right) _{i=1,\ldots ,I}$. Then, the cost functional $\Upsilon ^h_{\rho ,\delta }$ and the problem $\left( \mathcal {P}^h_{\rho ,\delta }\right)$ can be rewritten as

$$\begin{aligned} \min _{q\in \mathcal {Q}^h_{ad}} \bar{\Upsilon }^h_{\rho ,\delta } (q) := \min _{q\in \mathcal {Q}^h_{ad}} \left( \underbrace{\frac{1}{I}\sum \nolimits _{i=1}^I\int _\Omega q\nabla \left( \mathcal {N}^h_qj_\delta ^i- \mathcal {D}^h_qg_\delta ^i\right) \cdot \nabla \left( \mathcal {N}^h_qj_\delta ^i- \mathcal {D}^h_qg_\delta ^i\right) }_{:= \bar{\mathcal {J}}_\delta ^h (q)} + \rho \int _\Omega \sqrt{\left| \nabla q\right| ^2+\epsilon ^h}\right) , \quad \left( \bar{\mathcal {P}}^h_{\rho ,\delta }\right) \end{aligned}$$

which also attains a solution $\bar{q}^h_{\rho ,\delta }$. The Neumann boundary condition in the Eq. (5.9) is chosen in the same form of (5.11), i.e.

$$\begin{aligned} \begin{aligned} j^\dag _{(A,B,C,D)}&= A\cdot \chi _{(0,1]\times \{-1\}} - A\cdot \chi _{[-1,0]\times \{1\}} + B\cdot \chi _{(0,1]\times \{1\}} -B\cdot \chi _{[-1,0]\times \{-1\}} \\&~\quad +C\cdot \chi _{\{-1\}\times (-1,0]} - C\cdot \chi _{\{1\}\times (0,1)} + D\cdot \chi _{\{1\}\times (-1,0]} - D\cdot \chi _{\{-1\}\times (0,1)}, \end{aligned} \end{aligned}$$

(5.13)

that depends on the constants A, B, C and D. Let $g^\dag _{(A,B,C,D)} := \gamma \mathcal {N}_{q^\dag }j^\dag _{(A,B,C,D)}$ and assume that noisy observations are given by

$$\begin{aligned}&\left( j^{(A,B,C,D)}_{\delta _{\ell }}, g^{(A,B,C,D)}_{\delta _{\ell }} \right) \nonumber \\&\quad = \left( j^\dag _{(A,B,C,D)}+ \theta \cdot R_{j^\dag _{(A,B,C,D)}}, g^\dag _{(A,B,C,D)}+ \theta \cdot R_{g^\dag _{(A,B,C,D)}}\right) \quad \text{ with } \quad \theta >0, \end{aligned}$$

(5.14)

where $R_{j^\dag _{(A,B,C,D)}}$ and $R_{g^\dag _{(A,B,C,D)}}$ denote $\partial M^{h_\ell }\times 1$-matrices of random numbers on the interval $(-1,1)$.

Table 4 Numerical results for $\ell =64$, $\theta =0.1$, i.e. $\delta _\ell =0.3308$, and with multiple measurements $I=1,6,16$

Fig. 4

Computed numerical solution $q_\ell$ of the algorithm at the final iteration for $\ell =64$, $\theta =0.1$, i.e. $\delta _\ell =0.3308$, and with multiple measurements $I=1,6,16$, respectively

With $\theta =0.1$ and $\ell =64$ the last line of Table 3 displays the numerical results for the case $(A,B,C,D)=(1,2,3,4)$ and $I=1$, which is repeated in the first line of Table 4 for comparison.

We now fix $D=4$. Let (A, B, C) be equal to all permutations of the set $\{1,2,3\}$. Then, the Eqs. (5.13)–(5.14) generate $I=6$ measurements. Similarly, let (A, B, C, D) be all permutations of $\{1,2,3,4\}$ we get $I=16$ measurements. The numerical results for these two cases are presented in the two last lines of Table 4, respectively.

Finally, in Fig. 4 from left to right we show the computed numerical solution $q_\ell$ of the algorithm at the final iteration for $\ell =64$, $\theta =0.1$, i.e. $\delta _\ell =0.3308$, and $I=1,6,16$, respectively.

We observe that the use of multiple measurements improves the solution to yield an acceptable result even in the presence of relatively large noise.