ELECTRODE MODELLING: THE EFFECT OF CONTACT IMPEDANCE

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Dec 15, 2013 - the contact impedance on the finite element approximation of the .... more, |||V||| = 0 implies ∇v = 0, that is v = constant =: c and thus c = v|Em.
ELECTRODE MODELLING: THE EFFECT OF CONTACT IMPEDANCE ∗

arXiv:1312.4202v1 [math.AP] 15 Dec 2013

´ EMI ´ ´ JER DARDE

AND STRATOS STABOULIS



Abstract. Current-to-voltage measurements of electrical impedance tomography are accurately modelled by the complete electrode model which takes into account the contact impedance at the electrode/object interface. The formal limiting case is the shunt model, in which perfect contacts, that is, vanishing contact impedance is assumed. The main objective of this work is to study the relationship between these two models. In smooth geometry, we show that the energy norm discrepancy is almost of order square root of the contact impedance. In addition to this, we consider the discrepancy between the finite dimensional electrode potentials which are used as forward models in many practical applications: interpreting the shunt model as an orthogonal projection of the complete electrode model and applying a duality argument implies that this discrepancy decays twice as fast. We also demonstrate that in the limit, part of the Sobolev regularity of the spatial potential is lost; this possibly has the undesirable effect of slowing down the convergence of its numerical approximation. The theoretical results are backed up by two dimensional numerical experiments: one probing the asymptotic relationship between the models and another one testing the effect of the contact impedance on the finite element approximation of the complete electrode model.

1. Introduction. The modelling of current-to-voltage measurements is a fundamental part of electrical impedance tomography (EIT) in which the aim is to retrieve an image of the conductivity distribution inside a body by injecting current into it and recording voltage response on the boundary [1, 2, 4, 21]. The most accurate mathematical model for a practical EIT measurement is the complete electrode model (CEM) in which a key aspect is that together with taking the discrete nature of a reallife measurement into account, it introduces an extra parameter, known as the contact impedance, to characterize the resistive layers that arise at the electrode/object interfaces. It has been experimentally verified [19] that the CEM is capable of predicting experimental data to better than .1%. In order to motivate this work, let us consider the following. In a given object Ω ⊂ Rn , n = 2, 3 with conductivity σ = σ(x), x ∈ Ω, an application of static electric field induces an electric potential with amplitude u = u(x) determined via the conductivity equation ∇ · σ∇u = 0

in Ω.

(1.1)

The effect of contact impedance is modelled by z > 0 in a Robin-type boundary condition u + zν · σ∇u = U

(1.2)

on each electrode E ⊂ ∂Ω, with U ∈ R representing the voltage amplitude perceived by the electrode and ν denoting the outward unit normal of ∂Ω. As current injection is exclusively confined to the electrodes, i.e. ν · σ∇u = 0 on the rest of ∂Ω, one ends up with a mixed Neumann/Robin boundary value problem for (1.1). It is well known [6, 14] that in general the solution for such a problem satisfies at best u ∈ H 1+s (Ω), for any s < 1. The regularity is further reduced asymptotically as z tends to zero in the sense that for the solution of a mixed Neumann/Dirichlet problem [18], the aforementioned holds only for s < 21 . ∗ †

Institut de Math´ ematiques, Universit´ e de Toulouse, F-31062 Toulouse Cedex 9, France, Department of Mathematics and Systems Analysis, Aalto University, FI-00076 Aalto, Finland 1

2 In this paper, we study the CEM in the limit as z tends to zero. We show that the limit exists and that it coincides with the so called shunt model (SM) defined by the Neumann/Dirichlet boundary condition (z = 0 in (1.2)). Relying ultimately on some already existing regularity results [6, 18], we derive convergence rates in smooth geometry with respect to 0 < z → 0. In the natural H 1 (Ω)-norm, we obtain for u a convergence rate of order O(z s ) with an arbitrary s < 21 1 . As a means of further comparison between the models, we interpret the SM solution as an orthogonal projection of that of the CEM. As a consequence of this, application of an Aubin–Nitsche type duality argument yields a better — of order O(z 2s ) — rate in the modulus for the electrode potential U . A similar argument also applies to e.g. FE approximation by piecewise linears: the electrode potential U is more efficiently Galerkin-approximated than the spatial potential u. Our numerical experiments also indicate that the aforementioned dissolution of the regularity of u when approaching the SM can be expected to cause a slowdown in the convergence of a common finite element (FE) approximation of the CEM by piecewise linears. This has significance as in practice one usually aims for good contacts between the electrodes and the object whereas many reconstruction algorithms rely on repetitive (and accurate) applications of FE-based solvers [22, 12, 8]. As a conclusion, we claim that a very small contact impedance has as low as (almost) first order contribution to measurement data but it slows the convergence rate of the FE-approximation down to half of the optimum. The text is organized in the following way. In Sec. 2 the CEM and the relevant notation are introduced. Sec. 3 deals with the SM and its interpretation as an orthogonal projection of CEM. Sec. 4 is dedicated to deriving convergence rate between the models. The convergence rate as well as the effect of contact impedance to the convergence of a FE approximation are then numerically studied in Sec. 5. Finally, Sec. 6 is reserved for the concluding remarks. 2. Complete electrode model. Let Ω ⊂ Rn , n = 2, 3 be a bounded domain (open and simply connected) with Lipschitz regular boundary ∂Ω. The electrode patches are modelled by M ≥ 2 mutually disjoint, well-separated domains {Em }M m=1 on ∂Ω and the union of all electrodes is abbreviated by E. The conductivity σ : Ω → Rn×n is assumed to be symmetric and such that there are constants σ± > 0 satisfying σ− |ξ|2 ≤ ξ T σξ ≤ σ+ |ξ|2

(2.1)

for all ξ ∈ Rn almost everywhere in Ω, that is to say, σ is (possibly) anisotropic and somewhere between an ideal conductor and resistor. We assume that all electrodes are used for both current injection and voltage measurement, and we denote the amplitudes of the static net currents and voltage patterns by I, U ∈ RM , respectively. According to the current conservation law, in the absence of sinks and sources we have the necessary condition ( I∈

RM 

:=

V ∈R

M

:

M X

) Vm = 0 .

m=1

1

One may reflect this with the optimal s = 12 obtained in Ref. [6] for a generic two dimensional problem for Laplacian with fixed boundary data (1.2) (in our case U also depends on z).

3 The contact impedances at the electrode/object interfaces are characterized by positive real numbers 0 < z− ≤ zm ≤ z+ ,

m = 1, 2, . . . , M.

(2.2)

Remark 1. A time-harmonic current input would require taking the reactance into account, i.e., assuming σ, z as complex valued such that the real parts satisfy (2.1) and (2.2), respectively (see e.g. Ref. [19]). With the suitable modifications, most of the results in this paper can be generalized to the complex case. The boundary value problem corresponding to the CEM stands as follows: given an input current I ∈ RM  , find the induced potential pair 1 U := (u, U ) ∈ H 1 (Ω) ⊕ RM  =: H

that satisfies weakly  ∇ · σ∇u = 0        ν · σ∇u = 0

in Ω, on ∂Ω \ E,

u + zm ν · σ∇u = Um   Z     ν · σ∇u dS = Im , 

on Em ,

(2.3)

m = 1, 2, . . . M,

Em

where ν : ∂Ω → Rn is the exterior unit normal of ∂Ω. Note that since in practice only potential differences can be measured, we determine the ground level by fixing U ∈ RM  . Furthermore, here we have implicitly assumed σ to be regular enough so that all the above objects have a meaning in the sense of traces. For thorough physical justification of (2.3) we advise the reader to consult e.g. Refs. [2, 5]. Let us briefly recall some fundamental properties of the space H1 . As a closed subspace of H 1 (Ω) × RM , when endowed with the natural scalar product Z  V, W H 1 (Ω)×RM = (∇v · ∇w + vw) dx + V · W, (2.4) Ω

it forms a Hilbert space; from here on, the norm induced by (2.4) will be denoted by k · k. Perhaps the most natural norm of choice for studying the solvability of the CEM is the one defined via Z M Z X 2 |||W||| = |∇w|2 dx + |w − Wm |2 dS, (2.5) Ω

m=1

Em

which can be interpreted as an energy norm. Lemma 2.1. The norms k · k and ||| · ||| are equivalent in H1 . Proof. Obviously, ||| · ||| is absolutely homogeneous and sub-additive. Furthermore, |||V||| = 0 implies ∇v = 0, that is v = constant =: c and thus c = v|Em = Vm for 1 every m. The condition V ∈ RM  leads to c = 0. Hence ||| · ||| is a norm on H . The rest of the proof is somewhat analogous to that of Lemma 3.2 of Ref. [19] but for the sake of completeness, we decide to carry it out explicitly. The part ||| · ||| . k · k is a direct consequence of the Trace theorem2 (for Sobolev spaces defined on Lipschitz boundaries, see Refs. [24, 16]). We prove the converse part 2

In the rest of the paper we utilize the common . -notation for ‘left-hand side function bounded by a constant times the right-hand side one’ to avoid explicitly writing several generic constants, depending only on the geometry and (possibly) on σ.

4 (j) by contradiction: suppose the existence of a sequence {V (j) }∞ , V (j) ) ⊂ H1 j=1 = (v such that kV (j) k = 1 and |||V (j) ||| → 0 as j → ∞. As the sequence is bounded in H1 , we may assume by Banach–Alaoglu theorem that it converges weakly to a certain V = (v, V ) ∈ H1 . Obviously, we actually have that V (j) strongly converges to V . Moreover, by Rellich–Kondrachov theorem [10], we can assume that {v (j) }M j=1 converges strongly to v in L2 (Ω). By weak convergence, continuity of modulus and Cauchy–Schwartz inequality, we have Z ∇v · ∇v (j) dx ≤ k∇vkL2 (Ω) lim |||V (j) ||| = 0 k∇vk2L2 (Ω) = lim j→∞

j→∞



implying v = constant =: c. Similarly, kv − Vm k2L2 (E

m)

≤ kv − Vm kL2 (E

m)

lim |||V (j) ||| = 0

j→∞

implies Vm = c for any m = 1, 2, . . . , M . Consequently, the condition V ∈ RM  enforces c = 0. Eventually, this leads to the contradiction 1 = lim kV (j) k2 ≤ kvk2L2 (Ω) + lim |||V (j) ||| + |V |2 = 0. j→∞

j→∞

Hence, k · k . ||| · |||. The bilinear form B = B(σ, z) : H1 × H1 → R associated with (2.3) is defined by [19] Z M X 1 B(V, W) = σ∇v · ∇w dx + (v − Vm )(w − Wm ) dS z Ω m=1 m Em Z

(2.6)

where the boundary restrictions of the appearing functions are identified with their corresponding traces. For any φ ∈ (H1 )0 , we consider a general variational problem for B: find U = U(φ) ∈ H1 so that B(U, W) = φ(W)

∀ W ∈ H1 .

(2.7)

The unique solvability of (2.7) follows from a standard application of the Lax– Milgram lemma. Lemma 2.2. For an arbitrary φ ∈ (H1 )0 there is a unique U = U(φ) ∈ H1 solving (2.7). In particular, the solution satisfies −1 kUk . max(σ− , z+ )kφk(H1 )0 .

(2.8)

Note that the constant in (2.8) is uniformly bounded as z+ → 0. Proof. By Lemma 2.1 we have −1 kWk2 . |||W||| . max(σ− , z+ )|B(W, W)| 2

which means that B is coercive. Trace theorem and Cauchy–Schwartz inequality imply −1 |B(V, W)| . max(σ+ , z− )kVkkWk

5 so B is also bounded. Given these two properties, the Lax–Milgram lemma [9] then guarantees the existence of a unique U ∈ H1 solving (2.7). The uniform norm estimate (2.8) is a consequence of coercivity and (2.7) with the choice W = U. Let us next return to the boundary value problem (2.3). In order to have a weak version of the co-normal derivative (denoted below by γ1 (σ)) in hand, we set H s (Ω, ∇ · σ∇) := {v ∈ H s (Ω) : ∇ · σ∇v ∈ L2 (Ω)},

s ∈ R,

and define the continuous map γ1 (σ) : H 1 (Ω, ∇ · σ∇) → H −1/2 (∂Ω) through Z Z hγ1 (σ)v, wi = σ∇v · ∇w dx + (∇ · σ∇v)w dx. Ω

(2.9)

(2.10)



In the above formula w ∈ H 1/2 (∂Ω) is identified with its arbitrary H 1 (Ω)-extension because the right hand side is defined only up to addition of an H01 (Ω)-function to w. Indeed, this follows by density since the weak (distributional) definition of the differential operator is Z h∇ · σ∇v, ϕi = − σ∇v · ∇ϕ dx Ω 1

for ϕ ∈ C0∞ (Ω) and H01 (Ω) = C0∞ (Ω)H (Ω) . Using the Green’s formula, it is easy to check that γ1 (σ) coincides with the standard conormal derivative for smooth enough functions and boundaries. We conclude this section by observing the connection between (2.3) and (2.7). Theorem 2.3. Let U = U(φI ) ∈ H1 be the solution of (2.7) with the secondmember φI (W) := I·W . Then U = (u, U ) satisfies (2.3) with the co-normal derivative replaced with γ1 (σ)u. The converse statement also holds. Proof. Taking any test function w ∈ C0∞ (Ω) in (2.7), we obtain ∇ · σ∇u = 0 in the sense of distributions. As a consequence we have from (2.7) that Z M X 1 hγ1 (σ)u, wi + (u − Um )(w − Wm ) dS = I · W z m=1 m Em

(2.11)

for an arbitrary W ∈ H1 . Hence we have Z M X 1 hγ1 (σ)u, wi + (u − Um )w dS = 0 z m=1 m Em

(2.12)

for arbitrary w ∈ H 1/2 (∂Ω), implying γ1 (σ)u = 0 on ∂Ω \ E and the desired Robin condition on each electrode (as a consequence γ1 (σ)u is in fact an element of L2 (∂Ω)). Denoting Z 1 (u − Um ) dS, m = 1, 2, . . . , M, Jm = − zm Em it follows from (2.11) with w ≡ 0 that J · W = I · W for all W ∈ RM  , that is, J = I + c for some c ∈ R. On the other hand, applying (2.10) and (2.12) with w ≡ 1 yields Z σ∇u · ∇1 dx = hγ1 (σ)u, 1i = −

0= Ω

M Z X m=1

Em

M X 1 (u − Um ) dS = Jm zm m=1

6 and hence c = 0. We therefore have obtained Z Z 1 (u − Um ) dS = γ1 (σ)u dS Im = Jm = − zm Em Em and hence U is the solution of (2.3) when interpreting ‘ν · σ∇ = γ1 (σ)’. The converse follows with a minor effort from the definition (2.10) and thus we omit the details. We will now focus on the behaviour of U when the contact impedances tend to zero. A natural candidate for the limit is the solution of the SM, which roughly correspond to the CEM problem (2.3) with vanishing contact impedances. 3. Convergence to the shunt model. 3.1. Shunt model. In EIT, the SM models the idealistic case of perfect conduction between the body and the electrodes. Mathematically speaking, this corresponds to replacing the Robin condition u + zm ν · σ∇u = Um in (2.3) by the corresponding Dirichlet condition u = Um , m = 1, 2, . . . , M . This change causes a drop of a half Sobolev smoothness index in the solution (see next section for the details). Therefore, the SM has a slightly more complicated definition than the CEM; in particular, the last equation of (2.3) does not hold anymore as a standard integral. For this reason, we will focus on the variational formulation of the SM, equivalent to the standard formulation, but easier to handle and sufficient for our purposes. For any closed subspace V ⊂ H1 , the Lax–Milgram lemma (cf. Lemma 2.2) guarantees the existence of a unique element UV = UV (φ) ∈ V satisfying B(UV , W) = φ(W)

∀ W ∈ V,

(3.1)

for φ ∈ V0 . Note that UV is the B-orthogonal projection of U onto the closed subspace V. It turns out that the solution of the SM problem is precisely the solution of (3.1), denoted by U0 (φI ), with V = H10 := {W ∈ H1 : w|Em = Wm , m = 1, 2, . . . , M }

(3.2)

and φ = φI from Theorem 2.3. Without going into details, we emphasize that this can, in essence, be shown by following the same lines of reasoning as in the proof of Theorem 2.3. However, as was mentioned above, we remind the reader that the lack of regularity of U0 (φI ) gives rise to the need for understanding the net current boundary condition in a weaker sense (cf. (4.15) for the case of smooth geometry). Notice, in particular, that U0 is independent of z although B is not. Before moving on to study the relationship between the CEM and SM, we recall a basic approximation result closely related to (3.1). In point of view of Galerkin approximation, it is interesting to remark that, in a sense, the electrode potential 1 part U ∈ RM  is better approximated than the spatial part u ∈ H (Ω) of the solution U = (u, U ) = U(φ) to (2.7). Proposition 3.1 (Aubin–Nitsche). Let U = U(φ) ∈ H1 be the solution to (2.7) and UV the solution of (3.1) corresponding to a given closed subspace V ⊂ H1 . Then for any ψ ∈ (H1 )0 it holds −1 |ψ(U − UV )| . max(σ+ , z− )kU − UV kkU ∗ − UV∗ k

(3.3)

where U ∗ = U ∗ (ψ) ∈ H1 is the dual solution satisfying B(W, U ∗ ) = ψ(W)

∀ W ∈ H1 .

(3.4)

7

Proof. Since σ is symmetric it follows that (3.4) is equivalent to B(U ∗ , W) = ψ(W)

∀W ∈ H1

(3.5)

and thus by Lemma 2.2 the dual problem is uniquely solvable. Given the projection UV∗ ∈ V corresponding to the variational problem defined by (3.5), we may write |ψ(U − UV )| = |B(U − UV , U ∗ )| = |B(U − UV , U ∗ − UV∗ )| where the rightmost equality follows from the fact that B(U, W) = B(UV , W) for any W ∈ V. The continuity of B concludes the proof. An important special case of Proposition 3.1 is the estimate for the error |U − UV | + ku − uV kL2 (Ω) corresponding to the choice ψ(W) =

U − UV ·W − |U − UV |

Z Ω

u − uV w dx. ku − uV kL2 (Ω)

In Sec. 5 we return numerically to this subject by considering the manifestation of Proposition 3.1 within the convergence of FE approximations of the CEM by piecewise linears. However, the fact that the constant in (3.3) blows up in the limit z+ → 0 makes (3.3) unsuitable for studying U − U0 asymptotically which we move on to do next. 3.2. Convergence result. For the rest of this section, we are interested in the limit (if it exists) of the solution corresponding to the CEM when the contact M impedances {zm }m=1 tend to zero on all electrodes. We actually prove a slightly more general result: For any given closed subspace V ⊂ H1 , UV always converges as z tends to zero, and its limit is UV0 ∈ V0 , where V0 := V ∩ H10 = {W ∈ V : w|Em = Wm , m = 1, 2, . . . , M },

(3.6)

and UV0 verifies B(UV0 , W) = φ(W) for all W ∈ V0 . Proposition 3.2. Let V ⊂ H1 be a closed subspace and φ ∈ V0 . With UV = UV (φ) ∈ V and UV0 = UV0 (φ) ∈ V0 being defined as above, we have lim UV = UV0

(3.7)

z→0

in the space H1 . Proof. For clarity, we abbreviate the solutions by V(z) = UV , z ∈ (0, ∞)M and the bilinear form by B(z). We perform an indirect argument and assume that (3.7) (j) ∞ is false, i.e., there exists a sequence sequence {V (j) }∞ )}j=1 that does not j=1 := {V(z M (j) converge to UV0 where (0, ∞) 3 z → 0 as j → ∞. According to the uniform (with respect to z) bound (2.8), the function z 7→ kV(z)k is bounded. Hence, by the Banach–Alaoglu theorem, we may assume that {V (j) }∞ j=1 converges weakly to a certain V ∈ V. First we show that actually V = UV0 . Using zm ≤ z+ for any m = 1, 2, . . . , M and (2.1), we obtain from (3.1) that M Z X m=1

Em

|v(z) − Vm (z)|2 dS . z+ kφkV0 → 0

as z+ → 0.

(3.8)

8 Hence the (weak) continuity of the trace operator yields M Z X m=1

2

|v − Vm | dS = lim

j→∞

Em

M Z X m=1

|v (j) − Vm(j) ||v − Vm | dS

Em

and subsequently, Cauchy–Schwarz inequality and (3.8) give v = Vm on every Em , that is, V ∈ V0 . Similarly, by weak convergence, an arbitrary W ∈ V0 satisfies Z Z σ∇v · ∇w dx = lim σ∇v (j) · ∇w dx = lim B(z (j) )(V (j) , W) = φ(W), j→∞



j→∞



where the middle equality is a consequence of the vanishing boundary term. Thus uniqueness guarantees the claimed V = UV0 . We are ready to derive the contradiction. By coercivity we estimate kV (j) − UV0 k2 . |B(z (j) )(V (j) − UV0 , V (j) − UV0 )| = |φ(V (j) − UV0 )|

(3.9)

with the underlying constant being independent of j. Note that the equality in (3.9) follows by symmetricity of σ and vanishing of suitable boundary terms. Therefore, by weak convergence the right-hand side of (3.9) converges to zero as j → ∞, implying strong convergence for the sequence {V (j) }∞ j=1 . This is contradicts the assumption. There are certain special cases of Proposition 3.2 that are of particular interest. First of all, in the case V = H1 it follows that that the CEM solution converges to that of the SM with an unspecified rate as (0, ∞)M 3 z → 0. Secondly, any Galerkin approximation of the CEM converges to that of the SM problem as the contact impedances tend to zero. 4. Regularity and convergence rates in the smooth setting. 4.1. Regularity of the spatial part. In this section we move onwards to study the Sobolev smoothness of the spatial potentials of CEM and SM in the case when all the predetermined attributes are smooth enough. We show in both cases that the H 1 -regularity of the spatial potential is not the optimal. To avoid extra technicality, we assume that ∂Ω, ∂Em , m = 1, 2, . . . , M are all in the C ∞ -class. We also suppose that in addition to satisfying (2.1), the conductivity σ belongs to C ∞ (Ω; Rn×n ). At this point, we need to enlarge the domain and range of the co-normal derivative γ1 (σ)u in smooth domains. In what follows, we use the generic h · , · i to denote the dual evaluation between any pair H −s (∂Ω) and H s (∂Ω), s ≥ 0; if there is danger of confusion, we give further specifications. By density (see §2 Theorem 7.3 of Ref. [15]), the operator γ1 (σ) : C ∞ (Ω) → C ∞ (∂Ω), ϕ 7→ ν · σ∇ϕ|∂Ω extends to a bounded operator γ1 (σ) : H s (Ω, ∇ · σ∇) → H s−3/2 (∂Ω)

(4.1)

for any 0 < s < 2. Next we introduce a reference problem which will be used to infer the extra regularity of u, u0 ∈ H 1 (Ω). Suppose Γ ⊂ ∂Ω is simply connected, non-empty and that ∂Γ is of class C ∞ . Then, it is well known that for any pair of data g ∈ H 1/2 (Γ), f ∈ L2 (Ω) and parameter β ≥ 0, there exists a unique vβ ∈ H 1 (Ω) satisfying weakly ∇ · σ∇vβ = f

in Ω,

γ1 (σ)vβ = 0

on ∂Ω \ Γ,

vβ + βγ1 (σ)vβ = g

on Γ (4.2)

9 with the case β = 0 in the rightmost constraint interpreted as a Dirichlet condition. Using the properties of the Dirichlet-to-Neumann map and interpolation of Sobolev spaces, we obtain the following regularity estimate: Theorem 4.1. The solution vβ ∈ H 1 (Ω) of (4.2) satisfies for all s ∈ (− 12 , 12 ) kvβ kH 1+s (Ω) . (kgkH 1/2+s (Γ) + kf kL2 (Ω) )

(4.3)

with a constant C > 0 independent of β. Proof. For simplicity, we assume that f = 0 as the below reasoning can be adjusted to the general case with a few straightforward modifications. The idea is essentially based on the proof of Corollary 4.4 of Ref. [6]. In the case β = 0, the unique solution of (4.2) exists and satisfies [18, 6] the estimate kv0 kH 1+s (Ω) . kgkH 1/2+s (Γ) + kf kL2 (Ω)

(4.4)

for all s ∈ (− 21 , 12 ). We define the associated (partial) Dirichlet-to-Neumann map by Λ = ΛΓ (σ) : H s+1/2 (Γ) → (H 1/2−s (Γ))0 ,

g 7→ (γ1 (σ)v0 )|Γ ,

(4.5)

where (γ1 (σ)v0 )|Γ : w 7→ hγ1 (σ)v0 , wi, e

w e ∈ H 1/2−s (∂Ω), w| eΓ=w

(4.6)

and v0 is the solution of (4.2) corresponding to g and f = 0. Note that as a consequence of the zero-Neumann condition, (4.6) is independent of the choice of the extension w. e By the standard characterization of H 1/2−s (Γ) by restrictions (see §1 Theorem 9.2 of Ref. [15]), (4.1) and (4.4), we see that Λ is bounded. Suppose for now that β > 0 and vβ solves (4.2) for some g ∈ L2 (Γ) and let Id : L2 (Γ) → L2 (Γ) denote the identity map. As a consequence of the fact that vβ trivially solves the corresponding mixed Dirichlet/Neumann problem, we can write (Id + βΛ)(vβ |Γ ) = g.

(4.7)

The rest of the proof is analogous to that of Corollary 4.4 of Ref. [6]; it relies on studying continuity properties of the operator3 (Id + βΛ)−1 : (H 1/2−s (Γ))0 → H 1/2+s (Γ),

β>0

(− 21 , 12 )

via interpolation of Sobolev spaces [15] and utilization of the for all s ∈ continuity of (4.5). Remark 2. In fact, the interpolation argument of the proof of Corollary 4.4 of Ref. [6] yields also a convergence rate kvβ − v0 kH 1+s (Ω) = O(β t−s ) for any t ∈ [s, 12 ). However, this result is not directly applicable to ku − u0 kH 1+s (Ω) because in the CEM the electrode boundary data depends on z. Moreover, note that we cannot expand (Id + βΛ)−1 in terms of the Neumann series because powers Λj , j ≥ 2, are not well defined (see Theorem 3.1 of Ref. [6]). The next theorem shows that (4.3) can nevertheless be used to get an analogous norm estimate for u, u0 ∈ H 1 (Ω). Theorem 4.2. The functions u, u0 ∈ H 1 (Ω), i.e. the spatial parts of CEM and SM solutions, satisfy −1 kukH 1+s (Ω) . max(σ− , z+ )|I|, 3

−1 ku0 kH 1+s (Ω) . σ− |I|

For the existence of the inverse operater between these spaces, we refer to Refs. [18, 6].

(4.8)

10 for any given s < 12 . Proof. Here we consider only u (u0 can be handled with straightforward modifications). The idea is to use a suitable partition of unity to get local estimates 2 from Theorem 4.1. We claim that there exists a partition of unity {ϕp }M p=1 ⊂ C (Ω) satisfying M X

ϕp |Em = δpm ,

ϕp = 1 in Ω,

ν · σ∇ϕp = 0 on ∂Ω.

(4.9)

p=1

This set functions can be constructed in the following way. First use e.g. the converse of trace theorem [16] to select functions ϕ bp ∈ C 2 (Ω) such that ϕ bp satisfies the latter two conditions of (4.9). Then defining the functions ϕp ∈ C 2 (Ω) by ϕ1 := 1 −

M X

ϕ bp ,

ϕp = ϕ bp , p = 2, 3, . . . , M,

p=2

gives a set of functions satisfying also the remaining summability condition (recall that E m ∩ E p = ∅ if m 6= p). We find out that up := uϕp satisfies a boundary value problem of form (4.2). Obviously we have ∇ · σ∇up = 2σ∇u · ∇ϕp + u∇ · σ∇ϕp ∈ L2 (Ω)

(4.10)

in the weak sense with the underlying norm estimate k∇ · σ∇up kL2 (Ω) ≤ CkukH 1 (Ω) . In addition, a straightforward calculation based on (2.11) reveals Z hγ1 (σ)up , gi = hγ1 (σ)u, gϕp i + (ν · σ∇ϕp )gu dS ∂Ω

for any g ∈ H 1/2 (∂Ω). Therefore, by (4.9) and (2.7) we have γ1 (σ)up = −

1 (u − Up )χp , zp p

(4.11)

where χp is the indicator function of Ep . Consequently, Theorem 4.2 can be applied to the boundary value problem defined by (4.10) and (4.11). Therefore, for any given s < 21 , the solution up satisfies the norm estimate kup kH 1+s (Ω) . kUp χp kH 1/2+s (E

p)

+ k∇ · σ∇up kL2 (Ω)

. |U | + kukH 1 (Ω) −1 . max(σ− , z+ )|I|

where the middle estimate follows from (4.10). Eventually, using the summability condition of (4.9) and triangle inequality, the proof is concluded. Thanks to Theorem 4.2 and the regularity of Neumann problem, an extra 12 degree of smoothness can be obtained for the CEM solution. Corollary 4.3. The function u belongs to H 3/2+s (Ω) with −s− kukH 3/2+s (Ω) = O(z− )

as z− → 0

(4.12)

11 for arbitrary s ∈ [0, 12 ),  ∈ (0, 1 − s). Note that the right-hand side goes to infinity as z− goes to zero. Proof. We abbreviate gm :=

1 (U − u)|Em ∈ H 1/2 (Em ), zm m

m = 1, 2, . . . , M

and let gem denote the extension of gm to ∂Ω by zero; the continuity of the extension (see §1 Theorem 7.4 of Ref. [15]) implies

γ1 (σ)u =

M X

gem ∈ H 1/2− (∂Ω),

M X

kγ1 (σ)ukH 1/2− (∂Ω) .

m=1

kgm kH 1/2− (E

m)

.

m=1

By regularity of the Neumann problem (cf. §2 Remark 7.2 of Ref. [15]) and interpolation between spaces H t (Em ), (H t (Em ))0 , t ≥ 0 (see §1 Theorem 12.5 of Ref. [15]), we obtain inf ku + ckH 3/2+s (Ω) .

c∈R

M X

kgm kH s (Em )

m=1

.

M X

kgm kθH 1− (E

m)

kgm k1−θ  (H (E

m ))

0

m=1

R where θ = s+. Since gm ∈ L2 (Em ) is identified with w 7→ E gm w dS in (H  (Em ))0 , m we have kgm k(H  (Em ))0 . ke gm kH − (∂Ω) . Henceforth, the fact that g˜m do not overlap each other, the continuity of γ1 (σ) from H 3/2− (Ω, ∇ · σ∇) to H − (∂Ω) (see (4.1)) and equation (4.8) yield inf ku + ckH 3/2+s (Ω) . |I|1−θ

c∈R

M X

−θ zm ku − Um kθH 1− (E

m)

(4.13)

m=1

with θ = s +  and  ∈ [0, 12 ). By triangle inequality, trace theorem [15] and the fact that Um ∈ R is a constant, we may further estimate the right hand side of (4.13) using ku − Um kH 1− (E

m)

. kukH 3/2− (Ω) + |Um |.

Thus, by Theorem 4.2 and (2.8), we get inf ku + ckH 3/2+s (Ω) . |I|

c∈R

M X

−s− zm .

(4.14)

m=1

In order to manipulate the quotient norm in (4.14), we recall that by basic properties of Sobolev inner product [24], there holds Z (w, 1)H 3/2+s (Ω) = (w, 1)L2 (Ω) =

w dx Ω

12 for all w ∈ H 3/2+s (Ω). Thus we have

2 Z

1

inf ku + ck2H 3/2+s (Ω) = inf u − u dx + c

3/2+s c∈R c∈R |Ω| Ω H (Ω) ) (

2 Z

1 + c2 |Ω| = inf u− u dx

3/2+s c∈R |Ω| Ω H (Ω)

2

Z

1

= u − u dx

3/2+s |Ω| Ω H (Ω) 2 Z 1 2 = kukH 3/2+s (Ω) − u dx |Ω| Ω ≥ kuk2H 3/2+s (Ω) − kuk2L2 (Ω) where the last estimate is a direct consequence of Cauchy–Schwarz inequality. Applying the above estimate, (4.14) and (2.8), we obtain kukH 3/2+s (Ω) . kukL2 (Ω) + |I|

M X

−s− −1 zm . |I| max(σ− , z+ ) + |I|

M X

−s− zm

m=1

m=1

which implies the claim. Identifying γ1 (σ)u with the boundary current density we can deduce (cf. §1 Theorem 9.8 of Ref. [15]) that γ1 (σ)u|Em ∈ H 1+s (Em ) ⊂ C 0 (Em ),

s ∈ [0, 21 ), Ω ⊂ Rn , n = 2, 3.

Therefore, in particular, we have γ1 (σ)u ∈ L∞ (∂Ω). This is not true for γ1 (σ)u0 since it even falls outside of L2 (∂Ω). In the special case σ ≡ 1, n = 2 for (4.2), the drop in Sobolev regularity was characterized in Ref. [6] (see also Ref. [17]) by classifying the type of the singularities of vβ at the transition points of boundary conditions. Using a singular decomposition technique, in the case β > 0, it was demonstrated that the most severe singularity is of type r log r whereas in the case β = 0 it is r1/2 with (r, θ) denoting the polar coordinates centered at the transition point in question. 4.2. Convergence rates. In order to take advantage of the regularity provided by (4.8) in deriving convergence rates, we need the following lemma related to the approximation of trivially extended Sobolev functions by bump functions: Lemma 4.4. Let Γ ⊂ ∂Ω be a connected set with a C ∞ -boundary. Suppose that g ∈ H s (Γ) for some s ∈ [0, 1/2) and denote by ge ∈ L2 (∂Ω) the extension of g to ∂Ω by zero. Then ge ∈ H s (∂Ω) and there exists a sequence of C ∞ (∂Ω)-functions supported in Γ that converges to ge in H s (∂Ω). Proof. By the density of compactly supported functions (see e.g. §1 Theorem 11.1 of Ref. [15]) for any Sobolev exponent s ∈ [0, 1/2], it is possible to fix a sequence of ∞ s functions (ϕj )∞ j=1 ⊂ C0 (Γ) which converges to g in the norm of H (Γ). As the zero ∞ extension ϕ ej remains in C (∂Ω), the continuity (see §1 Theorem 7.4 of Ref. [15]) of the zero extension operator for s ∈ [0, 1/2) implies ge ∈ H s (∂Ω) and that the smooth functions ϕ ej converge to ge in H s (∂Ω). Considering the smoothness given by Theorem 4.2, the net current condition for the SM can be interpreted in the following way. Proposition 4.5. In the C ∞ -setting U0 ∈ H1 satisfies hγ1 (σ)u0 , χm i = Im ,

m = 1, 2, . . . , M

(4.15)

13 where the dual evaluation can be taken between H −s (∂Ω) and H s (∂Ω) for arbitrary s ∈ (0, 21 ) and χm is the indicator function of Em . Proof. Let g ∈ C ∞ (∂Ω) and s ∈ (0, 12 ) be arbitrary. According to Lemma 4.4 we ∞ can pick a sequence (ϕj )∞ j=1 ⊂ C0 (∂Ω \ E) such that in H s (∂Ω).

lim ϕj = χ∂Ω\E

j→∞

(4.16)

By basic properties of Sobolev norm and (4.16) we have lim kgϕj − gχ∂Ω\E kH s (∂Ω) ≤ kgkC 1 (∂Ω) lim kϕj − χ∂Ω\E kH s (∂Ω) = 0

j→∞

j→∞

and hence by continuity hγ1 (σ)u0 , gχ∂Ω\E i = lim hγ1 (σ)u0 , gϕj i = 0 j→∞

where the last equality is a consequence of the variational problem in H10 defining U0 (cf. (3.1)) and the fact supp gϕj ⊂ ∂Ω \ E. Therefore, it holds hγ1 (σ)u0 , gi = hγ1 (σ)u0 , gχE i

(4.17)



for any g ∈ C (∂Ω). Choosing suitable test functions g that are constants on the electrodes, and recalling (3.1) and that the electrodes do not overlap, we arrive at the alleged result. Equation (4.15) allows us to estimate kU − U0 k by using the coercivity of B to obtain the following: Theorem 4.6. The discrepancy between the CEM and SM solutions satisfies the estimate s kU − U0 k . |I|z+

(4.18)

1 2 ).

for any s ∈ [0, Proof. As a consequence of (4.15) and (4.17) we write Z B(U − U0 , W) = − σ∇u0 · ∇w dx + I · W Ω

=−

M X

hγ1 (σ)u0 , wχm i +

M X

hγ1 (σ)u0 , Wm χm i

m=1

m=1

=−

M X

hγ1 (σ)u0 , (w − Wm )χm i

(4.19)

m=1

for all W ∈ H1 where the middle equality follows from the definition (2.10) of the conormal derivative. The choice W = U − U0 further leads to B(U − U0 , U − U0 ) = −

M X

hγ1 (σ)u0 , (u − Um )χm i.

(4.20)

m=1

Taking the coercivity of B into account, it is sufficient to obtain a bound of the desired form for the right hand side of (4.20). By the continuity of γ1 (σ) we can estimate |hγ1 (σ)u0 , (u − Um )χm i| . kγ1 (σ)u0 kH t−1/2 (∂Ω) k(u − Um )χm kH 1/2−t (∂Ω) . ku0 kH 1+t (Ω) ku − Um kH 1/2−t (E

m)

(4.21)

14 for an arbitrary t ∈ (0, 21 ). Applying the Robin boundary condition suitably (cf. proof of Corollary 4.3), we may use the continuity of γ1 (σ) to yield ku − Um k(H  (Em ))0 = kzm γ1 (σ)ukH − (∂Ω) . zm kukH 3/2− (Ω) for any  ∈ (0, t + 21 ]. Therefore, by interpolation and trace theorem, we get ku − Um kH 1/2−t (E

m)

. ku − Um k1−θ  (H (E

m ))

0

ku − Um kθH 1− (E

1−θ . zm ku − Um kθH 3/2− (Ω) kuk1−θ 3/2− H

m)

(Ω)

(4.22)

with θ = 12 +  − t. Finally, applying (4.8) and taking the square root, we obtain 1+2(t−) . (4.18) with s = 1−θ 2 = 4 Before concluding the section, we point out that convergence rates can be obtained also in other norms. In particular, the next corollary reveals that the electrode voltages U ∈ RM  converge twice as fast as the potential inside the body. Corollary 4.7. For the solutions U, U0 ∈ H1 there holds 2s ku − u0 kL2 (Ω) + |U − U0 | . |I|z+

(4.23)

for any s ∈ [0, 21 ). Furthermore, the spatial components satisfy 1/2−s−

ku − u0 kH 1+s (Ω) . z+

(4.24)

for any s ∈ [0, 12 ) and  ∈ (0, 1/2 − s). Proof. We prove the first part using a dual technique similar to the one used to prove Proposition 3.3. Define U ∗ ∈ H1 as the unique solution to the dual problem Z B(W, U ∗ ) = w(u − u0 ) dx + W · (U − U0 ) (4.25) Ω 1

for all W ∈ H . Consequently we may write ku − u0 k2L2 (Ω) + |U − U0 |2 = B(U − U0 , U ∗ ) = B(U − U0 , U ∗ − U0∗ ).

(4.26)

The idea of the proof is to derive a bound for the right-hand quantity without using the coercivity of B. Instead, we will apply Sobolev regularity and interpolation to obtain 2s |B(U − U0 , U ∗ − U0∗ )| . |I|z+ (ku − u0 k2L2 (Ω) + |U − U0 |2 )1/2

(4.27)

for all s ∈ [0, 12 ). Combining this with (4.26) and cancelling out the extra 21 -power will yield an estimate equivalent to (4.23) without the need of taking the square root 2s of z+ . Let us then demonstrate how to obtain (4.27). Since by (4.19) the modulus of the rightmost expression of (4.26) is bounded by M X m=1

∗ |hγ1 (σ)u0 , (u∗ − Um )χm i|,

(4.28)

15 it is sufficient to find a suitable bound for this quantity. To this end, we note it is not difficult to show (applying Theorem 4.1) that results analogous to (4.8) and (4.18) remain valid even if there is a non-zero source term f ∈ L2 (Ω) in the partial differential equation; if that is the case, the factor |I| appearing in the estimates is replaced with |I| + kf kL2 (Ω) . Referring to this fact, we have kU ∗ k . ku − u0 kL2 (Ω) + |U − U0 | and ku∗ kH 3/2− (Ω) . ku − u0 kL2 (Ω) + |U − U0 | for all  > 0 since the PDE defining U ∗ has u − u0 ∈ L2 (Ω) as the source term. Subsequently, according to (4.21) and (4.22), we deduce ∗ ∗ |hγ1 (σ)u0 , (u∗ − Um )χm i| . ku0 kH 1+t (Ω) ku∗ − Um kH 1/2−t (E

.

1−θ ku0 kH 1+t (Ω) ku∗ zm



m)

∗ θ kH 3/2− (Ω) ku∗ k1−θ Um 3/2− H (Ω)

1−θ . zm |I|(ku − u0 kL2 (Ω) + |U − U0 |)

(4.29)

for t, , θ as in the proof of Theorem 4.6. Combining (4.26) and (4.29) we get 2s ku − u0 k2L2 (Ω) + |U − U0 |2 . |I|(ku − u0 kL2 (Ω) + |U − U0 |)z+

where s = 1+2(t−) can be chosen freely from the interval [0, 21 ). This is clearly 4 equivalent to the claim. The second part of the claim is again an application of interpolation. Utilizing the partition of unity (4.9) with Theorem 4.1 (β = 0) we get ku − u0 kH 1+s (Ω) . ku − u0 kH 1 (Ω) +

M X

ku − u0 kH 1/2+s (E

m)

m=1

. kU − U0 k +

M X

ku − Um kH 1/2+s (E

m)

m=1

for any s ∈ [0, 12 ); note that the bottom estimate is merely based on trivial estimation and the electrode boundary condition of u0 . By interpolation (cf. (4.22)) we further estimate ku − Um kH 1/2+s (E

m)

1−θ . zm |I|

with θ = 12 + s +  and any  ∈ (0, 1/2 − s). The claim is a direct consequence of this since by (4.18) we can estimate kU − U0 k as required. To conclude the section, we remark that it is a rather straightforward task to generalize the above results to the case where zm → 0 possibly only for m in a subset of {1, 2, . . . , M }.

16 5. Numerical tests. We proceed with two numerical examples related to some of the results presented in Sec. 3 and 4. In Sec. 5.1 we test whether a convergence rate indicated by (4.18) is apparent in the corresponding FE approximations by piecewise linears and we are also interested whether the electrode voltages converge noticeably faster than the spatial potentials (cf. (4.23)). The other example is presented in Sec. 5.2. There we numerically study what kind of an effect different values of z have on the convergence (with respect to the maximal triangle diameter h) of the FE approximation of the CEM. Although this question is of practical interest on its own, it can also be understood as an indirect numerical verification of the observed regularity drop (see (4.8), (4.12)). 5.1. Convergence test. In the first numerical example, the test object Ω is a regular hexadecagon with all of the 16 corners lying on the unit sphere. The conductivity is constant σ ≡ 1 and there are M = 8 identical, equidistant electrodes each of which covers exactly one boundary edge of Ω. We compute approximate solutions using the FE method with piecewise linear basis functions. Denoting the space defined by the triangle-wise linear polynomials by P1 , we select V = P1 ⊕ RM  ; this Galerkin space is used to compute UV i.e. a FE approximation of the CEM. The corresponding V0 is defined as in (3.6) and we use it to approximate the SM. For a detailed description of the assembly and computation of the system matrices, we refer to Ref. [22].

0

10

−1

10

X

ku − uV0 kH 1 (Ω) X V kU − UV0 k X V ku − uV0 kL2 (Ω) X V kRV − RV0 kM ×M

−2

10

−3

10 −4 10

−3

10

−2

10

O(β 0.4567 ) O(β 0.4582 ) O(β 0.7163 ) O(β 0.8011 )

−1

10

Fig. 5.1: On the left: discrepancy as a function of z ≡ β = constant. On the right: convergence rates in different norms. The error in the measurement map is calculated in the operator norm of RM ×M . All the computations were performed using a fixed triangulation such that away from the boundary the mesh parameter was h = 0.079 and near the boundary h = 0.005.

In Fig. 5.1 the discrepancies between the CEM and SM are visualized as the constant contact impedance z = [β, β, . . . , β]T ∈ (0, ∞)M approaches 0. The ex(2) (2) amined solutions UV = UV and UV0 = UV0 are computed using the input current I (2) = [cos(2πm/M )]M m=1 but we emphasize that similar rates were also obtained for other input currents. We have also considered the measurement matrix RV ∈ RM ×M

17 (RV0 respectively) defined as the unique matrix having the following two properties: M 1 it maps every I ∈ RM  to UV ∈ R , where UV = (uV , UV ) ∈ H is the corresponding solution to (3.1), and its null space is spanned by [1, 1, . . . , 1]T ∈ RM . Note that the I (2) defined above is an eigenvector of RV (RV0 respectively) corresponding to the second smallest eigenvalue (see e.g. Ref. [19]). First of all, we observe that the convergence indicated by Proposition 3.2 appears to take place. The estimated convergence rates in the tabular of Fig. 5.1 are obtained by a least squares fit of linear functions in log β. Although the results fall below the rates predicted by Theorem 4.6 and Corollary 4.7, it is reasonable to hypothesize that for any s ∈ [0, 12 ) there exists a fine enough triangulation of Ω such that an (infinitely precise) numerical scheme will detect the rates ku − u0 kH 1 (Ω) = O(β s ),

kU − U0 k = O(β s ),

ku − u0 kL2 (Ω) = O(β 2s ),

kR − R0 kM ×M = O(β 2s ).

We further observe that qualitatively the obtained estimates are fairly well in accordance with the theory in the sense that kUV − UV0 k ≈ O(β 0.4582 ) is far from linear whereas kRV −RV0 kM ×M ≈ O(β 0.8011 ) decays roughly twice as fast in the limit β → 0. 5.2. The effect on the convergence of FE approximation. We continue working in the same geometry as in the previous example. However, in this case we do not fix the triangulation of Ω but instead use a set of gradually sharpening uniform triangulations4 to estimate the convergence rate of the FE approximation by V = P1 ⊕ RM  . More precisely, for each member of a set of constant contact impedances z ≡ β > 0, we compute for UV an estimated convergence rate with respect to the ever decreasing mesh parameter 0 < h → 0+. In order to derive a priori estimates with respect to h, we note that for any given v ∈ H 3/2+s (Ω), s ∈ [0, 21 ), it can be shown using a suitable polynomial interpolator [3, 20], that inf kv − wkH 1 (Ω) . h1/2+s kvkH 3/2+s (Ω) ,

w∈P1

(5.1)

where the hidden constant depends on s. As a consequence, the hypothesis that u satisfies (4.12)5 leads to an O(h1/2+s ) upper bound for kU − UV k and to the nearly quadratic one O(h1+2s ) for |U − UV |. Indeed, by C´ea’s lemma and the fact that V = P1 ⊕ RM  , we have kU − UV k . inf kU − Wk . inf ku − wkH 1 (Ω) . |I|h1/2+s W∈V

w∈P1

(5.2)

with the rightmost hidden constant being of order O(β −1/2−s− ) for any  ∈ (0, 1 − s). As an implication of (3.3) we additionally obtain that ku − uV kL2 (Ω) + |U − UV | . |I|h1+2s 4

(5.3)

The authors admit that this is not reasonable in practical applications. Due to the high regularity of u away from ∂Ω it is advisable to use adaptive meshing (cf. e.g. Ref. [11]). 5 It is well known that in polygonal domains this is not the case e.g. if the boundary has concave angles. For a detailed discussion on the topic, see for example Ref. [6] and the references therein.

18 with a hidden constant of order O(β −3/2−s− ). Let us emphasize that in both estimates (5.2) and (5.3), due to the underlying interpolation space argument, the constant depends also on s and . Because of the constants’ explosion in the limit β → 0+, one may anticipate that (when using uniform triangulations) the computational detection of rates corresponding to s close to 1/2 becomes increasingly demanding. Remark 3. (a) Let CN denote the condition number of the matrix corresponding to the discretization of the bilinear form (2.6) by finite elements. A simple computation shows that CN ≥

C(Ω, h) σ+ β

with C(Ω, h) being a constant independent of β. As β goes to zero, the inversion of the linear system becomes more and more unstable, hence it is justified to expect that the quality of the computed approximation will eventually deteriorate. (b) According to (5.3), in order to simulate measurement data U by using UV , a triangulation much coarser than what one would need for approximating u by uV is sufficient. In gradient based reconstruction algorithms of EIT, an approximation of U 0 , i.e., the Fr´echet derivative of U with respect to a finite dimensional σ (often among other parameters [12, 8]), depends also on uV . In that case it is well motivated to use a finer triangulation for the approximation of U 0 than for the mere simulation of the electrode data. 2.0

1.5

1.0

0.5

0 −5 10

−4

10

−3

10

−2

10

−1

10

0

10

Fig. 5.2: Convergence rate of the FE approximation by piecewise linears as a function of β. On the vertical axis is the estimated (by least squares) slope in log h. The L2 and H 1 -errors are computed over Ω for the interior potential, and the error in the measurement map is measured in the operator norm of RM ×M . The horizontal lines illustrate the respective estimates obtained for the SM (dashed, meas. map; dotted, L2 (Ω)-norm; solid, H 1 (Ω)-norm).

ˆh ˆ with a mesh parameter h We approximate the ’exact’ solution by taking V considerably smaller than the ones for any of the explored V. In Fig. 5.2 the estimated h-convergence rates are illustrated in different norms as a function of β. The applied

19 current inputs are chosen as above in Sec. 6. Again, each one of the estimated rates is obtained from a least squares fit of a linear function in log h. For comparison, ˆ 0 as Galerkin we performed the calculation also for the SM case i.e. using V0 and V spaces, respectively. 6. Conclusions. We have shown that the CEM converges to the SM as the contact impedance tends to zero. In smooth domains, we derived a discrepancy of the order O(z s ), 0 ≤ s < 21 , between the two models. Moreover, using a duality argument, it was possible to demonstrate that, in theory, the difference between the corresponding electrode measurement maps is O(z s ), 0 ≤ s < 1. The first numerical experiment verified these rates to a certain extent. We also pointed out that the spatial part of the SM solution has Sobolev regularity of a half degree less than that of the CEM. Given that typical convergence estimates for the h-FEM rely on Sobolev smoothness, the results of the latter numerical experiment support this drop in regularity, and point out that the FEM gives a more accurate approximation for the CEM when z  0. 7. Acknowledgements. The authors would like to thank professor Nuutti Hyv¨ onen for carefully reading the manuscript and suggesting improvements. REFERENCES [1] D. C. Barber and B. H. Brown, Applied potential tomography, J. Phys. E: Sci. Instrum., 17, (1984) 723–733. [2] L. Borcea, Electrical impedance tomography, Inv. Prob. 18, (2002) R99–R136. [3] S. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Method, Springer– Verlag, 2008. [4] M. Cheney, D. Isaacson and J. C. Newell, Electrical Impedance Tomography, SIAM Rev., 41, (1999) 85–101. [5] K.-S. Cheng, D. Isaacson, J. S. Newell and J. C. Newell, Electrode models for electric current computed tomography, IEEE Trans. Biomed. Engrg. 36, (1989) 918–924. [6] M. Costabel and M. Dauge, A singularly perturbed mixed boundary value problem, Commun. Partial Differential Equations, 21, (1996) 1919–1949. [7] M. Costabel, M. Dauge and M. Suri, Numerical approximation of a singularly perturbed contact problem, Comput. Methods Appl. Mech. Engrg. 157, (1998) 349–363. ´, N. Hyvo ¨ nen, A. Seppa ¨ nen and S. Staboulis, Simultaneous recovery of admittivity [8] J. Darde and body shape in electrical impedance tomography: An experimental evaluation, Inv. Prob, 29, (2013) 085004. [9] R. Dautray and J-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 2, Springer–Verlag, Berlin, 1988. [10] L. C. Evans, Partial Differential Equations, 2nd ed., Amer. Math. Soc., 2010. ¨ nen and T. Tuominen, On hp-adaptive solution of complete electrode for[11] H. Hakula, N. Hyvo ward problems of electrical impedance tomography, J. Comput. Appl. Math., 236, (2012) 4635–4659. [12] L. M. Heikkinen, T. Vilhunen, R. M. West and M. Vauhkonen, Simultaneous reconstruction of electrode contact impedances and internal electrical properties: II. Laboratory experiments, Meas. Sci. and Technol., 13, (2002) 1855–1861. ¨ nen, Complete electrode model of electrical impedance tomography: Approximation [13] N. Hyvo properties and characterization of inclusions, SIAM J. App. Math., 64, (2004) 902–931. ¨ nen, Approximating idealized boundary data of electric impedance tomography by [14] N. Hyvo electrode measurements, Mathematical Models and Methods in Applied Sciences, 19, (2009) 1185–1202. [15] J-L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Vol. I, Springer–Verlag, Berlin, 1972. ˇas, Direct Methods in the Theory of Elliptic Equations, Springer–Verlag, 2012. [16] J. Nec [17] M. Pidcock, S. Ciulli and S. Ispas, Singularities of mixed boundary value problems in electrical impedance tomography, Physiol. Meas, 16, (1995) A213–A218.

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