Eddy current approximation in Maxwell obstacle problems

. This paper analyzes the mathematical modeling of the transient eddy current approximation in the Maxwell obstacle problem. Here, the medium is assumed to be solely open, containing conducting and non-conducting materials with certain properties of anisotropy and non-smoothness. The proposed evolutionary PDE model preserves the Faraday law and excludes the displacement current from the governing Amp J ere–Maxwell variational inequality (VI). Our study strives to justify this model and delivers two main results: Global well-posedness of the model and its quantitative precision by uniform a priori estimates. The latter result yields an explicit bound for the smallness condition on the ratio between the electric permittivity and the electric conductivity in the region where the displacement current is disregarded. Below this threshold, the eddy current solution provides the desired reasonable approximation and justifies the proposed model.


Introduction
Maxwell obstacle problems describe the dynamics and propagation of electromagnetic (EM) fields under the influence of constraints. For instance, in electromagnetic shielding, certain magnetic and conducting materials may serve as a barrier to redirect or block EM fields in a specific domain of interest. From the mathematical perspective, this phenomenon falls into the class of Maxwell obstacle problems: In the free region, EM fields satisfy Maxwell's equations, while in the shielded area (unilateral or bilateral) constraints are imposed on the fields. The very first contribution to this research direction was made by Duvaut and Lions [11,Chapter 7,Section 8], who explored and analyzed the electromagnetic wave propagation in a polarizable medium through a Maxwell obstacle problem involving an electric constraint of the type jE.x; t /j Ä d.x/ a.e. in .0; T / for some obstacle d W ! OE0; 1. Based on the method of vanishing curl curl-viscosity and constraint penalization, they proved a well-posedness result [11,Chapter 7, Theorem 8.1] which was modified some years later by Milani [21,22] to the case of a time-dependent obstacle d D d.x; t /. More recently, building on [34], the second author [35] refined the developed theory by Duvaut and Lions [11] to allow a more general constraint structure. Let us formulate the Maxwell obstacle problem we focus on in this paper: Suppose that R 3 is an open set representing an anisotropic medium in which the electric field E W .0; T / ! R 3 and the magnetic field H W .0; T / ! R 3 are acting in a finite time interval .0; T /. Furthermore, let 0 2 K L 2 . / denote a convex and closed subset representing the underlying feasible set for the electric field. Given initial data . Here, "; ; W ! R 3 3 denote the electric permittivity, the magnetic permeability, and the electric conductivity, respectively. All these coefficients are allowed to be nonsmooth. Moreover, as the medium may contain different conducting and non-conducting materials, the electric conductivity is assumed to be merely positive semi-definite. The precise mathematical assumptions for all data involved in (P) are specified in Assumption 1.1. Note that (P) preserves the Faraday law but modifies the AmpJ ere-Maxwell equation into a variational inequality of the first kind. The global well-posedness of (P) for Á 0 is a special case of [35,Theorems 1 and 2] and can be extended to the case of a non-vanishing conductivity . The present paper aims to explore the eddy current (magneto-quasistatic) approximation to (P) and its justification. Our analysis is mainly motivated by the profound role of eddy current modeling in electrical engineering applications and low-frequency physics. Generally speaking, the eddy current model approximates the full Maxwell system by excluding the displacement current " d dt E while still preserving the Faraday law. Such approximations are widely used in the engineering community and are particularly reasonable if the electric permittivity is significantly smaller than the electric conductivity, and the corresponding wavelength is much larger than the diameter of . From among many other contributions to the eddy current model, we refer to the monographs by Alonso and Valli [3] and Bossavit [6,8], and papers [1,4,5,10,14,15,23,26,28,29,31]. While the mathematical and numerical analysis for the eddy current equations seems to have reached an advanced stage of development, so far, we are not aware of any previous study regarding the justification of eddy current modeling for (P).
We focus on an eddy current model allowing the displacement current to be disregarded in an open conducting subregion . More precisely, we look for a unique solution .E; H/ 2 W 1;1 ..0; T /; L 2 . n / L 2 . // \ L 1 ..0; T /; H 0 .curl/ L 2 . // to (P ec ) To justify the eddy current model (P ec ), there are two open mathematical questions to be rigorously addressed and answered. First, the model itself (see (P ec )) has to be reasonable in the sense that there exists a unique solution to it. Second, under a suitable condition, its solution must provide a good estimation for the original problem (see (P)). In particular, inspired by the time-harmonic case [3], (P ec ) should serve as a reasonable approximation if the quantity 1 k"k L 1 . / 3 3 is small enough, with > 0 denoting a uniform lower bound for the lowest eigenvalues of .x/ for almost all x 2 . This paper develops three novelties delivering positive answers to the issues mentioned above. The first novelty concerns the a priori analysis for the time-discrete approximation (P N ) of (P ec ) based on the Rothe method. Here, our analysis hinges on the mild compatibility assumption (1.8) for the initial data .E 0 ; H 0 / in the subset . With the compatibility condition, we prove the stability of (P N ) (see Theorem 2.1) through the use of special correction terms developed using the variational inequality structure of (P ec ). Then, the analysis for the time-discrete scheme (P N ) allows us to establish a well-posedness result for (P ec ) as the second novelty of this paper (Theorem 3.1). To be more precise, applying the stability result to a specific interpolation of (P N ) and passing to the limit in the time discretization, the weak-star limit of the interpolation turns to satisfy (P ec ), leading to an existence result for (P ec ). We note that the standard technique of passing to the weak-star limit of the piecewise linear interpolations fails to work, as (P N ) does not admit sufficient stability of its solutions in . This difficulty is overcome by considering the weak-star limit of the piecewise constant interpolations. Let us point out that (P ec ) does not exclude the displacement current " d dt E in n . In Section 3.1, we address the case described by (P 0 ec ) where the displacement current is entirely neglected both in the conducting and non-conducting regions ( and n ). It turns out that the proposed techniques for (P ec ) can be extended to (P 0 ec ), leading to a well-posedness result for (P 0 ec ) (see Theorem 3.2) under additional assumptions (see Assumption 3.2). Our final result is the justification for (P ec ) (see Theorem 4.1): If j n j ¤ 0, then the solution .E ec ; H ec / of (P ec ) approximates the solution .E; H/ to (P) through the following uniform a priori estimate: where L. / > 0 (resp. L. n / > 0) stands for the Lipschitz constant of f j (resp. f j n ), and ". n / > 0 denotes a uniform lower bound for the lowest eigenvalues of ".x/ for almost all x 2 n . If D , that is, if the displacement current is completely removed in the conducting medium , then the following precision is obtained for the eddy current approximation: We emphasize that in many electromagnetic applications (see, e.g., [3,20]), the ratio k"= k L 1 . / 3 3 is often negligibly small. For instance, stainless steel and copper admit the value 6:14 10 18 and 1:56 10 19 for the corresponding ratio, respectively. This property is in particular satisfied by every good conductor (see [20]) as the electric permittivity " is in this case very close to the one in a vacuum ( 8:85 10 12 ), and the electric conductivity is in the order of 10 6 -10 7 . Therefore, the achieved estimation reveals the desired approximation by the eddy current solution with a specific bound for the smallness condition on the quantity k"= k L 1 . / 3 3 . At the same time, it guarantees the strong convergence of (P ec ) towards (P) with a linear convergence rate in terms of k"k L 1 . / 3 3 . Last but not least, all theoretical results of this paper, in particular uniform estimates (1.1) and (1.2), also apply to the classical Maxwell equations by simply considering K D L 2 . /.
Another important application of the eddy current model arises in the context of type-II superconductivity. The corresponding model leads to parabolic obstacle problems with first-order gradient or curl constraints. We refer to [7,12,24,27,30,32,33] for contributions in this research direction. More recently, a unified analysis for non-linear parabolic obstacle problems, including those with curl-type constraints, has been recently developed by Miranda et al. [25].
The remainder of this paper is organized as follows: In the upcoming section, we introduce our notation as well as basic properties and present the required assumptions for our analysis. The subsequent section includes the formulation of the time-discrete scheme together with its associated a priori stability analysis (Theorem 2.1). Section 3 is devoted to the existence and uniqueness analysis (Theorems 3.1 and 3.2) for (P ec ) and its full eddy current version (P 0 ec ). In Section 4, we prove Theorem 4.1 for the justification of the eddy current model, and the final section features a numerical test verifying the a priori estimate and the predicted convergence rate (see Theorem 4.1).

Preliminaries
Given a real Hilbert space H , we denote by . ; / H and k k H its scalar product and induced norm, respectively. In the case of H D R d , we simply write a dot and j j for the Euclidean scalar product and norm, respectively. Note that given a 0 ; : : : ; a n 0 2 H for n 0 2 N, the binomial-type formula n 0 X nD1 .a n a n 1 ; a n / H D 1 2 holds. Discussing problems of Maxwell-type, there naturally arise function spaces of R 3valued functions. We will therefore use a bold typeface to indicate them. Given an open set R 3 , let L 2 . / denote the space of all (equivalence classes of) R 3 -valued Lebesgue square-integrable functions. We introduce the Hilbert space ndowed with its natural graph norm. Here the curl operator is to be understood in the sense of distributions. Furthermore, let C 1 0 . / denote the space of infinitely differentiable R 3 -valued functions with compact support in . The subspace H 0 .curl/ stands for the closure of C 1 0 . / with respect to the H.curl/ topology. We recall that H 0 .curl/ admits the useful characterization By L 1 sym . / 3 3 we denote the space of all (equivalence classes of) symmetric R 3 3valued Lebesgue measurable and essentially bounded functions with respect to the spectral norm, that is, j˛.x/ j < 1; 8˛2 L 1 sym . / 3 3 : For a given uniformly positive definite matrix-valued function˛2 L 1 sym . / 3 3 , that is, there exists a constant˛> 0 such that .x/ ˛j j 2 for a.e. x 2 and all 2 R 3 ; we denote by L 2 . / the vector space L 2 . / equipped with the weighted scalar product .˛ ; / L 2 . / . We close this section by presenting the mathematical assumptions for our analysis.
Assumption 1.1. We make the following assumptions in our analysis: (1) The set (2) Both the electric permittivity " W ! R 3 3 and the magnetic permeability W ! R 3 3 are of class L 1 sym . / 3 3 and uniformly positive definite, that is, there exist constants "; > 0 such that ".x/ "j j 2 and .x/ j j 2 for a.e. x 2 and all 2 R 3 : (1.6) The electric conductivity W ! R 3 3 is of class L 1 sym . / 3 3 and positive semidefinite. Furthermore, it is uniformly positive definite on , that is, there exists a constant > 0 such that .x/ j j 2 for a.e. x 2 and all 2 R 3 : (1.7) (3) The obstacle set K L 2 . / is assumed to be closed and convex containing 0.
(5) The initial value satisfies . If E 0 .x/ D 0 holds for a.e. x 2 and the first components of curl H 0 and f.0/ are non-positive a.e. in , then condition (1.8) is satisfied in the case of (1.11). Lastly, we note that in the case given by (P), that is to say D ;, (1.8) is always satisfied since all integrals over vanish. In particular, for (P) there is no restriction on the initial value other than .E 0 ; H 0 / 2 .K \ H 0 .curl// H.curl/.
2. Analysis of the time-discrete approximation to (P ec ) This section is devoted to the analysis of the time-discrete approximation to (P ec ) based on the Rothe method. Let us begin by introducing an equidistant partition of the time interval OE0; T as follows: Given N 2 N, we set WD T N ; 0 D t 0 < t 1 < : : : < t N D T with t n WD n ; n 2 ¹0; : : : ; N º: Furthermore, we introduce the backward Euler difference quotients ıE n WD E n E n 1 ; ıH n WD H n H n 1 ; 8n 2 ¹1; : : : ; N º (2.1) and set f n WD f.t n / 2 L 2 . / for all n 2 ¹0; : : : ; N º: Invoking these quantities, the timediscrete (Euler) approximation to (P ec ) reads as follows: Find elements ¹.E n ; H n /º N nD1 .K \ H 0 .curl// H.curl/ such that curl/ 8n 2 ¹1; : : : ; N ºI ıH n C curl E n D 0; 8n 2 ¹1; : : : ; N º: To derive an existence and uniqueness result for (P N ), let us consider a bounded and coercive bilinear form a W H 0 .curl/ H 0 .curl/ ! R given by n "E n 1 v dx; 8n 2 ¹1; : : : ; N º: In view of (2.1), (P N ) is equivalent to the problem of successively finding elements .E 1 ; H 1 /; : : : ; .
The well-posedness of (P N ) therefore follows from the classical theory of elliptic variational inequalities (see [18,Theorem 2.1] or [13, Theorem 3.1]), which we summarize in the following lemma: In the upcoming theorem, we prove our first main result on the stability for (P N ). For the convenience of the reader, we recall Gronwall's lemma in its discrete version in the following auxiliary lemma (see [9, page 280]): kD0 and ¹b k º 1 kD0 be sequences of non-negative real numbers satisfying a n Ä c C n 1 X kD0 a k b k ; 8n 2 N for some constant c > 0. Then, it holds that a n Ä c exp As pointed out in the introduction, our upcoming stability proof is based on the use of certain correction terms z 2 L 2 . n / and w 2 L 2 . / for the initial data, which are defined as follows: Theorem 2.1. Let Assumption 1.1 be satisfied. Then, there exists a positive real constant C 0 depending only on T; "; Proof. Let N 2 N be arbitrarily fixed and let ¹.
.K \ H 0 .curl// H.curl/ denote the unique solution to (P N ). Now let v 2 K \ H 0 .curl/ be arbitrarily fixed. Multiplying the above equation for z by v E 0 and integrating the resulting equality over n , we obtain that Then, combining the above equality with (1.8), it follows that and consequently, applying characterization (1.4) to the previous inequality, the initial data .
Now, the underlying system (see (2.3)) allows us to incorporate the initial data .E 0 ; H 0 / to the time-discrete scheme (P N ) and preserve its pivotal structure for our stability analysis.
To realize this, we employ the quantities and deduce from (2.3) that the unique solution to (P N ) fulfills curl/; 8n 2 ¹0; : : : ; N ºI ıH n C curl E n D w N n ; 8n 2 ¹0; : : : ; N º: (2.5) For every n 2 ¹1; : : : ; N º, setting v D E n 1 (resp. v D E n ) in the n-th inequality of (2.5) (resp. the .n 1/-th inequality of (2.5)) and then dividing the resulting inequalities by , On the other hand, the second equation in (2.5) yields that curl ıE n D 1 .ıH n ıH n 1 / C 1 .w N n w N n 1 /; 8n 2 ¹1; : : : ; N º: (2.8) Adding (2.6) and (2.7) together and then utilizing (2.8), as well as the positive semidefiniteness of , we get .w N n w N n 1 / ıH n dx; 8n 2 ¹1; : : : ; N º: Now let n 0 2 ¹1; : : : ; N º be arbitrarily fixed, and sum up inequality (2.9) over ¹1; : : : ; n 0 º. Then, applying binomial formula (1.3) along with Hölder's inequality and (1.7), it follows that Using Young's inequality together with an estimate of the type˛k k 2 L 2 . / Ä k k 2 L 2 . / and the Lipschitz property of f, the first and second terms in the right-hand side of (2.10) can be estimated by For the remaining terms on the right-hand side of (2.10), we find by Young's inequality and the triangle inequality that and analogously, Applying (2.11)-(2.14) to (2.10) along with ıE 0 D E 0 and ıH 0 D H 0 , it follows after some rearrangement that By virtue of Lemma 2.2, we eventually deduce that with a generic constant C > 0 depending only on T; L; "; , ; E 0 ; H 0 . In particular, it holds that ıE n 0 L 2 . n / C ıH n 0 L 2 . / Ä C: From (2.15) and the reversed triangle inequality, it follows by the definition of the difference quotients in (2.1) that immediately results from (2.15) along with the discrete Faraday law in (P N ). We are left with showing the estimate for E n 0 in L 2 . /. To do so, we test with v D 0 in (P N ) and use the positive semi-definiteness of to obtain Applying the estimate (1.7) and Hölder's inequality, we end up with where all the terms on the right-hand side are bounded due to the stability shown before. Since n 0 2 ¹1; : : : ; N º was chosen arbitrarily, (2.15), (2.16), (2.17) and (2.19) imply that the a priori estimate (2.2) is valid.

Well-posedness
This section is devoted to the well-posedness analysis for the eddy current obstacle problem (see (P ec )) based on the time-discrete approximation (P N ). As a preparation, for every N 2 N, we set up piecewise linear and piecewise constant (in time) interpolations out of the solution ¹.E n ; H n /º N nD1 .K \ H 0 .curl// H.curl/ of (P N ) as follows: H n 1 C .t t n 1 /ıH n if t 2 .t n 1 ; t n ; (3.1) and In view of (P N ), it follows immediately that the above interpolations satisfy  Indeed, the definition of the weak time derivative implies that and hence d dt E D . Analogous arguments are also valid for H, which concludes the proof of (3.7). Altogether, the weak-star limit .E; H/ enjoys the regularity property from which it follows that To this aim, let t 2 .0; T ; w 2 L 2 . n /, and 2 C 1 .OE0; t / be arbitrarily fixed. Integration by parts yields Now let v 2 L 2 ..0; T /; H 0 .curl// be arbitrarily fixed and satisfy v.t / 2 K for a.e. t 2 .0; T /. By standard properties of the limit superior, we deduce that (3.14) lim N !1 Our next step is to estimate the remaining terms on the right-hand side of (3.15). First of all, by the weak sequential lower semi-continuity of the squared norm, we infer that Using once again the weak sequential lower semi-continuity of the squared norm, we find that lim sup  and therefore lim sup for all v 2 L 2 ..0; T /; H 0 .curl// such that v 2 K a.e. on .0; T /. Finally, to show that the variational inequality in (P ec ) holds, let us assume the contrary, that is, Inserting v WD M z C .0;T /nM E into (3.21) immediately contradicts (3.22). In conclusion, .E; H/ satisfies the variational inequality in (P ec ). This completes the existence proof. Uniqueness and Lipschitz stability. Let .E 1 ; H 1 / and .E 2 ; H 2 / denote, respectively, solutions to (P ec ) associated with the initial data .E 1 0 ; H 1 0 /; .E 2 0 ; H 2 0 / and the right-hand sides f 1 ; f 2 satisfying Assumption 1.1. Setting v D E 2 .s/ in (P ec ) for E D E 1 (resp. v D E 1 .s/ in (P ec ) for E D E 2 ) and multiplying with 1, we have  By integration over the time interval .0; t / and rearrangement, it follows that Employing the Gronwall lemma, we then arrive at C .OE0;T ;L 2 . // ; 8t 2 OE0; T : In view of (3.26), we conclude that (P ec ) admits at most one solution.
Remark 3.1. Introducing the subset is Lipschitz continuous, as a consequence of (3.26).

The full eddy current case in the presence of a non-conducting region
Up to this point, the displacement current d dt E was only neglected in the region where is uniformly positive definite. In this section, we suppose that n is of non-zero Lebesgue measure and represents an insulating region, that is, D 0 a.e. in n : Our focus lies on the full eddy current case where the displacement current is completely removed in the whole domain containing the insulating region n . Here, the previously developed analysis serves as the foundation to cover this case with some additional assumptions as follows: Assumption 3.2. We make the following assumptions: (6) It holds that j n j ¤ 0 and D 0 a.e. in n ; f D 0 a.e. in .0; T / . n /; curl H 0 D 0 a.e. in n : (3.27) (7) The obstacle set K satisfies one of the following conditions: represents an insulating region such as air, it is physically reasonable to assume that no current source is present in the insulator. Condition (3.27) on the vanishing source and vanishing initial rotational magnetic field in the insulator is indeed common in the study of the eddy current problems (see, for example, [29, page 42] or [3, page 239]).
(ii) Condition (3.28a) is obviously satisfied if the obstacle set K is bounded in L 2 . /. A prominent example is the set K D ¹v 2 L 2 . / j jv.x/j Ä d.x/ for a.e. x 2 º for some electric obstacle d 2 L 2 . /. On the other hand, condition (3.28b) describes a physical medium with vanishing charge density, that is, the case where the electric field satisfies div."E/ Á 0.
Let us now state the full eddy current problem we focus on in this section: Note that in contrast to (P ec ), problem (P 0 ec ) comprises an elliptic VI for the electric field E and an evolutionary equation for the magnetic field H, which is why we do not impose any initial condition for E (cf. [23] for the case of the full eddy current equations with a constant and scalar conductivity > 0). The time-discrete approximation for (P 0 ec ) reads as finding ¹.E n ; H n /º N nD1 .K \ H 0 .curl// H.curl/ such that curl/; 8n 2 ¹1; : : : ; N ºI ıH n C curl E n D 0; 8n 2 ¹1; : : : ; N º: To prove the well-posedness of (P 0 N ), we reformulate it as a minimization problem in a Hilbert space as follows: Proof. First, using the discrete Faraday law, we rewrite problem (P 0 N ) as Next, let n 2 ¹1; : : : ; N º be arbitrarily fixed. For any v 2 H 0 .curl/, it holds that This shows that the objective functional associated with (3.29) is bounded from below. Therefore, since K \ H 0 .curl/ is non-empty, we have that there exists an infimal sequence ¹v n k º 1 kD1 K \ H 0 .curl/ for minimization problem (3.29). Thanks to (3.30), the infimal sequence ¹v n k º 1 for some constant C > 0, independent of k. Now, if (3.28a) is satisfied, then in view of (3.31) it follows that the infimal sequence ¹v n k º 1 kD1 is bounded in H 0 .curl/. On the other hand, if (3.28b) is satisfied, then it implies the Poincaré-Friedrichs-type inequality [2, Lemma 3.1] (3.32) which yields due to (3.31) the boundedness of ¹v n k º 1 kD1 in H 0 .curl/. In conclusion, for every n 2 ¹1; : : : ; N º, the existence of a minimizer to (3.29) follows by standard arguments as in the proof of the direct method of variational calculus. Finally, if (3.28b) holds true, then due to (3.32) the objective functional associated with (3.29) is strictly convex, and so minimization problem (3.29) admits a unique solution.
Proof. Let N 2 N be arbitrarily fixed and let ¹.E n ; H n /º N nD1 .K \ H 0 .curl// H.curl/ denote a solution to (P 0 N ). Further, let v 2 K \ H 0 .curl/ and n 0 2 ¹1; : : : ; N º be arbitrarily fixed. The lines of argumentation are similar to Theorem 2.1, where we simply set z to be zero, thanks to (3.27) and since " does not appear in (P 0 N ). Then, together with the fact that f D 0 a.e. in .0; T / . n /, by analogous argumentation to the proof of Theorem 2.1, it follows that In turn, this implies the estimate In view of the above estimate, we obtain kH n 0 k 2 L 2 . / Ä C and k curl E n 0 k L 2 . / Ä C , due to (2.16) and the discrete Faraday law (P 0 N ). The bound on kE n 0 k L 2 . / is obtained by testing with v D 0 in (P 0 N ) and proceeding as in (2.18) and (2.19). The bound on kE n 0 k L 2 . n / is an immediate result of (3.28a) or (3.28b) along with the Poincaré-Friedrichs-type inequality (3.32) and the estimate k curl E n 0 k L 2 . / Ä C .
In view of (P 0 N ), invoking again constructions (3.1) and (3.2), it follows that the interpolations satisfy which implies the initial condition H.0/ D H 0 a.e. in . Also, as in the proof of Theorem 3.1, the above weak-star convergence yields the feasibility E.t / 2 K for a.e. t 2 .0; T /. Ultimately, the final passage to the limit in ( z P 0 N ) follows again the same arguments as in the proof of Theorem 3.1. In conclusion, the weak-star limit .E; H/ satisfies (P 0 ec ). Let us now assume that (3.28b) is valid and let .E 1 ; H 1 /, and .E 2 ; H 2 / denote, respectively, solutions to (P 0 ec ). As H 1 .0/ D H 0 D H 2 .0/, the above inequality implies that H 1 D H 2 , which yields due to the Faraday law in (P 0 ec ) that curl.E 1 E 2 / D 0. As a result of the Poincaré-Friedrichstype inequality (3.32), it then follows that E 1 D E 2 . This completes the proof. Remark 3.6. Let us mention that even in the case of the eddy current equations, without assuming additional conditions such as (3.28b), uniqueness of the solution cannot be expected in general.
Remark 3.7. The analysis in this section with respect to condition (3.28b) reveals that a local Poincaré-Friedrichs-type inequality in the insulator n is sufficient to obtain an existence and uniqueness result for (P 0 ec ). This allows us to work with another obstacle set K as follows: Suppose again that is a bounded Lipschitz domain such that and n is connected. Then, for the obstacle set K, an alternative assumption to (3.28b) reads ."v; h/ L 2 . n / D 0 8 h 2 H¯; where H denotes the finite-dimensional vector space of Neumann fields related to topological quantities of the physical domain and the insulating region n (see [3, page 13] for its definition). As proven in [3,Lemma 2.2], the Poincaré-Friedrichs-type inequality below holds true: With (3.35) at hand, the existence of a unique solution to (P 0 ec ) is obtained under minor and obvious changes of this section. Our goal now is to justify the eddy current model (P ec ) in the sense that its unique solution .E ec ; H ec / is close to .E; H/ under a reasonable smallness condition on k"= k L 1 . / 3 3 . By proposing an additional assumption on the initial data (see Assumption 4.1), we are able to not only justify the eddy current model but also prove an a priori error estimate for the eddy current approximation with a linear convergence rate in terms of k"k L 1 . / 3 3 . In what follows, if j n j ¤ 0, the constant ". n / denotes a uniform lower bound for the lowest eigenvalues of " in n , that is, it satisfies

Justification of the eddy current model
".x/ ". n /j j 2 for a.e. x 2 n and all 2 R 3 : (4.5) Furthermore, let L. / and L. n / denote, respectively, the Lipschitz constants of f in and n , that is,  Proof. We split the proof into three parts.
Step 1: / ds with an upper bound being independent of " j . Setting v D E.s C h/ (resp. v D E.s/) in (P) for t D s (resp. t D s C h) and then adding the resulting inequalities, we obtain (similarly to the uniqueness proof for Theorem 3.1) by employing the Faraday law, the properties of , and Hölder's inequality that for a.e. s 2 .0; T / and a.e. h 2 .0; T s/. Integrating the above inequality over .0; t / and dividing by h 2 , we obtain that 1 2 for any t 2 .0; T / and h 2 .0; T t /. Note that if j n j D 0 then all integrals over n vanish, and we may simply set ". n / D 1 in the case of j n j D 0. Now, by Lipschitz property (4.6) and the regularity property E 2 W 1;1 ..0; T /; L 2 . //, it follows that Our goal now is to show the boundedness of the difference quotients at the point 0 appearing on the right-hand side of (4.10). Setting v D E 0 in (P) yields Applying (4.14) to (4.13), integrating the resulting inequality over .0; h/ and dividing by h 2 , we follow the same argumentation as before to deduce by the Hölder and Young inequalities as well as the properties of that and consequently, by the Lipschitz continuity of f as well as rearrangement, we arrive at ds; 8h 2 .0; T /: In conclusion, Gronwall's lemma delivers for all h 2 .0; T /. Going back to (4.10) and on account of (4.15), we attain 2L. / 2 T ; 8t 2 .0; T /: (4.18) Step 2: The proof of (4.7) for j n j ¤ 0. We start by inserting v D E.s/ in (P ec ) and v D E ec .s/ in (P) to obtain that for all t 2 .0; T /. In view of regularity properties (4.1) and (4.2), the above pointwise estimate leads immediately to uniform estimate (4.7).
Step 3: The proof of (4.8) for D . In this case, inequality (4.16) turns out to be The final claim (see (4.8)) follows therefore by applying (4.26) to (4.27).

Numerical verification
We close this paper with a brief numerical verification of our theoretical findings. In particular, our numerical test confirms the linear convergence rate with respect to " j for the eddy current approximation (see Theorem 4.1). Note that the following example is of merely academic nature as the conducting domain is chosen to be equal to the whole domain. So, for the test, we consider D . 1; 1/ 3 , T D 1, Á 1, Á 1 and D , with .0; 0/ as an initial value. For the applied current source, we choose f W OE0; 1 ! R 3 defined by f.t; x 1 ; x 2 ; x 3 / WD where P D ¹.x 1 ; x 2 ; x 3 / 2 R 3 j 0 Ä x 1 Ä 0:5; 0:3 Ä q x 2 2 C x 2 3 Ä 0:5º models a cylindrical pipe coil. Furthermore, the feasible set is set to be K D ® v 2 L 2 . / j jv.x/j 1 Ä 5 10 4 for a.e. x 2 !¯; with the obstacle region ! WD ® .x 1 ; x 2 ; x 3 / 2 j 0:25 Ä x 1 Ä 0:125; jx 2 j Ä 0:5; jx 3 j Ä 0:5¯: Note that the choice of the bound 5 10 4 in the obstacle set K is of no particular importance. With the choice of our bound, we strive to model the effects of electric shielding. Our numerical computation is based on the time-discrete (implicit Euler) scheme (P N ) along with the space discretization consisting of Nédélec's edge elements [17] for E and piecewise constant elements for H. The corresponding finite element approximation of the time-discrete problems in (P N ) (with roughly 829; 000 degrees of freedom) were solved by the primal dual active set algorithm (see [16]) implemented on the open-source platform FEniCS [19] (see Figures 1 and 2 for a visualization). We note that the primal dual method approximates the elliptic variational inequalities in (P N ) by equalities on the corresponding active and inactive sets that are iteratively updated. To verify the convergence of the eddy current approximation, we use the quantity with .E; H/ k being the numerical solution to (P) for " D 1 2 k and .E ec ; H ec / being the numerical solution to the eddy current model (P ec ). Furthermore, to check the experimental order of convergence with respect to ", we make use of the following quantity: EOC k D log.Error kC1 / log.Error k / log.2 .kC1/ / log.2 k / : Table 1 depicts the computed error and experimental order of convergence for the values k D 4; : : : ; 14. In agreement with our theoretical finding (Theorem 4.1), we observe that the eddy current approximation (P ec ) becomes closer and closer to (P) as " decreases. More importantly, the experimental order of convergence is readily very close to 1, which exactly confirms the linear convergence rate in a priori error estimate (4.8).  Table 1. Convergence behavior of the eddy current approximation.  Funding. This work was partially supported by the German Research Foundation (DFG Grants YO 159/4-1 and YO 159/2-2).