A unified variational formulation for the parabolic-elliptic ... - M1 - TUM
A UNIFIED VARIATIONAL FORMULATION FOR THE PARABOLIC-ELLIPTIC EDDY CURRENT EQUATIONS∗ LILIAN ARNOLD† AND BASTIAN HARRACH‡ Abstract. Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by the eddy current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell’s equations. The eddy current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behaviour), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behaviour). For fixed conductivity, the equations are well studied. However, little rigorous mathematical results are known for the solution’s dependence on the conductivity, in particular for the solution’s sensitivity with respect to the equation changing from elliptic to parabolic type. In this work, we derive a new unified variational formulation for the eddy current equations, that is uniformly coercive with respect to the conductivity. We then apply our new unified formulation to study the case when the conductivity approaches zero, and rigorously linearize the eddy current equations around a non-conducting domain with respect to the introduction of a conducting object. Key words. eddy current problem, parabolic-elliptic equation, unified variational formulation AMS subject classifications. 35M10, 35Q60, 35R05
1. Introduction. Transient excitation currents J(x, t) generate electromagnetic fields, E(x, t) and H(x, t), which can be described by Maxwell’s equations ∂E + σE + J, ∂t ∂H curl E = −µ , ∂t
curl H =
where the curl-operator acts on the three spatial coordinates, and (under the assumption of linear and isotropic time-independent material laws) σ(x), (x) and µ(x) are the conductivity, permittivity and permeability of the considered domain. For slowly varying electromagnetic fields, the displacement currents ∂E ∂t can be neglected. This leads to the transient eddy current equations curl H = σE + J, ∂H curl E = −µ , ∂t
(1.1) (1.2)
resp., after elimininating H, ∂t (σE) + curl
1 curl E µ
= −∂t J.
(1.3)
The eddy current model is well-established in the engineering literature, cf., e.g., Albanese and Rubinacci [2] or Dirks [13]. A rigorous mathematical justification has been derived by Alonso [3] and Ammari, Buffa and N´ed´elec [4]. ∗ This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HA 6158/1-1. † Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at M¨ unchen, Germany ([email protected]) ‡ Birth name: Bastian Gebauer, Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at M¨ unchen, Germany ([email protected])
1
2
L. Arnold and B. Harrach
In a typical application the domain under consideration consists of both, conducting regions (σ(x) > 0) and non-conducting regions (σ(x) = 0), which has two interesting consequences. The first one is that equation (1.3) is of parabolic-elliptic type. The physical interpretation is that the time-scale is different in the conducting and the insulating region. In the insulating regions, the field instantaneously adapts to the excitation (quasistationary behaviour), while, in the conducting regions, this adaptation takes some time (due to eddy currents induced by the varying electromagnetic fields). A particular consequence is that initial values are only meaningful in the conducting region. The second consequence is that equation (1.3), together with initial values, does only determine E up to the addition of a gauge field, which is a curl-free field that vanishes inside the conductor. However, in many applications, one is only interested in σE, or curl E, anyway. Several well-posed variational formulations have been proposed for the eddy current equations and used for the numerical solution, cf., e.g. [7, 6, 14, 5, 18, 19, 21, 1]. These approaches concentrate on solving the eddy current equations with a fixed conducting region in which the conductivity is assumed to be bounded from below by some positive constant. Accordingly, the variational formulations, with their underlying solution spaces and coercivity constants, depend, in some form or another, on this lower bound, on the support of the conductivity, or on both. A noteworthy exemption appears when σ is constant inside the conductor. Then, a solution of (1.3) can be found by considering the standard variational formulation of (1.3), see (2.3) below, in the space of divergence-free functions, where it is coercive. In the general case of spatially varying σ, however, the solution of (1.3) will not be divergence free in the conductor. In that case, restricting the standard variational formulation of (1.3) to the space of divergence-free functions will not yield the true solution up to a gauge field. Inverse problems, as well as sensitivity considerations, require a unified formulation with respect to σ. An important application is landmine detection, where a source current in an inductor coil is used to generate electromagnetic fields that, in turn, induce currents in a buried conductor. The resulting change in the magnetic field can then be measured by a receiver coil, so that, from one or several such measurements, one may try to reconstruct information about the buried object, cf., e.g., [17] for a formulation of the corresponding measurement operators. A natural approach to this inverse problems is to start by linearizing the problem with respect to σ around a homogeneous non-conducting state. Roughly speaking, this leads to the following question: how does the solution of the elliptic magnetostatic problem (i.e., (1.3) with σ = 0) change if the problem becomes a little bit parabolic? For a scalar analogue this question has been answered in [16]. To the knowledge of the authors, no rigorous linearization results are known for the eddy current model so far. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to σ. To be more precise, what we present is a variational formulation that is uniformly coercive (and hence uniquely solvable) in the space of divergence-free functions, and whose solution agrees with the true solution up to the addition of a gradient field. At this point, let us stress again, that, for spatially varying σ, the standard variational formulation of (1.3) restricted to divergence-free functions does not determine the solution up to a curl-free field. We apply our new unified formulation, to study the limit of the solutions of (1.3) for σ → 0 and prove convergence against their magnetostatic counterparts. We
A unified variational formulation for the parabolic-elliptic eddy current equations
3
then turn to the above mentioned question and rigorously determine the directional derivative of the solutions of (1.3) with σ = 0, with respect to σ, i.e., we linearize the solutions of the elliptic (magnetostatic) problem with respect to the problem becoming parabolic in some parts. This paper is organized as follows. In Section 2 we introduce the necessary notations, derive the standard variational formulation for equation (1.3), characterize well-posed initial conditions and prove uniqueness of solutions (up to gauge fields). Section 3 contains our main theoretical tool: a new uniformly coercive variational formulation that determines the solution up to the addition of a gradient field. This also proves solvability of the eddy current equations. Finally, in Section 4, we apply our new unified formulation to study the dependence of the solutions when the conductivity approaches zero and rigorously linearize (1.3) around a non-conducting domain with respect to the introduction of a conducting object. 2. The eddy current equation. 3 2.1. Notations and general assumptions. Let T > 0. Let µ ∈ L∞ + (R ), ∞ 3 ∞ 3 where we denote by L+ (R ) the space of L (R )-functions with positive (essential) 3 ∞ 3 infima. Let σ ∈ L∞ ≥ (R ) have bounded support Ω, where L≥ (R ) is the space of ∞ 3 L (R )-functions that are almost everywhere non-negative. D(R), D(R3 ), D(]0, T [), resp., D(R3 ×]0, T [) denote the space of C ∞ -functions in x, t, resp., (x, t), which are compactly supported in R, R3 , ]0, T [, resp., R3 ×]0, T [. We will also use the notation D([0, T [), resp., D(R3 × [0, T [) for the space of restrictions of functions from D(]0, T [), resp., D(R3 ×] − ∞, T [) to ]0, T [, resp., R3 ×]0, T [. D0 (R3 ) denotes the space of linear continuous mappings from D(R3 ) to R and 0 D (R3 )3 the one of D(R3 )3 to R. D0 (R3 ×]0, T [)3 is defined likewise. Let L2ρ (R3 ) and W denote the distributional spaces 2
1
L2ρ (R3 ) : = {E ∈ D0 (R3 ) | (1 + |x| )− 2 E ∈ L2 (R3 )}, W (curl) : = {E ∈ L2ρ (R3 )3 | curl E ∈ L2 (R3 )3 }. L2ρ (R3 )n , n = 1, 3, and W (curl) are Hilbert spaces with norms 2
1
k · kρ := k(1 + |x| )− 2 · kL2 (R3 )n , and k · k2W (curl) = k · k2ρ + k curl · k2L2 (R3 )3 . We introduce the Beppo-Levi spaces W 1 (R3 ) : = {E ∈ L2ρ (R3 ) | ∇E ∈ L2 (R3 )3 }, W 1 := W 1 (R3 )3 = {E ∈ L2ρ (R3 )3 | ∇E ∈ L2 (R3 )3×3 }. In the latter space, ∇E denotes the 3 × 3-Jacobian of E. For any bounded domain O ⊂ R3 , W 1 (R3 \ O) is defined likewise. These spaces are Hilbert spaces on which k∇ · kL2 defines an equivalent norm, cf., e.g., [10, IX, §1] and [11, XI, Part B, §1]. Note, that D(R3 ) is dense in L2ρ (R3 ) and in W 1 (R3 ) and that D(R3 )3 is dense in 2 Lρ (R3 )3 , W (curl) and in W 1 . We denote the dual space of a space H by H 0 and the dual pairing on H 0 × H by h·, ·iH 0 ×H . Throughout this work, we frequently use the dual pairing between W (curl)0 and W (curl), hence we write in this case hG, Ei := hG, EiW (curl)0 ×W (curl)
for G ∈ W (curl)0 , E ∈ W (curl).
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L. Arnold and B. Harrach
We write L2 (R3T ) instead of L2 (R3 ×]0, T [), and usually omit the arguments x and t and only use them where we expect them to improve readability. We consider the space L2 (0, T, W (curl)) as the space to look for a solution of (1.3). Generally, it is not the case that every E ∈ L2 (0, T, W (curl)) has some well-defined initial values. However, in the following we show, that at least every solution of (1.3) has well-defined initial values. Then, we derive a standard variational formulation and discuss, in what sense uniqueness can be expected. In this paper, we assume that we are given Jt ∈ L2 (0, T, W (curl)0 ) with div Jt = 0 and E0 ∈ L2 (R3 )3 with div(σE0 ) = 0. Theorem 2.1. Let E ∈ L2 (0, T, W (curl)). The eddy current problem reads 1 curl E(x, t) = −Jt (x, t) in R3 ×]0, T [, (2.1) ∂t (σ(x)E(x, t)) + curl µ(x) p p σ(x)E(x, 0) = σ(x)E0 (x) in R3 . (2.2) The following holds: a) For every solution E ∈ L2 (0, T, W (curl)) of (2.1) we have √ σE ∈ C(0, T, L2 (R3 )3 ). b) E ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) if and only if E solves Z TZ Z TZ 1 − curl E · curl Φ dx dt σE · ∂t Φ dx dt+ 0 R3 0 R3 µ Z T Z hJt , Φi dt + σE0 · Φ(0) dx =− 0
(2.3)
R3
for all Φ ∈ D(R3 × [0, T [)3 . √ c) Equations (2.1)–(2.2) uniquely determine curl E and σE. Moreover, if E ∈ L2 (0, T, W (curl)) solves then every F ∈ L2 (0, T, W (curl)) with √ (2.1)–(2.2), √ curl F = curl E and σF = σE also solves (2.1)–(2.2). We prove Theorem 2.1 in the following subsection. The initial condition (2.2) can be interpreted as stating that, wherever it makes sense to speak of initial values, they must agree with E0 . In Ω, the equation is parabolic and initial values are meaningful and necessary. Outside of Ω, where the equation is elliptic, initial conditions are meaningless and (2.2) does not contain any information. Let us stress again, in this section we only require that σ is nonnegative, bounded and has bounded support. 2.2. Initial values, a standard variational formulation and uniqueness. For E ∈ L2 (0, T, W (curl)) we have that E(t)|Ω , curl E(t) ∈ L2 (R3 )3 for t ∈]0, T [ a.e. and consequently the products 1 curl E(t), σE(t) ∈ L2 (R3 )3 µ are well-defined. Moreover, the assumption div(σE0 ) = 0 makes sense as E0 ∈ L2 (R3 )3 . As D(R3 )3 is dense in W (curl), we can also regard L2 (0, T, W (curl)0 ) as a subspace of D0 (R3 ×]0, T [)3 . Hence, div Jt is a well-defined in the sense of distributions.
A unified variational formulation for the parabolic-elliptic eddy current equations
5
Now, the transient eddy current equation (2.1) is equivalent to: Z
T
Z
Z
T
Z
σE · ∂t Φ dx dt +
− 0
R3
R3
0
1 curl E · curl Φ dx dt = − µ
Z
T
hJt , Φi dt
(2.4)
0
for all Φ ∈ D(R3 ×]0, T [)3 . In the rest of this subsection, we continue along the lines in [16, Sect. 2]. We first recall the definition of the time-derivative in the sense of vector-valued distributions: For two Banach spaces X, Y and a continuous injection ι : X ,→ Y , E ∈ L2 (0, T, X) has a time-derivative in L2 (0, T, Y ) in the sense of vector-valued distributions, if there exists a E˙ ∈ L2 (0, T, Y ) which fulfills Z
T
E˙ ϕ dt = −
Z
T
ιE ∂t ϕ dt
for all ϕ ∈ D(]0, T [)
0
0
ι
ι0
(cf., e.g., [12, XVIII, §1]). For a Gelfand triple V ,→ H ,→ V 0 of real separable Hilbert spaces V and H, the space n o W(0, T, V, V 0 ) := E ∈ L2 (0, T, V) | E˙ ∈ L2 (0, T, V 0 ) is defined by taking the time-derivative with respect to the injection ι0 ι : V ,→ V 0 . The image of the space W(0, T, V, V 0 ) under ι is continuously imbedded in C(0, T, H) and, for E, F ∈ W(0, T, V, V 0 ), the following integration by parts formula holds: T
Z
h i ˙ hE(t), F (t)iV 0 ×V + hF˙ (t), E(t)iV 0 ×V dt = (ιE(T ), ιF (T ))H − (ιE(0), ιF (0))H ,
0
where (·, ·)H denotes the inner product of H, cf., e.g., [12, XVIII, §1, Thms. 1,2]. In view of (2.1), we introduce the space . Wσ := E ∈ L2 (0, T, W (curl)) | (σE) ∈ L2 (0, T, W (curl)0 ) , where (σE). denotes the time-derivative of σE ∈ L2 (0, T, L2 (R3 )3 ) in the sense of vector-valued distributions with respect to the canonical injection L2 ,→ W (curl)0 . Moreover, we define the space L2σ by taking the closure of √ σE | E ∈ D(R3 )3 ⊂ L2 (R3 )3 Hilbert space equipped with the with respect to the L2 (R3 )3 -norm. L2σ is a separable √ standard L2 (R3 )3 -inner product. Note, that { σE | E ∈ L2 (R3 )3 } ⊂ L2σ . √ 2 Lemma 2.2. If E ∈ Wσ , then σE ∈ C(0, T, L (R3 )3 ). Additionally, for E, F ∈ Wσ the following integration by parts formula holds: Z 0
T
. h(σE) , F i dt +
Z 0
T
. h(σF ) , Ei dt =
Z σ(E(T ) · F (T ) − E(0) · F (0)) dx. (2.5) R3
Proof. In [16, Sect. 2] this lemma is proven for a scalar analog. We repeat the proof for the convenience of the reader. Both, W (curl) and L2σ are separable Hilbert spaces. We define the mapping I by √ I : W (curl) → L2σ , E 7→ σE,
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L. Arnold and B. Harrach
which is continuous and has dense range. We identify the Hilbert space L2σ with its dual. Then, after factoring out the nullspace N of I we obtain, that ι : W (curl)/N → L2σ ,
E + N 7→ IE
defines an injective, continuous mapping and hence a Gelfand triple ι0
ι
W (curl)/N ,→ L2σ ,→ (W (curl)/N )0 . For all G ∈ (W (curl)/N )0 the dual mapping ι0 is given by Z √ hι0 G, F + N i(W (curl)/N )0 ×W (curl)/N = G · σF dx for all F ∈ W (curl). (2.6) R3
Let E ∈ Wσ and G = (σE). ∈ L2 (0, T, W (curl)0 ) be its time-derivative with respect to I 0 I : W (curl) ,→ W (curl)0 . Now we show, that G = (E + N ). with respect 0 to ι0 ι : W (curl)/N ,→ (W (curl)/N ) . Let ϕ ∈ D(]0, T [). For F ∈ N we have Z T Z TZ hG(t), F iϕ(t) dt = − σE(t) · F dx ∂t ϕ(t) dt = 0 0
0
R3
and thus hG(t), F i = 0 for t ∈]0, T [ a.e.. Hence, G(t) ∈ N ⊥ and we can identify G with an element of L2 (0, T, (W (curl)/N )0 ). Then, for F + N ∈ W (curl)/N it follows Z T Z T hG(t), F + N i(W (curl)/N )0 ×W (curl)/N ϕ(t) dt = hG(t), F iϕ(t) dt 0
0
Z
T
Z
=−
σE(t) · F dx ∂t ϕ(t) dt R3
0
Z
T
hι0 ι(E(t) + N ), F + N i(W (curl)/N )0 ×W (curl)/N ∂t ϕ(t) dt
=− 0
and, accordingly, G √ = (E + N ). and E + N ∈ W(0, T, W (curl)/N, (W (curl)/N )0 ). Now, it follows that σE = ι(E + N ) ∈ C(0, T, L2σ ) and, together with (2.6), for all E, F ∈ Wσ the integration by parts formula (2.5). Lemma 2.3. Every solution E ∈ L2 (0, T, W (curl)) of (1.3) is in Wσ and thus has well-defined initial values p σ(x)E(x, 0) ∈ L2σ ⊂ L2 (R3 )3 . For t ∈]0, T [ a.e., (σE). (t) ∈ W (curl)0 is given by Z 1 . h(σE) (t), F i = −hJt (t), F i − curl E(t) · curl F dx for all F ∈ W (curl). (2.7) R3 µ Proof. Let E be a solution of (1.3). Define G(t) ∈ W (curl)0 by Z 1 hG(t), Ψi = −hJt (t), Ψi − curl E(t) · curl Ψ dx µ 3 R for all Ψ ∈ D(R3 )3 . Then G ∈ L2 (0, T, W (curl)0 ), and, due to the fact that E solves (2.4) with Φ = Ψϕ for all ϕ ∈ D(]0, T [) and all Ψ ∈ D(R3 )3 , it holds Z T Z TZ Z T hG(t), Ψiϕ(t) dt = − σE · Ψ dx∂t ϕ dt = − hσE(t), Ψi∂t ϕ(t) dt. (2.8) 0
0
R3
0
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A unified variational formulation for the parabolic-elliptic eddy current equations
Since D(R3 )3 is dense in W (curl) and both sides depend continuously on Ψ, we obtain that equation (2.8) holds for all Ψ ∈ W (curl). Now it follows from the fact, that W (curl) ⊗ D(]0, T [) is dense in L2 (0, T, W (curl)), that G = (σE). with respect to the canonical injection L2 (R3 )3 ,→ W (curl)0 . This shows that E ∈ Wσ . Lemma 2.3 shows, that it makes sense to search E ∈ L2 (0, T, W (curl)) that solves (2.1)–(2.2). Now, we give some equivalent variational formulation: Lemma 2.4. The following problems are equivalent: a) Find E ∈ L2 (0, T, W (curl)) that solves (2.1) and (2.2). b) Find E ∈ Wσ that solves (2.2) and T
Z
. h(σE) , F i dt +
0
Z
T
0
Z R3
1 curl E · curl F dx dt = − µ
T
Z
hJt , F i dt
(2.9)
0
for all F ∈ L2 (0, T, W (curl)). c) Find E ∈ L2 (0, T, W (curl)) that solves T
Z −
. h(σF ) , Ei dt +
0
Z
T
Z R3
0
1 curl E · curl F dx dt µ Z T Z =− hJt , F i dt +
σE0 · F (0) dx
R3
0
√ for all F ∈ Wσ with σF (T ) = 0. d) Find E ∈ L2 (0, T, W (curl)) that solves Z
T
Z
−
Z
T
Z
σE · ∂t Φ dx dt + 0
R3
0
R3
1 curl E · curl Φ dx dt µ Z T Z =− hJt , Φi dt + 0
σE0 · Φ(0) dx
R3
for all Φ ∈ D(R3 × [0, T [)3 . Proof. We start by showing a) ⇒ b). If E ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) it follows from Lemma 2.3 that E ∈ Wσ and (2.9) holds for all F (x, t) = G(x)ϕ(t) with G ∈ W (curl) and ϕ ∈ D(]0, T [). Since W (curl)⊗D(]0, T [) is dense in L2 (0, T, W (curl)) and both sides of (2.9) depend continuously on F ∈ L2 (0, T, W (curl)), b) follows. b) ⇒ c) follows from the integration by parts formula (2.5). c) ⇒ d) follows from the fact that for Φ ∈ D(R3 × [0, T [)3 the time-derivative (σΦ). ∈ L2 (0, T, W (curl)0 ) of σΦ ∈ L2 (0, T, L2 (R3 )3 ) with respect to the injection L2 ,→ W (curl)0 is the image of the classical time-derivative σ∂t Φ(t) under this injection, i.e. Z . h(σΦ) (t), E(t)i = σ∂t Φ(t) · E(t) dx for t ∈]0, T [ a.e. R3
To show d) ⇒ a) we use d) with Φ ∈ D(R3 ×]0, T [)3 . Then E ∈ L2 (0, T, W (curl)) solves (2.2) and Lemma 2.3 yields E ∈ Wσ . Now, the integration by parts formula 3 3 (2.5) applied √ √ on d) with Φ = Ψϕ, Ψ ∈ D(R ) , ϕ ∈ D([0, T [) with ϕ(0) = 1 implies σE0 = σE(0).
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Now, the proof of Theorem 2.1 reads: Proof of Theorem 2.1. a) This is Lemma 2.3. b) This is Lemma 2.4 d). √ c) Assume that E ∈ Wσ is a solution of (2.1)–(2.2) with σE(0) = 0 and Jt = 0. Using Lemma 2.4 b) and the integration by parts formula (2.5) implies Z TZ Z T 1 . h(σE) , Ei dt + 0= curl E · curl E dx dt 0 R3 µ 0 1 √ 1 ≥ k σE(T )k2L2 (R3 )3 + k curl Ek2L2 (R3 )3 . T 2 kµk∞ √ We obtain curl E = 0 and σE = 0. The second assertion is obvious. 3. A unified variational formulation. In this section we present a new, uniquely solvable and uniformly coercive variational formulation that determines the solution of the eddy current equations, (2.1) and (2.2), up to the addition of a gradient field. From this we obtain solvability of (2.1) and (2.2), and a continuity result that is uniform with respect to the conductivity σ. For this result we need stronger assumptions on σ. Let R > 0. In the following, we assume that 3 ∞ 3 ∞ σ ∈ L∞ R (R ) := {σ ∈ L≥ (R ) | Ω := supp σ ⊂ BR , σ ∈ L+ (Ω), and ∃ s ∈ N :
Ω = ∪si=1 Ωi , where Ωi , i = 1, . . . , s, are bounded Lipschitz domains such that Ωi ∪ Ωj = ∅ and R3 \ Ω is connected}, where BR denotes the open ball with radius R. The case of σ ≡ 0 is treated separately. Note, that our continuity results do not depend on the lower bound of σ. The fact that the curl of a solution is unique, but not the solution itself, leads to the idea to work with spaces where k curl · kL2 (R3 )3 defines a norm. Let the index ♦ of the space W 1 (resp., D(R3 × [0, T [)3 , D(R3 )3 ) denote its closed subspace of functions with vanishing divergence. Then, W♦1 equipped with the norm k · k♦ := k curl · kL2 (R3 )3 is a Hilbert space and, similar to the proof of [11, XI, Part B, §1, Lemma 1], we have kEkρ ≤ 2kEk♦
for all E ∈ W♦1 .
(3.1)
Lemma 3.1. There is a continuous linear map W 1 → H(curl 0, R3 ) := {E ∈ L2 (R3 )3 | curl E = 0},
E 7→ ∇uE ,
which yields E + ∇uE ∈ W (curl) and div(σ(E + ∇uE )) = 0
in R3 .
Obviously, this map extends to L2 (0, T, W 1 ) → L2 (0, T, H(curl 0, R3 )),
E 7→ ∇uE ,
where ∇uE (t) := ∇uE(t) for t ∈]0, T [ a.e. and E ∈ H 1 (0, T, W 1 ) implies ∇uE ∈ H 1 (0, T, H(curl 0, R3 ))
and
. (∇uE ) = ∇uE˙ .
(3.2)
A unified variational formulation for the parabolic-elliptic eddy current equations
9
Proof. Let E ∈ W 1 . Due to Poincare’s inequality (cf., e.g., [9, 4, IV, §7, Prop. 2]), the fact, that σ is positively bounded from below on Ω, and Lax-Milgram’s Theorem 1 (cf., e.g., [22, §8, Thm. 8.14]), there exists a unique uE ∈ H (Ω) that solves Z
Z σ∇u · ∇v dx = −
Ω 1 Here, H (Ω) :=
σE · ∇v dx for all v ∈ H 1 (Ω).
(3.3)
Ω
n o R v ∈ H 1 (Ω) | Ωi v dx = 0, i = 1, . . . , s . Further more uE depends
continuously on E|Ω and for any other solution u ∈ H 1 (Ω) it holds ∇u = ∇uE . We extend uE to an element of W 1 (R3 ) by solving ∆u = 0 on R3 \ Ω with u|∂Ω = uE |∂Ω for u ∈ W 1 (R3 \ Ω). Again, Lax-Milgram’s Theorem provides a unique solution, which depends continuously on uE |∂Ω and thus on E|Ω . Let uE , again, denote its extension. Then, uE ∈ W 1 (R3 ) and the mapping E 7→ ∇uE is well-defined, linear and continuous with a continuity constant that depends on the lower bound of σ. Moreover, E + ∇uE ∈ W (curl) and ∇uE solves (3.2). For the rest of this paper, let ∇uE denote the image of E under this mapping. √ Note, that there are different possibilities to construct this map, but σ∇uE is uniquely determined by the condition (3.2). Moreover, it holds √ √ k σ∇uE kL2 (R3 )3 ≤ k σEkL2 (R3 )3 .
(3.4)
We define the bilinear form a : L2 (0, T, W♦1 ) × D(R3 × [0, T [)3 → R by Z
T
Z
T
Z
a(E, Φ) := −
Z
σ(E + ∇uE ) · ∂t Φ dx dt + 0
R3
0
R3
1 curl E · curl Φ dx dt, (3.5) µ
and, cf. Lemma 2.4 d), a linear form l : D(R3 × [0, T [)3 → R: Z l(Φ) := −
T
Z hJt , Φi dt +
σE0 · Φ(0) dx. R3
0
Now we can state the main result of this section: Theorem 3.2. ˜ ∈ L2 (0, T, W 1 ). Then E ˜ + ∇u ˜ ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) a) Let E ♦ E ˜ solves if and only if E ˜ Φ) = l(Φ) a(E,
for all Φ ∈ D♦ (R3 × [0, T [)3 .
(3.6)
a|(D♦ (R3 ×[0,T [)3 )2 is uniformly coercive with respect to k · kL2 (0,T,W♦1 ) : a(Φ, Φ) ≥
1 kΦk2L2 (0,T,W 1 ) ♦ kµk∞
for all Φ ∈ D♦ (R3 × [0, T [)3 .
˜ ∈ L2 (0, T, W 1 ) of (3.6). E ˜ depends continuously b) There is a unique solution E ♦ √ on Jt and σE0 : √ √ ˜ L2 (0,T,W 1 ) ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ). kEk ♦ (3.7)
10
L. Arnold and B. Harrach
˜ + ∇u ˜ solves (2.1)–(2.2) and any other solution E ∈ L2 (0, T, W (curl)) of E E (2.1)–(2.2) fulfills √ √ ˜ + ∇u ˜ ). ˜ σE = σ(E (3.8) curl E = curl E, E √ √ curl E and σE depend continuously on Jt and σE0 : √ √ k curl EkL2 (R3T )3 ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ), p √ √ k σEkL2 (R3T )3 ≤ 4 1 + R2 k σk∞ k curl EkL2 (R3T )3 . ˜ ∈ L2 (0, T, W 1 ) of (2.1) and this c) For σ ≡ 0 there exists a unique solution E ♦ solution depends continuously on Jt : √ ˜ L2 (0,T,W 1 ) ≤ kµk∞ 5kJt kL2 (0,T,W (curl)0 ) . kEk ♦ The proof can be found in the following subsection. 3 ∞ 3 Corollary 3.3. Let (σn )n∈N ⊂ L∞ R (R ) ⊂ L (R ) be a bounded sequence. Let 2 1 ˜ (En )n∈N ⊂ L (0, T, W♦ ) be the sequence of corresponding unique solutions of (3.6). Then √ ˜n )n∈N , (√σn ∇u ˜ )n∈N ⊂ L2 (R3T )3 ˜n )n∈N ⊂ L2 (0, T, W♦1 ) and ( σ n E (E En are bounded. The bounds depend on the bound of (σn )n∈N . In particular, for any sequence (En )n∈N ⊂ L2 (0, T, W (curl)) of corresponding solutions of (2.1)–(2.2) the sequences √ (curl En )n∈N , ( σ n En )n∈N ⊂ L2 (R3T )3 are bounded. 3.1. Existence. To show Theorem 3.2 a), we will make use of a decomposition of D(R3 )3 , which is similar to a well known decomposition of L2 (R3 )3 , cf., e.g., Poincare’s Lemma [10, IX, §1, Lemma 4’]. Note, that the following can be shown similarly for D(R3 × [0, T [)3 . Lemma 3.4. D(R3 )3 = D♦ (R3 )3 ⊕ ∇D(R3 ). Proof. Obviously, D♦ (R3 )3 ⊕ ∇D(R3 ) ⊂ D(R3 )3 . For the other inclusion, let Φ ∈ D(R3 )3 . Then, Lax-Milgram’s Theorem yields a unique solution φ ∈ W 1 (R3 ) of ∆φ = div Φ
in R3 .
Moreover, div Φ ∈ C ∞ (R3 ) implies φ ∈ C ∞ (R3 ), cf., e.g., [8, II, §3, Prop. 1]. Define Ψ := Φ − ∇φ ∈ C ∞ (R3 )3 . Then div Ψ = 0, and it remains to show, that the support of φ (and thus the one of Ψ) is bounded. To this end, let 0 < r0 r such that supp Φ ( Br0 ( Br for the balls Br0 , Br . Then, φ solves ∆φ = 0
in R3 \ Br0 .
(3.9)
Let γ be a smooth cutoff-function, that equals 1 on R3 \ Br and that vanishes on Br0 , cf., e.g., [22, Thm. 5.3]. Then, we have for all v ∈ W 1 (R3 ) that Z Z Z Z ∂ν φ v|∂Br ds = ∆φ γv dx + ∇φ · ∇(γv) dx = Φ · ∇(γv) dx. ∂Br
Br
Br
Br \Br0
A unified variational formulation for the parabolic-elliptic eddy current equations
11
The right hand side vanishes and thus ∂ν φ|∂Br = 0. Hence φ|R3 \Br ∈ W 1 (R3 \ Br ) solves (3.9) with ∂ν φ|∂Br = 0 which (again from Lax-Milgram’s Theorem) admits the unique solution φ|R3 \Br = 0. We will prove the existence, Theorem 3.2 b), using Lions’s Projection Lemma. Lemma 3.5 (Lions’s Projection Lemma). Assume that H is Hilbert space and Φ is a subspace. Let a : H × Φ → R be a bilinear form satisfying the following properties: a) For every φ ∈ Φ, the linear form E 7→ a(E, φ) is continuous on H. b) There exists α > 0 such that a(φ, φ) ≥ αkφk2H for all φ ∈ Φ. Then for each continuous linear form l ∈ H0 , there exists El ∈ H such that a(El , φ) = hl, φi for all φ ∈ Φ and kEl kH ≤
1 klkH0 . α
The proof can be found in [20] (for an english translation see, for instance, [15]). Proof of Theorem 3.2. a) Obviously, for gradient fields ∇φ with φ ∈ D(R3 × [0, T [), a(·, ∇φ) as well as l(∇φ) vanish. Hence it follows from the decomposition of D(R3 × [0, T [)3 (cf. Lemma 3.4) and from the linearity of a and l, that ˜ Φ) = l(Φ) a(E, ˜ ∈ L2 (0, T, W 1 )) holds for all Φ ∈ D(R3 × [0, T [)3 if and only if it (for any E ♦ holds for all Φ ∈ D♦ (R3 × [0, T [)3 . Now, in consideration of div Jt = 0 and div(σE0 ) = 0, Lemma 2.4 d) yields the first assertion. For Φ ∈ D♦ (R3 × [0, T [)3 , the integration by parts formula (2.5) yields that Z TZ Z TZ 1 2 |curl Φ| dx dt a(Φ, Φ) = − σ(Φ + ∇uΦ ) · ∂t Φ dx dt + µ 3 3 0 R 0 R 1 √ 1 2 2 ≥ k σ(Φ + ∇uΦ )(0)kL2 (R3 )3 + kΦkL2 (0,T,W 1 ) (3.10) ♦ 2 kµk∞ and thus the second assertion. b) We define a Hilbert space H by √ H := {E ∈ L2 (0, T, W♦1 ) | σ(E + ∇uE )(0) ∈ L2 (R3 )3 }. A rigorous definition is obtained by taking the closure of D(R3 × [0, T ])3 in L2 (0, T, W 1 ) with respect to the graph norm √ kΦkL2 (0,T,W 1 ) + k σ(Φ + ∇uΦ )(0)kL2 (R3 )3 . This yields a Hilbert space that is continuously imbedded in L2 (0, T, W 1 ), so that E 7→ div E defines a linear continuous mapping. H is then defined as the kernel of this mapping. Note, that the norm on H reads √ kEk2H = kEk2L2 (0,T,W 1 ) + k σ(E + ∇uE )(0)k2L2 (R3 )3 . ♦
Let l, again, denote its continuous extension to H. We show, that a|H×D♦ (R3 ×[0,T [)3 and l fulfill the requirements of Lemma 3.5. Let E ∈ H and Φ ∈ D♦ (R3 × [0, T [)3 . First of all, (3.10) implies that a(Φ, Φ) ≥
1 kΦk2H . max(kµk∞ , 2)
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L. Arnold and B. Harrach
Moreover, it follows with C = max(kΦkL2 (0,T,W♦1 ) , k∂t ΦkL2 (R3T )3 ) from (3.1), 3 (3.4) and µ ∈ L∞ + (R ), that Z Z T 1 −σ(E + ∇uE ) · ∂t Φ + curl E · curl Φ dx dt |a(E, Φ)| = 0 R3 µ √ √ 1 ≤C 2k σk∞ k σEkL2 (R3T )3 + kEkL2 (0,T,W♦1 ) inf µ p 1 ≤C 4kσk∞ 1 + R2 + kEkL2 (0,T,W♦1 ) inf µ p 1 kEkH . ≤C 4kσk∞ 1 + R2 + inf µ Thus, a(·, Φ) is continuous on H. Again by use of (3.1), it holds Z l(E) = −
T
Z hJt , Ei dt +
σE0 · E(0) dx R3
0
≤ kJt kL2 (0,T,W (curl)0 ) kEkL2 (0,T,W (curl)) √ √ + k σE0 kL2 (R3 )3 k σ(E + ∇uE )(0)kL2 (R3 )3 √ √ ≤ max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 )kEkH . ˜ ∈ H that This yields l ∈ H 0 . Now, Lemma 3.5 yields the existence of an E fulfills (3.6) and that depends continuously on l. It follows ˜ L2 (0,T,W 1 ) ≤ kEk ˜ H kEk ♦
√ √ ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ).
˜ + ∇u ˜ ∈ L2 (0, T, W (curl)) is a solution of (2.1)–(2.2). Now, a) yields that E E ˜ and thus (3.8). From Theorem 2.1 c) we obtain the uniqueness of E c) For σ ≡ 0, E ∈ L2 (0, T, W♦1 ) is a solution of (2.1) if and only if E solves a0 (E, F ) = l0 (F )
for all F ∈ L2 (0, T, W♦1 ),
(3.11)
where a0 and l0 denote a(·, ·) and l(·) with σ ≡ 0. As µ ∈ L∞ + , Lax-Milgram’s Theorem yields the assertion. 3.2. On time regularity. We close this section by showing a result on time regularity of the solutions. Lemma 3.6. Let Jt ∈ H 1 (0, T, W (curl)0 ) and E0 ∈ W (curl) such that 1 curl E0 = −Jt (0) curl µ ˜ ∈ L2 (0, T, W 1 ) be the in addition to the general assumptions on Jt and E0 . Let E ♦ solution of (3.6). Then, the following holds: ˜ ∈ H 1 (0, T, W 1 ) and F = E ˜˙ is the solution of a) E ♦
Z a(F, Φ) = − 0
T
. h(Jt ) , Φi dt
for all Φ ∈ D♦ (R3 × [0, T [)3 .
(3.12)
A unified variational formulation for the parabolic-elliptic eddy current equations
13
b) For any solution E ∈ L2 (0, T, W (curl)) of (2.1)–(2.2) we have . ˜˙ E|Ω ∈ H 1 (0, T, W (curl)|Ω ), curl E ∈ H 1 (0, T, L2 (R3 )3 ), (curl E) = curl E. ˜˙ + ∇u ˙ ∈ L2 (0, T, W (curl)) solves Moreover, F = E ˜ E ∂t (σF ) + curl
1 curl F µ
. = −(Jt )
in R3 ×]0, T [
with zero initial conditions. ˜ ∈ L2 (0, T, W 1 ) be the solution c) The analogous assertion holds for σ ≡ 0: Let E ♦ ˜ ∈ H 1 (0, T, W 1 ) and F = E ˜˙ is the solution of of (3.11). Then, E ♦ 1 . curl F = −(Jt ) . curl µ Proof. a) It follows from Theorem 3.2 a) and b) that (3.12) has a unique solution F ∈ L2 (0, T, W♦1 ), so it only remains to show that Z Z(t) =
t
F (s) ds + E0 + ∇vE0 ∈ H 1 (0, T, W♦1 ).
0
solves (3.6). Here, vE0 ∈ W 1 (R3 ) is the unique element that solves div(E0 + ∇v) = 0. For any Φ ∈ D♦ (R3 ×[0, T [)3 choose Ψ ∈ D♦ (R3 ×[0, T [)3 such that ∂t Ψ = Φ. For instance, Ψ could be the restriction of Z t Z T Ψ(x, t) = Φ(x, τ ) dτ − γ(t) Φ(x, τ ) dτ 0
0
3
to R ×]0, T [, where γ ∈ D(R) is a smooth cutoff-function with γ = 1 in [0, T ] and γ = 0 outside of ] − 1, T + 1[. Now, the assertion can be easily shown by verifying (3.6) for Φ = ∂t Ψ by use of (Z + ∇uZ ). = F + ∇uF . b) Follows immediately from a) and Theorem 2.1 c). c) Follows likewise to a). 4. Sensitivity Analysis. In this section we analyze the solution(s) behaviour 3 if σ approaches zero. To this end, let (σn )n∈N ⊂ L∞ R (R ) be a sequence such that lim σn = 0
n→∞
in L∞ (R3 ).
Corresponding to (σn )n∈N , let (En )n∈N ⊂ L2 (0, T, W (curl)) denote any sequence ˜n )n∈N ⊂ L2 (0, T, W 1 ) denote the sequence of of solutions of (2.1)–(2.2) and let (E ♦ unique solutions of (3.6). For σ ≡ 0, let E ∈ L2 (0, T, W (curl)) denote any solution of ˜ ∈ L2 (0, T, W 1 ) the solution of (3.11). (2.1) and let E ♦ Our first result is that the solutions converge: Theorem 4.1. It holds, that √ . curl En → curl E, σn En → 0 in L2 (R3T )3 and (σn En ) * 0 in L2 (0, T, W (curl)0 ).
14
L. Arnold and B. Harrach
Moreover we show, that (under some regularity assumptions) the directional derivative of E with respect to σ exists and can be characterized in the following way: Theorem 4.2. Let Jt ∈ H 1 (0, T, W (curl)0 ), and E0 ∈ W (curl) such that 1 curl curl E0 = −Jt (0) µ 3 in addition to our general assumptions on Jt and E0 . Let d ∈ L∞ R (R ) and h > 0. Let 1 Ed ∈ H (0, T, W (curl)) be a solution of (2.1) with σ ≡ 0 that fulfills div(dEd ) = 0 and F ∈ L2 (0, T, W (curl)) be a solution of 1 curl F = −dE˙ d in R3 ×]0, T [. curl µ
Let Eh ∈ L2 (0, T, W (curl)) be a solution of (2.1)–(2.2) with σ = hd. Then it holds 1 (curl Eh − curl E) * curl F ∈ L2 (R3T )3 h
(h → 0+ ).
Note, that such Ed and F exist and that they are, as well as Eh , not unique. For ˜ + ∇u ˜ , where ∇u ˜ is the image of E ˜ under the mapping defined in instance, Ed = E E E Lemma 3.1 with σ = d. F exists from Theorem 3.2 b). However, this theorem holds for every choice of Eh , Ed and F . These two theorems are proven in the following. √ 4.1. Convergence. Obviously, σn E0 → 0 in L2 (R3 )3 . Lemma 4.3. It holds, that √ ˜ √ ˜ n →E ˜ in L2 (0, T, W♦1 ) E and σn En , σn ∇uE˜n → 0 in L2 (R3T )3 . ˜n * E. ˜ To prove this it suffices to show that Proof. First, we show that E ˜ ˜ From every subsequence of (En ) has a subsequence that converges weakly against E. ˜ Corollary 3.3 we know that (En ) is bounded, so that Alaoglu’s Theorem, cf., e.g., ˜n ) contains a subsequence (that [22, Thm. 6.62], yields that every subsequence of (E ˜ we still denote by (En ) for ease of notation) that converges weakly against some ˜ 0 ∈ L2 (0, T, W 1 ). We show that all these weak limits are identical to E. ˜ E ˜n * E ˜0 E ♦ implies √ ˜ √ ˜n * curl E ˜0 curlE σn En → 0 σn ∇uE˜n → 0 in L2 (R3T )3 wherefrom we obtain the second and third assertion and moreover, that for every ˜n , Φ) of (3.6) with σ = σn converges Φ ∈ D♦ (R3 × [0, T [)3 the left hand side a(E ˜ 0 , Φ). Clearly, the right hand side of (3.6) with σ = σn converges against against a0 (E ˜ 0 solves (3.11) and thus uniqueness provides E ˜=E ˜0. l0 (Φ). Hence, E ˜n yields Now, a short computation shows that the weak convergence of E 1 ˜n k2 2 3 3 ≤ kµ− 21 curl Ek ˜ 22 3 3 lim sup kµ− 2 curl E L (R ) L (R )
n→∞
T
T
˜n → E. ˜ wherefrom we obtain E Proof of Theorem 4.1. The precedent lemma provides curl En → curl E and ˜n ) σn En → 0. The weak convergence of (σn En ). against 0 follows from that of (E . ˜ against E by the explicit form (2.7) of (σn En ) in Lemma 2.3.
15
A unified variational formulation for the parabolic-elliptic eddy current equations
4.2. Linearization results. To characterize the directional derivative of E with respect to σ, some more time-regularity is needed. To this end, we assume in addition, that Jt ∈ H 1 (0, T, W (curl)0 ), and E0 ∈ W (curl) such that 1 curl curl E0 = −Jt (0). µ Lemma 4.4. For every n, En − E ∈ L2 (0, T, W (curl)) solves 1 curl(En − E) = −σn E˙ n in R3 ×]0, T [. curl µ Moreover, there is a constant C such that lim sup
˜n − Ek ˜ L2 (0,T,W 1 ) kE ♦ kσn k∞
n→∞
≤ C.
˜n , E, ˜ u ˜ and Proof. From Lemma 3.6 we know that the time derivatives of E En ˜n − E ˜ solves for all Φ ∈ D♦ (R3 ×]0, T [)3 En |Ωn exist. Then, one can easily see, that E Z TZ ˜˙ n + ∇u ˙ ) · Φ dx dt, ˜ ˜ a0 (En − E, Φ) = − σn (E ˜ E 0
n
R3
˜n + ∇u ˜ )|Ω = En |Ω . Now, the and the first assertion follows from the identity (E n n En coercivity of a0 , (3.1) and (3.4) yield ˜n − Ek ˜ 22 ˜n − Ek ˜ L2 (0,T,L2 (B )3 ) ˜˙ kE )k 2 3 3 kE 1 ) ≤kµk∞ kσn (En + ∇u ˜ R L (0,T,W♦ E˙ n L (RT ) p √ √ ˜˙ n kL2 (R3 )3 kE ˜n − Ek ˜ L2 (0,T,W 1 ) ≤4 1 + R2 kµk∞ k σn k∞ k σn E T ♦ ˜˙ n kL2 (0,T,W 1 ) kE ˜n − Ek ˜ L2 (0,T,W 1 ) . ≤8(1 + R2 )kµk∞ kσn k∞ kE ♦ ♦ ˜˙ n )n∈N is bounded ˜˙ n solves (3.12) with σ = σn , Corollary 3.3 yields that (E As every E and thus the second assertion follows. 3 2 1 Lemma 4.5. Let d ∈ L∞ R (R ) and h > 0. Let F ∈ L (0, T, W♦ ) be the solution of T
Z
Z
a0 (F, Φ) = − R3
0
˜˙ + ∇u ˙ ) · Φ dx dt d(E ˜ E
for all Φ ∈ D♦ (R3 × [0, T [)3 ,
(4.1)
˜˙ under the mapping defined in Lemma 3.1 with σ = d. where ∇uE˜˙ is the image of E ˜h ∈ L2 (0, T, W 1 ) be the solution of (3.6) corresponding to σ = hd. Then Let E ♦ 1 ˜ ˜ * F ∈ L2 (0, T, W♦1 ) (Eh − E) h
(h → 0+ ).
Proof. From Lemma 4.4 it follows that the sequence is bounded and thus similar to Lemma 4.3, that every subsequence has a subsequence that convergences weakly against some F˜ ∈ L2 (0, T, W♦1 ). Again, Lemma 4.4 yields for every h a0
1 ˜ ˜ Φ (Eh − E), h
Z
T
Z
=− 0
R3
˜˙ h + ∇u ˙ ) · Φ dx dt for all Φ ∈ D♦ (R3 ×]0, T [)3 , d(E ˜ E h
16
L. Arnold and B. Harrach
˜˙ h under mapping defined in Lemma 3.1 with σ = hd. where ∇uE˜˙ is the image of E h This mapping does not change, if we take it instead with σ = d. Consequently, as ˜˙ h * E ˜˙ implies ∇u ˙ * ∇u ˙ and thus F˜ solves (4.1). Again, (4.1) is d is fixed, E ˜h ˜ E E uniquely solvable in L2 (0, T, W 1 ), thus F˜ = F and the assertion follows. ♦
Proof of Theorem 4.2. Theorem 4.2 immediately follows from the last lemma. References. [1] R. Acevedo, S. Meddahi, and R. Rodr´ıguez, An e-based mixed formulation for a time-dependent eddy current problem, Math. of Comp., 78 (2009), pp. 1929– 1949. [2] R. Albanese and G. Rubinacci, Fomulation of the eddy-current problem, IEE Proc. A., 137 (1990), pp. 16–22. [3] A. Alonso, A mathematical justication of the low-frequency heterogeneous time-harmonic maxwell equations, Math. Models Methods Appl. Sci., 9 (1999), pp. 475–489. ´ de ´lec, A justification of eddy currents [4] H. Ammari, A. Buffa, and J.-C. Ne model for the maxwell equations, SIAM J. Appl. Math., 60 (2000), pp. 1805–1823. ¨ berl, Numerical analysis of nonlin[5] F. Bachinger, U. Langer, and J. Scho ear multiharmonic eddy current problems, Numerische Mathematik, 100 (2005), pp. 594–616. [6] R. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of maxwell’s equations, Surveys Math. Indust., 8 (1999), pp. 271–312. [7] R. Beck, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 159–182. [8] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology - Volume 1: Physical Origins and Classical Methods, Springer-Verlag, Berlin, 2000. , Mathematical Analysis and Numerical Methods for Science and Technology [9] - Volume 2: Functional and Variational Methods, Springer-Verlag, Berlin, 2000. [10] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 3: Spectral Theory and Applications, Springer-Verlag, Berlin, 2000. [11] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 4: Integral Equations and Numerical Methods, Springer-Verlag, Berlin, 2000. [12] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 5: Evolution Problems I, Springer-Verlag, Berlin Heidelberg, 2000. [13] H. K. Dirks, Quasi-stationary fields for microelectronic applications, Electr. Engng., 79 (1996), pp. 145–155. [14] B. Flemisch, Y. Maday, F. Rapetti, and B. I. Wohlmuth, Coupling scalar and vector potentials on nonmatching grids for eddy currents in a moving conductor, J. Comput. Appl. Math., 168 (2004), pp. 191–205. ¨ hauf, B. Gebauer, and O. Scherzer, Detecting interfaces in a [15] F. Fru parabolic-elliptic problem from surface measurements, SIAM J. Appl. Math., 45 (2007), pp. 810–836. [16] B. Gebauer, Sensitivity analysis of a parabolic-elliptic problem, Quart. Appl. Math., 65 (2007), pp. 591–604. [17] B. Gebauer, M. Hanke, and C. Schneider, Sampling methods for low-
A unified variational formulation for the parabolic-elliptic eddy current equations
[18] [19] [20] [21] [22]
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frequency electromagnetic imaging, Inverse Problems, 24 (2008), p. 015007 (18pp). R. Hiptmair and O. Sterz, Current and voltage excitations for the eddy current model, Int. J. Numer. Model., 18 (2005), pp. 1–21. J. Lang and D. Teleaga, Towards a fully space-time adaptive fem for magnetoquasistatics, IEEE Transactions on Magnetics, 44 (2008), pp. 1238–1241. J. L. Lions, Equations differenti´elles op´erationelles et probl`emes aux limites, Springer-Verlag, Berlin, 1961. S. Meddahi and V. Selgas, An h-based fem-bem formulation for a time dependent eddy current problem, Appl. Numer. Math., 58 (2008), pp. 1061–1083. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics 13, Springer-Verlag, New York, 1993.
1. Introduction. Transient excitation currents J(x, t) generate electromagnetic fields, E(x, t) and H(x, t), which can be described by Maxwell’s equations ∂E + σE + J, ∂t ∂H curl E = −µ , ∂t
curl H =
where the curl-operator acts on the three spatial coordinates, and (under the assumption of linear and isotropic time-independent material laws) σ(x), (x) and µ(x) are the conductivity, permittivity and permeability of the considered domain. For slowly varying electromagnetic fields, the displacement currents ∂E ∂t can be neglected. This leads to the transient eddy current equations curl H = σE + J, ∂H curl E = −µ , ∂t
(1.1) (1.2)
resp., after elimininating H, ∂t (σE) + curl
1 curl E µ
= −∂t J.
(1.3)
The eddy current model is well-established in the engineering literature, cf., e.g., Albanese and Rubinacci [2] or Dirks [13]. A rigorous mathematical justification has been derived by Alonso [3] and Ammari, Buffa and N´ed´elec [4]. ∗ This work was supported by the Deutsche Forschungsgemeinschaft (DFG) under grant HA 6158/1-1. † Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at M¨ unchen, Germany ([email protected]) ‡ Birth name: Bastian Gebauer, Fakult¨ at f¨ ur Mathematik, Technische Universit¨ at M¨ unchen, Germany ([email protected])
1
2
L. Arnold and B. Harrach
In a typical application the domain under consideration consists of both, conducting regions (σ(x) > 0) and non-conducting regions (σ(x) = 0), which has two interesting consequences. The first one is that equation (1.3) is of parabolic-elliptic type. The physical interpretation is that the time-scale is different in the conducting and the insulating region. In the insulating regions, the field instantaneously adapts to the excitation (quasistationary behaviour), while, in the conducting regions, this adaptation takes some time (due to eddy currents induced by the varying electromagnetic fields). A particular consequence is that initial values are only meaningful in the conducting region. The second consequence is that equation (1.3), together with initial values, does only determine E up to the addition of a gauge field, which is a curl-free field that vanishes inside the conductor. However, in many applications, one is only interested in σE, or curl E, anyway. Several well-posed variational formulations have been proposed for the eddy current equations and used for the numerical solution, cf., e.g. [7, 6, 14, 5, 18, 19, 21, 1]. These approaches concentrate on solving the eddy current equations with a fixed conducting region in which the conductivity is assumed to be bounded from below by some positive constant. Accordingly, the variational formulations, with their underlying solution spaces and coercivity constants, depend, in some form or another, on this lower bound, on the support of the conductivity, or on both. A noteworthy exemption appears when σ is constant inside the conductor. Then, a solution of (1.3) can be found by considering the standard variational formulation of (1.3), see (2.3) below, in the space of divergence-free functions, where it is coercive. In the general case of spatially varying σ, however, the solution of (1.3) will not be divergence free in the conductor. In that case, restricting the standard variational formulation of (1.3) to the space of divergence-free functions will not yield the true solution up to a gauge field. Inverse problems, as well as sensitivity considerations, require a unified formulation with respect to σ. An important application is landmine detection, where a source current in an inductor coil is used to generate electromagnetic fields that, in turn, induce currents in a buried conductor. The resulting change in the magnetic field can then be measured by a receiver coil, so that, from one or several such measurements, one may try to reconstruct information about the buried object, cf., e.g., [17] for a formulation of the corresponding measurement operators. A natural approach to this inverse problems is to start by linearizing the problem with respect to σ around a homogeneous non-conducting state. Roughly speaking, this leads to the following question: how does the solution of the elliptic magnetostatic problem (i.e., (1.3) with σ = 0) change if the problem becomes a little bit parabolic? For a scalar analogue this question has been answered in [16]. To the knowledge of the authors, no rigorous linearization results are known for the eddy current model so far. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to σ. To be more precise, what we present is a variational formulation that is uniformly coercive (and hence uniquely solvable) in the space of divergence-free functions, and whose solution agrees with the true solution up to the addition of a gradient field. At this point, let us stress again, that, for spatially varying σ, the standard variational formulation of (1.3) restricted to divergence-free functions does not determine the solution up to a curl-free field. We apply our new unified formulation, to study the limit of the solutions of (1.3) for σ → 0 and prove convergence against their magnetostatic counterparts. We
A unified variational formulation for the parabolic-elliptic eddy current equations
3
then turn to the above mentioned question and rigorously determine the directional derivative of the solutions of (1.3) with σ = 0, with respect to σ, i.e., we linearize the solutions of the elliptic (magnetostatic) problem with respect to the problem becoming parabolic in some parts. This paper is organized as follows. In Section 2 we introduce the necessary notations, derive the standard variational formulation for equation (1.3), characterize well-posed initial conditions and prove uniqueness of solutions (up to gauge fields). Section 3 contains our main theoretical tool: a new uniformly coercive variational formulation that determines the solution up to the addition of a gradient field. This also proves solvability of the eddy current equations. Finally, in Section 4, we apply our new unified formulation to study the dependence of the solutions when the conductivity approaches zero and rigorously linearize (1.3) around a non-conducting domain with respect to the introduction of a conducting object. 2. The eddy current equation. 3 2.1. Notations and general assumptions. Let T > 0. Let µ ∈ L∞ + (R ), ∞ 3 ∞ 3 where we denote by L+ (R ) the space of L (R )-functions with positive (essential) 3 ∞ 3 infima. Let σ ∈ L∞ ≥ (R ) have bounded support Ω, where L≥ (R ) is the space of ∞ 3 L (R )-functions that are almost everywhere non-negative. D(R), D(R3 ), D(]0, T [), resp., D(R3 ×]0, T [) denote the space of C ∞ -functions in x, t, resp., (x, t), which are compactly supported in R, R3 , ]0, T [, resp., R3 ×]0, T [. We will also use the notation D([0, T [), resp., D(R3 × [0, T [) for the space of restrictions of functions from D(]0, T [), resp., D(R3 ×] − ∞, T [) to ]0, T [, resp., R3 ×]0, T [. D0 (R3 ) denotes the space of linear continuous mappings from D(R3 ) to R and 0 D (R3 )3 the one of D(R3 )3 to R. D0 (R3 ×]0, T [)3 is defined likewise. Let L2ρ (R3 ) and W denote the distributional spaces 2
1
L2ρ (R3 ) : = {E ∈ D0 (R3 ) | (1 + |x| )− 2 E ∈ L2 (R3 )}, W (curl) : = {E ∈ L2ρ (R3 )3 | curl E ∈ L2 (R3 )3 }. L2ρ (R3 )n , n = 1, 3, and W (curl) are Hilbert spaces with norms 2
1
k · kρ := k(1 + |x| )− 2 · kL2 (R3 )n , and k · k2W (curl) = k · k2ρ + k curl · k2L2 (R3 )3 . We introduce the Beppo-Levi spaces W 1 (R3 ) : = {E ∈ L2ρ (R3 ) | ∇E ∈ L2 (R3 )3 }, W 1 := W 1 (R3 )3 = {E ∈ L2ρ (R3 )3 | ∇E ∈ L2 (R3 )3×3 }. In the latter space, ∇E denotes the 3 × 3-Jacobian of E. For any bounded domain O ⊂ R3 , W 1 (R3 \ O) is defined likewise. These spaces are Hilbert spaces on which k∇ · kL2 defines an equivalent norm, cf., e.g., [10, IX, §1] and [11, XI, Part B, §1]. Note, that D(R3 ) is dense in L2ρ (R3 ) and in W 1 (R3 ) and that D(R3 )3 is dense in 2 Lρ (R3 )3 , W (curl) and in W 1 . We denote the dual space of a space H by H 0 and the dual pairing on H 0 × H by h·, ·iH 0 ×H . Throughout this work, we frequently use the dual pairing between W (curl)0 and W (curl), hence we write in this case hG, Ei := hG, EiW (curl)0 ×W (curl)
for G ∈ W (curl)0 , E ∈ W (curl).
4
L. Arnold and B. Harrach
We write L2 (R3T ) instead of L2 (R3 ×]0, T [), and usually omit the arguments x and t and only use them where we expect them to improve readability. We consider the space L2 (0, T, W (curl)) as the space to look for a solution of (1.3). Generally, it is not the case that every E ∈ L2 (0, T, W (curl)) has some well-defined initial values. However, in the following we show, that at least every solution of (1.3) has well-defined initial values. Then, we derive a standard variational formulation and discuss, in what sense uniqueness can be expected. In this paper, we assume that we are given Jt ∈ L2 (0, T, W (curl)0 ) with div Jt = 0 and E0 ∈ L2 (R3 )3 with div(σE0 ) = 0. Theorem 2.1. Let E ∈ L2 (0, T, W (curl)). The eddy current problem reads 1 curl E(x, t) = −Jt (x, t) in R3 ×]0, T [, (2.1) ∂t (σ(x)E(x, t)) + curl µ(x) p p σ(x)E(x, 0) = σ(x)E0 (x) in R3 . (2.2) The following holds: a) For every solution E ∈ L2 (0, T, W (curl)) of (2.1) we have √ σE ∈ C(0, T, L2 (R3 )3 ). b) E ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) if and only if E solves Z TZ Z TZ 1 − curl E · curl Φ dx dt σE · ∂t Φ dx dt+ 0 R3 0 R3 µ Z T Z hJt , Φi dt + σE0 · Φ(0) dx =− 0
(2.3)
R3
for all Φ ∈ D(R3 × [0, T [)3 . √ c) Equations (2.1)–(2.2) uniquely determine curl E and σE. Moreover, if E ∈ L2 (0, T, W (curl)) solves then every F ∈ L2 (0, T, W (curl)) with √ (2.1)–(2.2), √ curl F = curl E and σF = σE also solves (2.1)–(2.2). We prove Theorem 2.1 in the following subsection. The initial condition (2.2) can be interpreted as stating that, wherever it makes sense to speak of initial values, they must agree with E0 . In Ω, the equation is parabolic and initial values are meaningful and necessary. Outside of Ω, where the equation is elliptic, initial conditions are meaningless and (2.2) does not contain any information. Let us stress again, in this section we only require that σ is nonnegative, bounded and has bounded support. 2.2. Initial values, a standard variational formulation and uniqueness. For E ∈ L2 (0, T, W (curl)) we have that E(t)|Ω , curl E(t) ∈ L2 (R3 )3 for t ∈]0, T [ a.e. and consequently the products 1 curl E(t), σE(t) ∈ L2 (R3 )3 µ are well-defined. Moreover, the assumption div(σE0 ) = 0 makes sense as E0 ∈ L2 (R3 )3 . As D(R3 )3 is dense in W (curl), we can also regard L2 (0, T, W (curl)0 ) as a subspace of D0 (R3 ×]0, T [)3 . Hence, div Jt is a well-defined in the sense of distributions.
A unified variational formulation for the parabolic-elliptic eddy current equations
5
Now, the transient eddy current equation (2.1) is equivalent to: Z
T
Z
Z
T
Z
σE · ∂t Φ dx dt +
− 0
R3
R3
0
1 curl E · curl Φ dx dt = − µ
Z
T
hJt , Φi dt
(2.4)
0
for all Φ ∈ D(R3 ×]0, T [)3 . In the rest of this subsection, we continue along the lines in [16, Sect. 2]. We first recall the definition of the time-derivative in the sense of vector-valued distributions: For two Banach spaces X, Y and a continuous injection ι : X ,→ Y , E ∈ L2 (0, T, X) has a time-derivative in L2 (0, T, Y ) in the sense of vector-valued distributions, if there exists a E˙ ∈ L2 (0, T, Y ) which fulfills Z
T
E˙ ϕ dt = −
Z
T
ιE ∂t ϕ dt
for all ϕ ∈ D(]0, T [)
0
0
ι
ι0
(cf., e.g., [12, XVIII, §1]). For a Gelfand triple V ,→ H ,→ V 0 of real separable Hilbert spaces V and H, the space n o W(0, T, V, V 0 ) := E ∈ L2 (0, T, V) | E˙ ∈ L2 (0, T, V 0 ) is defined by taking the time-derivative with respect to the injection ι0 ι : V ,→ V 0 . The image of the space W(0, T, V, V 0 ) under ι is continuously imbedded in C(0, T, H) and, for E, F ∈ W(0, T, V, V 0 ), the following integration by parts formula holds: T
Z
h i ˙ hE(t), F (t)iV 0 ×V + hF˙ (t), E(t)iV 0 ×V dt = (ιE(T ), ιF (T ))H − (ιE(0), ιF (0))H ,
0
where (·, ·)H denotes the inner product of H, cf., e.g., [12, XVIII, §1, Thms. 1,2]. In view of (2.1), we introduce the space . Wσ := E ∈ L2 (0, T, W (curl)) | (σE) ∈ L2 (0, T, W (curl)0 ) , where (σE). denotes the time-derivative of σE ∈ L2 (0, T, L2 (R3 )3 ) in the sense of vector-valued distributions with respect to the canonical injection L2 ,→ W (curl)0 . Moreover, we define the space L2σ by taking the closure of √ σE | E ∈ D(R3 )3 ⊂ L2 (R3 )3 Hilbert space equipped with the with respect to the L2 (R3 )3 -norm. L2σ is a separable √ standard L2 (R3 )3 -inner product. Note, that { σE | E ∈ L2 (R3 )3 } ⊂ L2σ . √ 2 Lemma 2.2. If E ∈ Wσ , then σE ∈ C(0, T, L (R3 )3 ). Additionally, for E, F ∈ Wσ the following integration by parts formula holds: Z 0
T
. h(σE) , F i dt +
Z 0
T
. h(σF ) , Ei dt =
Z σ(E(T ) · F (T ) − E(0) · F (0)) dx. (2.5) R3
Proof. In [16, Sect. 2] this lemma is proven for a scalar analog. We repeat the proof for the convenience of the reader. Both, W (curl) and L2σ are separable Hilbert spaces. We define the mapping I by √ I : W (curl) → L2σ , E 7→ σE,
6
L. Arnold and B. Harrach
which is continuous and has dense range. We identify the Hilbert space L2σ with its dual. Then, after factoring out the nullspace N of I we obtain, that ι : W (curl)/N → L2σ ,
E + N 7→ IE
defines an injective, continuous mapping and hence a Gelfand triple ι0
ι
W (curl)/N ,→ L2σ ,→ (W (curl)/N )0 . For all G ∈ (W (curl)/N )0 the dual mapping ι0 is given by Z √ hι0 G, F + N i(W (curl)/N )0 ×W (curl)/N = G · σF dx for all F ∈ W (curl). (2.6) R3
Let E ∈ Wσ and G = (σE). ∈ L2 (0, T, W (curl)0 ) be its time-derivative with respect to I 0 I : W (curl) ,→ W (curl)0 . Now we show, that G = (E + N ). with respect 0 to ι0 ι : W (curl)/N ,→ (W (curl)/N ) . Let ϕ ∈ D(]0, T [). For F ∈ N we have Z T Z TZ hG(t), F iϕ(t) dt = − σE(t) · F dx ∂t ϕ(t) dt = 0 0
0
R3
and thus hG(t), F i = 0 for t ∈]0, T [ a.e.. Hence, G(t) ∈ N ⊥ and we can identify G with an element of L2 (0, T, (W (curl)/N )0 ). Then, for F + N ∈ W (curl)/N it follows Z T Z T hG(t), F + N i(W (curl)/N )0 ×W (curl)/N ϕ(t) dt = hG(t), F iϕ(t) dt 0
0
Z
T
Z
=−
σE(t) · F dx ∂t ϕ(t) dt R3
0
Z
T
hι0 ι(E(t) + N ), F + N i(W (curl)/N )0 ×W (curl)/N ∂t ϕ(t) dt
=− 0
and, accordingly, G √ = (E + N ). and E + N ∈ W(0, T, W (curl)/N, (W (curl)/N )0 ). Now, it follows that σE = ι(E + N ) ∈ C(0, T, L2σ ) and, together with (2.6), for all E, F ∈ Wσ the integration by parts formula (2.5). Lemma 2.3. Every solution E ∈ L2 (0, T, W (curl)) of (1.3) is in Wσ and thus has well-defined initial values p σ(x)E(x, 0) ∈ L2σ ⊂ L2 (R3 )3 . For t ∈]0, T [ a.e., (σE). (t) ∈ W (curl)0 is given by Z 1 . h(σE) (t), F i = −hJt (t), F i − curl E(t) · curl F dx for all F ∈ W (curl). (2.7) R3 µ Proof. Let E be a solution of (1.3). Define G(t) ∈ W (curl)0 by Z 1 hG(t), Ψi = −hJt (t), Ψi − curl E(t) · curl Ψ dx µ 3 R for all Ψ ∈ D(R3 )3 . Then G ∈ L2 (0, T, W (curl)0 ), and, due to the fact that E solves (2.4) with Φ = Ψϕ for all ϕ ∈ D(]0, T [) and all Ψ ∈ D(R3 )3 , it holds Z T Z TZ Z T hG(t), Ψiϕ(t) dt = − σE · Ψ dx∂t ϕ dt = − hσE(t), Ψi∂t ϕ(t) dt. (2.8) 0
0
R3
0
7
A unified variational formulation for the parabolic-elliptic eddy current equations
Since D(R3 )3 is dense in W (curl) and both sides depend continuously on Ψ, we obtain that equation (2.8) holds for all Ψ ∈ W (curl). Now it follows from the fact, that W (curl) ⊗ D(]0, T [) is dense in L2 (0, T, W (curl)), that G = (σE). with respect to the canonical injection L2 (R3 )3 ,→ W (curl)0 . This shows that E ∈ Wσ . Lemma 2.3 shows, that it makes sense to search E ∈ L2 (0, T, W (curl)) that solves (2.1)–(2.2). Now, we give some equivalent variational formulation: Lemma 2.4. The following problems are equivalent: a) Find E ∈ L2 (0, T, W (curl)) that solves (2.1) and (2.2). b) Find E ∈ Wσ that solves (2.2) and T
Z
. h(σE) , F i dt +
0
Z
T
0
Z R3
1 curl E · curl F dx dt = − µ
T
Z
hJt , F i dt
(2.9)
0
for all F ∈ L2 (0, T, W (curl)). c) Find E ∈ L2 (0, T, W (curl)) that solves T
Z −
. h(σF ) , Ei dt +
0
Z
T
Z R3
0
1 curl E · curl F dx dt µ Z T Z =− hJt , F i dt +
σE0 · F (0) dx
R3
0
√ for all F ∈ Wσ with σF (T ) = 0. d) Find E ∈ L2 (0, T, W (curl)) that solves Z
T
Z
−
Z
T
Z
σE · ∂t Φ dx dt + 0
R3
0
R3
1 curl E · curl Φ dx dt µ Z T Z =− hJt , Φi dt + 0
σE0 · Φ(0) dx
R3
for all Φ ∈ D(R3 × [0, T [)3 . Proof. We start by showing a) ⇒ b). If E ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) it follows from Lemma 2.3 that E ∈ Wσ and (2.9) holds for all F (x, t) = G(x)ϕ(t) with G ∈ W (curl) and ϕ ∈ D(]0, T [). Since W (curl)⊗D(]0, T [) is dense in L2 (0, T, W (curl)) and both sides of (2.9) depend continuously on F ∈ L2 (0, T, W (curl)), b) follows. b) ⇒ c) follows from the integration by parts formula (2.5). c) ⇒ d) follows from the fact that for Φ ∈ D(R3 × [0, T [)3 the time-derivative (σΦ). ∈ L2 (0, T, W (curl)0 ) of σΦ ∈ L2 (0, T, L2 (R3 )3 ) with respect to the injection L2 ,→ W (curl)0 is the image of the classical time-derivative σ∂t Φ(t) under this injection, i.e. Z . h(σΦ) (t), E(t)i = σ∂t Φ(t) · E(t) dx for t ∈]0, T [ a.e. R3
To show d) ⇒ a) we use d) with Φ ∈ D(R3 ×]0, T [)3 . Then E ∈ L2 (0, T, W (curl)) solves (2.2) and Lemma 2.3 yields E ∈ Wσ . Now, the integration by parts formula 3 3 (2.5) applied √ √ on d) with Φ = Ψϕ, Ψ ∈ D(R ) , ϕ ∈ D([0, T [) with ϕ(0) = 1 implies σE0 = σE(0).
8
L. Arnold and B. Harrach
Now, the proof of Theorem 2.1 reads: Proof of Theorem 2.1. a) This is Lemma 2.3. b) This is Lemma 2.4 d). √ c) Assume that E ∈ Wσ is a solution of (2.1)–(2.2) with σE(0) = 0 and Jt = 0. Using Lemma 2.4 b) and the integration by parts formula (2.5) implies Z TZ Z T 1 . h(σE) , Ei dt + 0= curl E · curl E dx dt 0 R3 µ 0 1 √ 1 ≥ k σE(T )k2L2 (R3 )3 + k curl Ek2L2 (R3 )3 . T 2 kµk∞ √ We obtain curl E = 0 and σE = 0. The second assertion is obvious. 3. A unified variational formulation. In this section we present a new, uniquely solvable and uniformly coercive variational formulation that determines the solution of the eddy current equations, (2.1) and (2.2), up to the addition of a gradient field. From this we obtain solvability of (2.1) and (2.2), and a continuity result that is uniform with respect to the conductivity σ. For this result we need stronger assumptions on σ. Let R > 0. In the following, we assume that 3 ∞ 3 ∞ σ ∈ L∞ R (R ) := {σ ∈ L≥ (R ) | Ω := supp σ ⊂ BR , σ ∈ L+ (Ω), and ∃ s ∈ N :
Ω = ∪si=1 Ωi , where Ωi , i = 1, . . . , s, are bounded Lipschitz domains such that Ωi ∪ Ωj = ∅ and R3 \ Ω is connected}, where BR denotes the open ball with radius R. The case of σ ≡ 0 is treated separately. Note, that our continuity results do not depend on the lower bound of σ. The fact that the curl of a solution is unique, but not the solution itself, leads to the idea to work with spaces where k curl · kL2 (R3 )3 defines a norm. Let the index ♦ of the space W 1 (resp., D(R3 × [0, T [)3 , D(R3 )3 ) denote its closed subspace of functions with vanishing divergence. Then, W♦1 equipped with the norm k · k♦ := k curl · kL2 (R3 )3 is a Hilbert space and, similar to the proof of [11, XI, Part B, §1, Lemma 1], we have kEkρ ≤ 2kEk♦
for all E ∈ W♦1 .
(3.1)
Lemma 3.1. There is a continuous linear map W 1 → H(curl 0, R3 ) := {E ∈ L2 (R3 )3 | curl E = 0},
E 7→ ∇uE ,
which yields E + ∇uE ∈ W (curl) and div(σ(E + ∇uE )) = 0
in R3 .
Obviously, this map extends to L2 (0, T, W 1 ) → L2 (0, T, H(curl 0, R3 )),
E 7→ ∇uE ,
where ∇uE (t) := ∇uE(t) for t ∈]0, T [ a.e. and E ∈ H 1 (0, T, W 1 ) implies ∇uE ∈ H 1 (0, T, H(curl 0, R3 ))
and
. (∇uE ) = ∇uE˙ .
(3.2)
A unified variational formulation for the parabolic-elliptic eddy current equations
9
Proof. Let E ∈ W 1 . Due to Poincare’s inequality (cf., e.g., [9, 4, IV, §7, Prop. 2]), the fact, that σ is positively bounded from below on Ω, and Lax-Milgram’s Theorem 1 (cf., e.g., [22, §8, Thm. 8.14]), there exists a unique uE ∈ H (Ω) that solves Z
Z σ∇u · ∇v dx = −
Ω 1 Here, H (Ω) :=
σE · ∇v dx for all v ∈ H 1 (Ω).
(3.3)
Ω
n o R v ∈ H 1 (Ω) | Ωi v dx = 0, i = 1, . . . , s . Further more uE depends
continuously on E|Ω and for any other solution u ∈ H 1 (Ω) it holds ∇u = ∇uE . We extend uE to an element of W 1 (R3 ) by solving ∆u = 0 on R3 \ Ω with u|∂Ω = uE |∂Ω for u ∈ W 1 (R3 \ Ω). Again, Lax-Milgram’s Theorem provides a unique solution, which depends continuously on uE |∂Ω and thus on E|Ω . Let uE , again, denote its extension. Then, uE ∈ W 1 (R3 ) and the mapping E 7→ ∇uE is well-defined, linear and continuous with a continuity constant that depends on the lower bound of σ. Moreover, E + ∇uE ∈ W (curl) and ∇uE solves (3.2). For the rest of this paper, let ∇uE denote the image of E under this mapping. √ Note, that there are different possibilities to construct this map, but σ∇uE is uniquely determined by the condition (3.2). Moreover, it holds √ √ k σ∇uE kL2 (R3 )3 ≤ k σEkL2 (R3 )3 .
(3.4)
We define the bilinear form a : L2 (0, T, W♦1 ) × D(R3 × [0, T [)3 → R by Z
T
Z
T
Z
a(E, Φ) := −
Z
σ(E + ∇uE ) · ∂t Φ dx dt + 0
R3
0
R3
1 curl E · curl Φ dx dt, (3.5) µ
and, cf. Lemma 2.4 d), a linear form l : D(R3 × [0, T [)3 → R: Z l(Φ) := −
T
Z hJt , Φi dt +
σE0 · Φ(0) dx. R3
0
Now we can state the main result of this section: Theorem 3.2. ˜ ∈ L2 (0, T, W 1 ). Then E ˜ + ∇u ˜ ∈ L2 (0, T, W (curl)) solves (2.1)–(2.2) a) Let E ♦ E ˜ solves if and only if E ˜ Φ) = l(Φ) a(E,
for all Φ ∈ D♦ (R3 × [0, T [)3 .
(3.6)
a|(D♦ (R3 ×[0,T [)3 )2 is uniformly coercive with respect to k · kL2 (0,T,W♦1 ) : a(Φ, Φ) ≥
1 kΦk2L2 (0,T,W 1 ) ♦ kµk∞
for all Φ ∈ D♦ (R3 × [0, T [)3 .
˜ ∈ L2 (0, T, W 1 ) of (3.6). E ˜ depends continuously b) There is a unique solution E ♦ √ on Jt and σE0 : √ √ ˜ L2 (0,T,W 1 ) ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ). kEk ♦ (3.7)
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L. Arnold and B. Harrach
˜ + ∇u ˜ solves (2.1)–(2.2) and any other solution E ∈ L2 (0, T, W (curl)) of E E (2.1)–(2.2) fulfills √ √ ˜ + ∇u ˜ ). ˜ σE = σ(E (3.8) curl E = curl E, E √ √ curl E and σE depend continuously on Jt and σE0 : √ √ k curl EkL2 (R3T )3 ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ), p √ √ k σEkL2 (R3T )3 ≤ 4 1 + R2 k σk∞ k curl EkL2 (R3T )3 . ˜ ∈ L2 (0, T, W 1 ) of (2.1) and this c) For σ ≡ 0 there exists a unique solution E ♦ solution depends continuously on Jt : √ ˜ L2 (0,T,W 1 ) ≤ kµk∞ 5kJt kL2 (0,T,W (curl)0 ) . kEk ♦ The proof can be found in the following subsection. 3 ∞ 3 Corollary 3.3. Let (σn )n∈N ⊂ L∞ R (R ) ⊂ L (R ) be a bounded sequence. Let 2 1 ˜ (En )n∈N ⊂ L (0, T, W♦ ) be the sequence of corresponding unique solutions of (3.6). Then √ ˜n )n∈N , (√σn ∇u ˜ )n∈N ⊂ L2 (R3T )3 ˜n )n∈N ⊂ L2 (0, T, W♦1 ) and ( σ n E (E En are bounded. The bounds depend on the bound of (σn )n∈N . In particular, for any sequence (En )n∈N ⊂ L2 (0, T, W (curl)) of corresponding solutions of (2.1)–(2.2) the sequences √ (curl En )n∈N , ( σ n En )n∈N ⊂ L2 (R3T )3 are bounded. 3.1. Existence. To show Theorem 3.2 a), we will make use of a decomposition of D(R3 )3 , which is similar to a well known decomposition of L2 (R3 )3 , cf., e.g., Poincare’s Lemma [10, IX, §1, Lemma 4’]. Note, that the following can be shown similarly for D(R3 × [0, T [)3 . Lemma 3.4. D(R3 )3 = D♦ (R3 )3 ⊕ ∇D(R3 ). Proof. Obviously, D♦ (R3 )3 ⊕ ∇D(R3 ) ⊂ D(R3 )3 . For the other inclusion, let Φ ∈ D(R3 )3 . Then, Lax-Milgram’s Theorem yields a unique solution φ ∈ W 1 (R3 ) of ∆φ = div Φ
in R3 .
Moreover, div Φ ∈ C ∞ (R3 ) implies φ ∈ C ∞ (R3 ), cf., e.g., [8, II, §3, Prop. 1]. Define Ψ := Φ − ∇φ ∈ C ∞ (R3 )3 . Then div Ψ = 0, and it remains to show, that the support of φ (and thus the one of Ψ) is bounded. To this end, let 0 < r0 r such that supp Φ ( Br0 ( Br for the balls Br0 , Br . Then, φ solves ∆φ = 0
in R3 \ Br0 .
(3.9)
Let γ be a smooth cutoff-function, that equals 1 on R3 \ Br and that vanishes on Br0 , cf., e.g., [22, Thm. 5.3]. Then, we have for all v ∈ W 1 (R3 ) that Z Z Z Z ∂ν φ v|∂Br ds = ∆φ γv dx + ∇φ · ∇(γv) dx = Φ · ∇(γv) dx. ∂Br
Br
Br
Br \Br0
A unified variational formulation for the parabolic-elliptic eddy current equations
11
The right hand side vanishes and thus ∂ν φ|∂Br = 0. Hence φ|R3 \Br ∈ W 1 (R3 \ Br ) solves (3.9) with ∂ν φ|∂Br = 0 which (again from Lax-Milgram’s Theorem) admits the unique solution φ|R3 \Br = 0. We will prove the existence, Theorem 3.2 b), using Lions’s Projection Lemma. Lemma 3.5 (Lions’s Projection Lemma). Assume that H is Hilbert space and Φ is a subspace. Let a : H × Φ → R be a bilinear form satisfying the following properties: a) For every φ ∈ Φ, the linear form E 7→ a(E, φ) is continuous on H. b) There exists α > 0 such that a(φ, φ) ≥ αkφk2H for all φ ∈ Φ. Then for each continuous linear form l ∈ H0 , there exists El ∈ H such that a(El , φ) = hl, φi for all φ ∈ Φ and kEl kH ≤
1 klkH0 . α
The proof can be found in [20] (for an english translation see, for instance, [15]). Proof of Theorem 3.2. a) Obviously, for gradient fields ∇φ with φ ∈ D(R3 × [0, T [), a(·, ∇φ) as well as l(∇φ) vanish. Hence it follows from the decomposition of D(R3 × [0, T [)3 (cf. Lemma 3.4) and from the linearity of a and l, that ˜ Φ) = l(Φ) a(E, ˜ ∈ L2 (0, T, W 1 )) holds for all Φ ∈ D(R3 × [0, T [)3 if and only if it (for any E ♦ holds for all Φ ∈ D♦ (R3 × [0, T [)3 . Now, in consideration of div Jt = 0 and div(σE0 ) = 0, Lemma 2.4 d) yields the first assertion. For Φ ∈ D♦ (R3 × [0, T [)3 , the integration by parts formula (2.5) yields that Z TZ Z TZ 1 2 |curl Φ| dx dt a(Φ, Φ) = − σ(Φ + ∇uΦ ) · ∂t Φ dx dt + µ 3 3 0 R 0 R 1 √ 1 2 2 ≥ k σ(Φ + ∇uΦ )(0)kL2 (R3 )3 + kΦkL2 (0,T,W 1 ) (3.10) ♦ 2 kµk∞ and thus the second assertion. b) We define a Hilbert space H by √ H := {E ∈ L2 (0, T, W♦1 ) | σ(E + ∇uE )(0) ∈ L2 (R3 )3 }. A rigorous definition is obtained by taking the closure of D(R3 × [0, T ])3 in L2 (0, T, W 1 ) with respect to the graph norm √ kΦkL2 (0,T,W 1 ) + k σ(Φ + ∇uΦ )(0)kL2 (R3 )3 . This yields a Hilbert space that is continuously imbedded in L2 (0, T, W 1 ), so that E 7→ div E defines a linear continuous mapping. H is then defined as the kernel of this mapping. Note, that the norm on H reads √ kEk2H = kEk2L2 (0,T,W 1 ) + k σ(E + ∇uE )(0)k2L2 (R3 )3 . ♦
Let l, again, denote its continuous extension to H. We show, that a|H×D♦ (R3 ×[0,T [)3 and l fulfill the requirements of Lemma 3.5. Let E ∈ H and Φ ∈ D♦ (R3 × [0, T [)3 . First of all, (3.10) implies that a(Φ, Φ) ≥
1 kΦk2H . max(kµk∞ , 2)
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L. Arnold and B. Harrach
Moreover, it follows with C = max(kΦkL2 (0,T,W♦1 ) , k∂t ΦkL2 (R3T )3 ) from (3.1), 3 (3.4) and µ ∈ L∞ + (R ), that Z Z T 1 −σ(E + ∇uE ) · ∂t Φ + curl E · curl Φ dx dt |a(E, Φ)| = 0 R3 µ √ √ 1 ≤C 2k σk∞ k σEkL2 (R3T )3 + kEkL2 (0,T,W♦1 ) inf µ p 1 ≤C 4kσk∞ 1 + R2 + kEkL2 (0,T,W♦1 ) inf µ p 1 kEkH . ≤C 4kσk∞ 1 + R2 + inf µ Thus, a(·, Φ) is continuous on H. Again by use of (3.1), it holds Z l(E) = −
T
Z hJt , Ei dt +
σE0 · E(0) dx R3
0
≤ kJt kL2 (0,T,W (curl)0 ) kEkL2 (0,T,W (curl)) √ √ + k σE0 kL2 (R3 )3 k σ(E + ∇uE )(0)kL2 (R3 )3 √ √ ≤ max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 )kEkH . ˜ ∈ H that This yields l ∈ H 0 . Now, Lemma 3.5 yields the existence of an E fulfills (3.6) and that depends continuously on l. It follows ˜ L2 (0,T,W 1 ) ≤ kEk ˜ H kEk ♦
√ √ ≤ max(kµk∞ , 2) max( 5kJt kL2 (0,T,W (curl)0 ) , k σE0 kL2 (R3 )3 ).
˜ + ∇u ˜ ∈ L2 (0, T, W (curl)) is a solution of (2.1)–(2.2). Now, a) yields that E E ˜ and thus (3.8). From Theorem 2.1 c) we obtain the uniqueness of E c) For σ ≡ 0, E ∈ L2 (0, T, W♦1 ) is a solution of (2.1) if and only if E solves a0 (E, F ) = l0 (F )
for all F ∈ L2 (0, T, W♦1 ),
(3.11)
where a0 and l0 denote a(·, ·) and l(·) with σ ≡ 0. As µ ∈ L∞ + , Lax-Milgram’s Theorem yields the assertion. 3.2. On time regularity. We close this section by showing a result on time regularity of the solutions. Lemma 3.6. Let Jt ∈ H 1 (0, T, W (curl)0 ) and E0 ∈ W (curl) such that 1 curl E0 = −Jt (0) curl µ ˜ ∈ L2 (0, T, W 1 ) be the in addition to the general assumptions on Jt and E0 . Let E ♦ solution of (3.6). Then, the following holds: ˜ ∈ H 1 (0, T, W 1 ) and F = E ˜˙ is the solution of a) E ♦
Z a(F, Φ) = − 0
T
. h(Jt ) , Φi dt
for all Φ ∈ D♦ (R3 × [0, T [)3 .
(3.12)
A unified variational formulation for the parabolic-elliptic eddy current equations
13
b) For any solution E ∈ L2 (0, T, W (curl)) of (2.1)–(2.2) we have . ˜˙ E|Ω ∈ H 1 (0, T, W (curl)|Ω ), curl E ∈ H 1 (0, T, L2 (R3 )3 ), (curl E) = curl E. ˜˙ + ∇u ˙ ∈ L2 (0, T, W (curl)) solves Moreover, F = E ˜ E ∂t (σF ) + curl
1 curl F µ
. = −(Jt )
in R3 ×]0, T [
with zero initial conditions. ˜ ∈ L2 (0, T, W 1 ) be the solution c) The analogous assertion holds for σ ≡ 0: Let E ♦ ˜ ∈ H 1 (0, T, W 1 ) and F = E ˜˙ is the solution of of (3.11). Then, E ♦ 1 . curl F = −(Jt ) . curl µ Proof. a) It follows from Theorem 3.2 a) and b) that (3.12) has a unique solution F ∈ L2 (0, T, W♦1 ), so it only remains to show that Z Z(t) =
t
F (s) ds + E0 + ∇vE0 ∈ H 1 (0, T, W♦1 ).
0
solves (3.6). Here, vE0 ∈ W 1 (R3 ) is the unique element that solves div(E0 + ∇v) = 0. For any Φ ∈ D♦ (R3 ×[0, T [)3 choose Ψ ∈ D♦ (R3 ×[0, T [)3 such that ∂t Ψ = Φ. For instance, Ψ could be the restriction of Z t Z T Ψ(x, t) = Φ(x, τ ) dτ − γ(t) Φ(x, τ ) dτ 0
0
3
to R ×]0, T [, where γ ∈ D(R) is a smooth cutoff-function with γ = 1 in [0, T ] and γ = 0 outside of ] − 1, T + 1[. Now, the assertion can be easily shown by verifying (3.6) for Φ = ∂t Ψ by use of (Z + ∇uZ ). = F + ∇uF . b) Follows immediately from a) and Theorem 2.1 c). c) Follows likewise to a). 4. Sensitivity Analysis. In this section we analyze the solution(s) behaviour 3 if σ approaches zero. To this end, let (σn )n∈N ⊂ L∞ R (R ) be a sequence such that lim σn = 0
n→∞
in L∞ (R3 ).
Corresponding to (σn )n∈N , let (En )n∈N ⊂ L2 (0, T, W (curl)) denote any sequence ˜n )n∈N ⊂ L2 (0, T, W 1 ) denote the sequence of of solutions of (2.1)–(2.2) and let (E ♦ unique solutions of (3.6). For σ ≡ 0, let E ∈ L2 (0, T, W (curl)) denote any solution of ˜ ∈ L2 (0, T, W 1 ) the solution of (3.11). (2.1) and let E ♦ Our first result is that the solutions converge: Theorem 4.1. It holds, that √ . curl En → curl E, σn En → 0 in L2 (R3T )3 and (σn En ) * 0 in L2 (0, T, W (curl)0 ).
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L. Arnold and B. Harrach
Moreover we show, that (under some regularity assumptions) the directional derivative of E with respect to σ exists and can be characterized in the following way: Theorem 4.2. Let Jt ∈ H 1 (0, T, W (curl)0 ), and E0 ∈ W (curl) such that 1 curl curl E0 = −Jt (0) µ 3 in addition to our general assumptions on Jt and E0 . Let d ∈ L∞ R (R ) and h > 0. Let 1 Ed ∈ H (0, T, W (curl)) be a solution of (2.1) with σ ≡ 0 that fulfills div(dEd ) = 0 and F ∈ L2 (0, T, W (curl)) be a solution of 1 curl F = −dE˙ d in R3 ×]0, T [. curl µ
Let Eh ∈ L2 (0, T, W (curl)) be a solution of (2.1)–(2.2) with σ = hd. Then it holds 1 (curl Eh − curl E) * curl F ∈ L2 (R3T )3 h
(h → 0+ ).
Note, that such Ed and F exist and that they are, as well as Eh , not unique. For ˜ + ∇u ˜ , where ∇u ˜ is the image of E ˜ under the mapping defined in instance, Ed = E E E Lemma 3.1 with σ = d. F exists from Theorem 3.2 b). However, this theorem holds for every choice of Eh , Ed and F . These two theorems are proven in the following. √ 4.1. Convergence. Obviously, σn E0 → 0 in L2 (R3 )3 . Lemma 4.3. It holds, that √ ˜ √ ˜ n →E ˜ in L2 (0, T, W♦1 ) E and σn En , σn ∇uE˜n → 0 in L2 (R3T )3 . ˜n * E. ˜ To prove this it suffices to show that Proof. First, we show that E ˜ ˜ From every subsequence of (En ) has a subsequence that converges weakly against E. ˜ Corollary 3.3 we know that (En ) is bounded, so that Alaoglu’s Theorem, cf., e.g., ˜n ) contains a subsequence (that [22, Thm. 6.62], yields that every subsequence of (E ˜ we still denote by (En ) for ease of notation) that converges weakly against some ˜ 0 ∈ L2 (0, T, W 1 ). We show that all these weak limits are identical to E. ˜ E ˜n * E ˜0 E ♦ implies √ ˜ √ ˜n * curl E ˜0 curlE σn En → 0 σn ∇uE˜n → 0 in L2 (R3T )3 wherefrom we obtain the second and third assertion and moreover, that for every ˜n , Φ) of (3.6) with σ = σn converges Φ ∈ D♦ (R3 × [0, T [)3 the left hand side a(E ˜ 0 , Φ). Clearly, the right hand side of (3.6) with σ = σn converges against against a0 (E ˜ 0 solves (3.11) and thus uniqueness provides E ˜=E ˜0. l0 (Φ). Hence, E ˜n yields Now, a short computation shows that the weak convergence of E 1 ˜n k2 2 3 3 ≤ kµ− 21 curl Ek ˜ 22 3 3 lim sup kµ− 2 curl E L (R ) L (R )
n→∞
T
T
˜n → E. ˜ wherefrom we obtain E Proof of Theorem 4.1. The precedent lemma provides curl En → curl E and ˜n ) σn En → 0. The weak convergence of (σn En ). against 0 follows from that of (E . ˜ against E by the explicit form (2.7) of (σn En ) in Lemma 2.3.
15
A unified variational formulation for the parabolic-elliptic eddy current equations
4.2. Linearization results. To characterize the directional derivative of E with respect to σ, some more time-regularity is needed. To this end, we assume in addition, that Jt ∈ H 1 (0, T, W (curl)0 ), and E0 ∈ W (curl) such that 1 curl curl E0 = −Jt (0). µ Lemma 4.4. For every n, En − E ∈ L2 (0, T, W (curl)) solves 1 curl(En − E) = −σn E˙ n in R3 ×]0, T [. curl µ Moreover, there is a constant C such that lim sup
˜n − Ek ˜ L2 (0,T,W 1 ) kE ♦ kσn k∞
n→∞
≤ C.
˜n , E, ˜ u ˜ and Proof. From Lemma 3.6 we know that the time derivatives of E En ˜n − E ˜ solves for all Φ ∈ D♦ (R3 ×]0, T [)3 En |Ωn exist. Then, one can easily see, that E Z TZ ˜˙ n + ∇u ˙ ) · Φ dx dt, ˜ ˜ a0 (En − E, Φ) = − σn (E ˜ E 0
n
R3
˜n + ∇u ˜ )|Ω = En |Ω . Now, the and the first assertion follows from the identity (E n n En coercivity of a0 , (3.1) and (3.4) yield ˜n − Ek ˜ 22 ˜n − Ek ˜ L2 (0,T,L2 (B )3 ) ˜˙ kE )k 2 3 3 kE 1 ) ≤kµk∞ kσn (En + ∇u ˜ R L (0,T,W♦ E˙ n L (RT ) p √ √ ˜˙ n kL2 (R3 )3 kE ˜n − Ek ˜ L2 (0,T,W 1 ) ≤4 1 + R2 kµk∞ k σn k∞ k σn E T ♦ ˜˙ n kL2 (0,T,W 1 ) kE ˜n − Ek ˜ L2 (0,T,W 1 ) . ≤8(1 + R2 )kµk∞ kσn k∞ kE ♦ ♦ ˜˙ n )n∈N is bounded ˜˙ n solves (3.12) with σ = σn , Corollary 3.3 yields that (E As every E and thus the second assertion follows. 3 2 1 Lemma 4.5. Let d ∈ L∞ R (R ) and h > 0. Let F ∈ L (0, T, W♦ ) be the solution of T
Z
Z
a0 (F, Φ) = − R3
0
˜˙ + ∇u ˙ ) · Φ dx dt d(E ˜ E
for all Φ ∈ D♦ (R3 × [0, T [)3 ,
(4.1)
˜˙ under the mapping defined in Lemma 3.1 with σ = d. where ∇uE˜˙ is the image of E ˜h ∈ L2 (0, T, W 1 ) be the solution of (3.6) corresponding to σ = hd. Then Let E ♦ 1 ˜ ˜ * F ∈ L2 (0, T, W♦1 ) (Eh − E) h
(h → 0+ ).
Proof. From Lemma 4.4 it follows that the sequence is bounded and thus similar to Lemma 4.3, that every subsequence has a subsequence that convergences weakly against some F˜ ∈ L2 (0, T, W♦1 ). Again, Lemma 4.4 yields for every h a0
1 ˜ ˜ Φ (Eh − E), h
Z
T
Z
=− 0
R3
˜˙ h + ∇u ˙ ) · Φ dx dt for all Φ ∈ D♦ (R3 ×]0, T [)3 , d(E ˜ E h
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L. Arnold and B. Harrach
˜˙ h under mapping defined in Lemma 3.1 with σ = hd. where ∇uE˜˙ is the image of E h This mapping does not change, if we take it instead with σ = d. Consequently, as ˜˙ h * E ˜˙ implies ∇u ˙ * ∇u ˙ and thus F˜ solves (4.1). Again, (4.1) is d is fixed, E ˜h ˜ E E uniquely solvable in L2 (0, T, W 1 ), thus F˜ = F and the assertion follows. ♦
Proof of Theorem 4.2. Theorem 4.2 immediately follows from the last lemma. References. [1] R. Acevedo, S. Meddahi, and R. Rodr´ıguez, An e-based mixed formulation for a time-dependent eddy current problem, Math. of Comp., 78 (2009), pp. 1929– 1949. [2] R. Albanese and G. Rubinacci, Fomulation of the eddy-current problem, IEE Proc. A., 137 (1990), pp. 16–22. [3] A. Alonso, A mathematical justication of the low-frequency heterogeneous time-harmonic maxwell equations, Math. Models Methods Appl. Sci., 9 (1999), pp. 475–489. ´ de ´lec, A justification of eddy currents [4] H. Ammari, A. Buffa, and J.-C. Ne model for the maxwell equations, SIAM J. Appl. Math., 60 (2000), pp. 1805–1823. ¨ berl, Numerical analysis of nonlin[5] F. Bachinger, U. Langer, and J. Scho ear multiharmonic eddy current problems, Numerische Mathematik, 100 (2005), pp. 594–616. [6] R. Beck, P. Deuflhard, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of maxwell’s equations, Surveys Math. Indust., 8 (1999), pp. 271–312. [7] R. Beck, R. Hiptmair, R. H. W. Hoppe, and B. Wohlmuth, Residual based a posteriori error estimators for eddy current computation, M2AN Math. Model. Numer. Anal., 34 (2000), pp. 159–182. [8] R. Dautray and J.L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology - Volume 1: Physical Origins and Classical Methods, Springer-Verlag, Berlin, 2000. , Mathematical Analysis and Numerical Methods for Science and Technology [9] - Volume 2: Functional and Variational Methods, Springer-Verlag, Berlin, 2000. [10] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 3: Spectral Theory and Applications, Springer-Verlag, Berlin, 2000. [11] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 4: Integral Equations and Numerical Methods, Springer-Verlag, Berlin, 2000. [12] , Mathematical Analysis and Numerical Methods for Science and Technology - Volume 5: Evolution Problems I, Springer-Verlag, Berlin Heidelberg, 2000. [13] H. K. Dirks, Quasi-stationary fields for microelectronic applications, Electr. Engng., 79 (1996), pp. 145–155. [14] B. Flemisch, Y. Maday, F. Rapetti, and B. I. Wohlmuth, Coupling scalar and vector potentials on nonmatching grids for eddy currents in a moving conductor, J. Comput. Appl. Math., 168 (2004), pp. 191–205. ¨ hauf, B. Gebauer, and O. Scherzer, Detecting interfaces in a [15] F. Fru parabolic-elliptic problem from surface measurements, SIAM J. Appl. Math., 45 (2007), pp. 810–836. [16] B. Gebauer, Sensitivity analysis of a parabolic-elliptic problem, Quart. Appl. Math., 65 (2007), pp. 591–604. [17] B. Gebauer, M. Hanke, and C. Schneider, Sampling methods for low-
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frequency electromagnetic imaging, Inverse Problems, 24 (2008), p. 015007 (18pp). R. Hiptmair and O. Sterz, Current and voltage excitations for the eddy current model, Int. J. Numer. Model., 18 (2005), pp. 1–21. J. Lang and D. Teleaga, Towards a fully space-time adaptive fem for magnetoquasistatics, IEEE Transactions on Magnetics, 44 (2008), pp. 1238–1241. J. L. Lions, Equations differenti´elles op´erationelles et probl`emes aux limites, Springer-Verlag, Berlin, 1961. S. Meddahi and V. Selgas, An h-based fem-bem formulation for a time dependent eddy current problem, Appl. Numer. Math., 58 (2008), pp. 1061–1083. M. Renardy and R.C. Rogers, An Introduction to Partial Differential Equations, Texts in Applied Mathematics 13, Springer-Verlag, New York, 1993.
