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Analysis of the A, V--A--[psi] potential formulation for the eddy current problem in a bounded domain.

Abstract. The aim of this paper is to provide a mathematical analysis of the well-known A, V--A--[psi] potential formulation for the eddy current problem. The resulting variational problem is proved to be well posed and error estimates are settled for a numerical method based on standard nodal finite elements.

Key words. eddy currents, potential formulation, well-posedness, finite elements, error estimates

AMS subject classifications. 78M10, 65N30

1. Introduction. The mathematical and numerical analysis of Maxwell's equations has experienced important developments in different areas of applied mathematics and engineering during the last thirty years. We refer the reader to the books by Bossavit [17], Monk [29] and Silvester and Ferrari [32], as a representative sampling of text books devoted to numerical solution of electromagnetic problems.

Among the numerical methods found in the literature to approximate Maxwell's equations, the finite element method is the most extended. See for instance [13] for a survey on this subject including a large list of references. Nowadays, it is the basis of several commer cial codes such us Ansys, Femlab, Flux, Magnet, MSC/Emas, Opera, etc. We refer the reader to [33] for a description of most of these codes and further references.

The eddy current problem is obtained from Maxwell's equations by assuming that all fields are harmonic and the frequency is low enough as to neglect the electric displacement in Ampere's Law. Such a situation happens, for instance, in problems related to electric machines working at power frequencies and in non-destructive materials testing.

In most practical situations, it is necessary to solve the electromagnetic problem in a bounded domain which contains conducting and non-conducting material (dielectrics), the equations in these two parts being typically of different kind. Moreover, the treatment of multiply connected conductors or dielectrics in three-dimensional domains presents special difficulties. The choice of the unknowns in each subdomain is a crucial point for analysis of the problem in domains with a general topology.

An important number of formulations and finite element methods for solving the eddy current problem in three-dimensional bounded domains can be found in the literature. There is a group of papers devoted to solving the problem in terms of certain scalar and vector potentials [1, 2, 14, 15, 26, 30, 31] and another group using formulations in terms of the magnetic field [4, 6, 7, 9, 10, 12, 34] or the electric field [3, 11, 27].

A thorough mathematical analysis of the formulations in terms of the magnetic or the electric field has been recently performed. This is not the case, however, for formulations in terms of scalar and vector potentials. Indeed, in spite of the fact that the latter are the most frequently used in applications, there are only a very small number of papers dealing with their mathematical analysis. Among them, we mention a paper by Alonso et al. [5], where the well-posedness of some of these formulations is analyzed, and another one by Biro and Valli [16] with the analysis of one such formulation in a general topological setting.

Different potentials have been used for the eddy current problem: a vector potential A for the magnetic field, a scalar potential V for the electric field in the conducting domain, a scalar magnetic potential [psi] in dielectric domains, etc. A hierarchy of formulations involving these potentials has been discussed by Biro and Preis [15]. In particular, they conclude that the so-called A, V--A--[psi] formulation, which involves all of them, is the most convenient in terms of computer cost. Numerical experiments illustrating the performance of this approach are also reported in this reference.

The aim of this paper is to provide a rigorous mathematical analysis of this formulation. Under rather general topological conditions, we prove that it leads to a well-posed problem, which can be numerically approximated by standard nodal finite elements. We also prove error estimates for the resulting numerical method. These estimates are valid as long as the three potentials are sufficiently smooth.

The smoothness of the scalar potentials V and [psi] only relies on that of the original physical variables of the problem: the magnetic and the electric fields. However, the smoothness of the vector potential A also depends on the geometry of the domain chosen to define this non-physical variable. In principle this domain can be chosen freely, as far as it contains the conductors and the source currents. However, when it is chosen so that its connected components are either convex polyhedra or simply connected domains with smooth boundaries, the smoothness of A is mainly determined by the regularity of another physical variable: the magnetic induction field.

Because of this, we make such a choice for the domain of A, which is not restrictive in practice. However, it is convenient to choose it as small as possible, because the magnetic field is written in terms of the more economical scalar potential [psi] outside this domain. Thus, in the applications, the domain of A typically consists of a union of disjoint boxes, as small as possible, containing the current source and the conductors.

The outline of the paper is as follows: We introduce the eddy current problem and discuss the topological setting in Section 2. The A, V--A--[psi] potential formulation is introduced in Section 3. The corresponding variational problem is obtained in Section 4, where we also prove its well-posedness. Finally, in Section 5, we prove error estimates for a standard finite element method to solve the problem numerically. We also discuss in this section the convenience of choosing a domain with convex connected components for the vector potential.

2. Eddy current problem. We consider a standard eddy current problem: to determine the electromagnetic fields induced in a three-dimensional conducting domain [[OMEGA].sub.C] by a given source current density [J.sub.d]. We assume that the support of [J.sub.d] is compact and disjoint with [[OMEGA].sub.C]. The eddy current problem is in principle posed in the whole space. However, we restrict it to a bounded domain [OMEGA] containing both, [[OMEGA].sub.C] and the support of [J.sub.d], such that adequate boundary conditions can be imposed on its boundary. To this aim, we choose the geometry of [OMEGA] as simple as possible (e.g., simply connected with a connected boundary). See Fig. 2.1 for a two-dimensional sketch.

Let [[OMEGA].sub.C] [subset] [R.sup.3] be an open and bounded set with boundary [[GAMMA].sub.C]. Let [OMEGA] [subset] [R.sub.3] be a simply connected bounded domain with a connected boundary [GAMMA], such that [[OMEGA].sub.C] [subset] [OMEGA]. We suppose that both, [OMEGA] and [[OMEGA].sub.C], are either Lipschitz polyhedra or domains with [C.sup.1,1] boundaries. We denote by n and [n.sub.C] the outward unit normal vectors to [OMEGA] and [[OMEGA].sub.C], respectively, and by [[OMEGA].sub.D] := [OMEGA] \ [[OMEGA].sub.C] the subdomain of [OMEGA] occupied by dielectric material, which icludes the support of the source current; see Fig. 2.1. We will use standard notation for Sobolev spaces and norms.

[FIGURE 2.1 OMITTED]

The eddy current problem reads as follows:

Find E [member of] H(curl; [[OMEGA].sub.C]) and H [member of] H(curl; [OMEGA]) such that:

(2.1) curl H = [sigma]E in [[OMEGA].sub.C],

(2.2) i[omega][micro]H + curl E = 0 in [[OMEGA].sub.C],

(2.3) curl H = [J.sub.d] in [[OMEGA].sub.C],

(2.4) div([micro]H) = 0 in [OMEGA],

(2.5) H x n = [f.sub.d] in [GAMMA].

The unknowns E and H are the magnetic and electric fields, respectively. The magnetic permeability [micro] and the conductivity a are bounded functions satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Let us remark that the magnetic field has to satisfy the following coupling conditions:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

In fact, the latter is a conseque[n.sub.C]e of (2.4), whereas the former follows from the fact that H must belong to H(curl; [OMEGA]).

The data of the problem are the source current density [J.sub.d] [member of] [L.sup.2][([OMEGA]).sup.3], for which we assume

supp [J.sub.d] [subset] [[OMEGA].sub.D] and div [J.sub.d] = 0 in [[OMEGA].sub.D],

and the tangential trace of the magnetic field [f.sub.d]. Precise assumptions on [f.sub.d] will be made in Section 4 below; they essentially mean that [f.sub.d] has to be the tangential trace of a curl-free vector field (recall that curl H vanishes in the neighborhood of [GAMMA]).

Equations (2.1)-(2.5) are enough to determine E and H only if the topology of the conducting domain [[OMEGA].sub.C] is trivial. Otherwise, additional constraints must be imposed. To do this, we reduce our analysis to domains satisfying a standard topological assumption; see for insta[n.sub.C]e Amrouche et al. [8]. We assume that there exists MD connected open surfaces [[SIGMA].sub.k] (so called "cuts") contained in [[OMEGA].sub.D], such that:

(i) each surface [[SIGMA].sub.k] is an open part of a smooth manifold,

(ii) the boundary of each [[SIGMA].sub.k] is contained in [[GAMMA].sub.C] , (iii) the intersection [[SIGMA].sub.i] [intersection] [[SIGMA].sub.k] is empty for i [not equal to] j,

(iv) the open set [[OMEGA].sub.D] := [[OMEGA].sub.D] [[union].sub.k] [[SIGMA].sub.k] is pseudo-Lipschitz and simply connected.

Under this assumption, since [GAMMA] is connected, the space of harmonic fields

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

satisfies dim [H.sub.[micro]], ([GAMMA], [[GAMMA].sub.C]) = [m.sub.D]; see, for insta[n.sub.C]e Fernandez and Gilardi [24, Proposition 5.6]. A basis for this space is given by [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.], where each [[phi].sub.j] [member of] [H.sup.1.sub.[GAMMA]] ([[OMEGA].sub.D] \ [[SIGMA].sub.j]) is the solution of the following elliptic problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

the expression above [parallel] * [parallel] denotes the jump across [[SIGMA].sub.j]. Here and thereafter the subscript [GAMMA] in [H.sup.1.sub.[GAMMA]](*) refers to function in [H.sup.1](*) with a vanishing trace on [GAMMA].

Notice that although in principle grad [[phi].sub.j] [member of] LZ ([[OMEGA].sub.C] \ [[SIGMA].sub.j]), the last equation implies that [micro] grad [[phi].sub.j] is a divergence-free function in the whole [[OMEGA].sub.D] (not only in [[OMEGA].sub.D] \ [[SIGMA].sub.j]). Moreover, since the jump [parallel][[phi].sub.j][parallel][[SIGMA].sub.j]; is constant, grad [[phi].sub.j] has also a vanishing curl in the whole [[OMEGA].sub.D] (and not only in [[OMEGA].sub.D] \ [[SIGMA].sub.j], again). Thus, grad [[phi].sub.j] [member of] [H.sub.[micro]] ([GAMMA], [[GAMMA].sub.C]).

To determine a unique solution of the eddy current problem (2.1)-(2.5), it is enough to add the following constraints (see Alonso et al. [5]):

(2.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Let us remark that the second integral above has a weak sense for E [member of] H(curl; [[OMEGA].sub.C]) and grad [[phi].sub.j] [member of] H (curl; [[OMEGA].sub.D]), as was shown by Buffa and Ciarlet [18, 19] for Lipschitz polyhedra and by Buffa et al. [20] for arbitrary Lipschitz domains; see Section 4 below for a precise definition.

3. The A, V--A--[psi] potential formulation. In this section we recall a classical formulation of the eddy current problem in terms of three potentials, A, V and [psi], which was introduced by Leonard and Rodger [28]. We refer to Biro and Preis [15] for a detailed discussion, which includes numerical tests showing the efficiency of this approach.

First, we introduce a magnetic vector potential A defined in a subdomain [[OMEGA].sub.A] of [OMEGA], which contains the conducting domain and the support of the source current. This subdomain does not need to be connected, but each of its connected components will be chosen either convex or simply connected with a smooth boundary. The reason for such a choice will be discussed in Section 5 below. On the other hand, for the sake of discretization, it is convenient to choose a polyhedral domain [[OMEGA].sub.A]; moreover, outside [[OMEGA].sub.A], we will use a scalar potential, which will consequently require much less degrees of freedom for its discretization. Because of this, [[OMEGA].sub.A] will be chosen as small as possible, but with convex polyhedral connected components containing [[OMEGA].sub.C] and supp [J.sub.d]; see Fig. 3.1.

[FIGURE 3.1 OMITTED]

Let [[OMEGA].sub.C] [subset] [R.sup.3] be an open set satisfying

(3.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

We denote by [[OMEGA].sup.j.sub.A] = 1, ..., [m.sub.A], the connected components of [[OMEGA].sub.A]. We assume that each [[OMEGA].sup.j.sub.A] is either a convex polyhedron or a simply connected domain with a [C.sup.1,1] boundary, and that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] are mutually disjoint. We denote by [[GAMMA].sub.A] the boundary of [[OMEGA].sub.A] and by [n.sub.A] its outward unit normal vector; see Fig. 3.1.

As a consequence of [25, Theorem 1.3.5.], equation (2.4) implies that there exist unique [A.sub.j] [member of] H(curl; [[OMEGA].sup.j.sub.A]) such that

(3.2) [micro]H = curl [A.sub.j] in [[OMEGA].sup.j.sub.A],

(3.3) div [A.sub.j] = 0 in [[OMEGA].sup.j.sub.A],

(3.4) [A.sub.j] * [n.sub.A] = 0 on [delta][[OMEGA].sup.j.sub.A].

Thus, if we define A : [[OMEGA].sub.A] [right arrow] C by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

then A belongs to the space

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

whose natural norm is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Next, according to Biro and Preis [15] (see also Biro [14] and Biro and Valli [16]) we introduce an electric scalar potential V [member of] [H.sup.1]([[OMEGA].sub.C]), such that

(3.5) E = -i[omega]A - i[omega]grad V in [[OMEGA].sub.C].

Let us remark that (2.6) is a necessary condition for a global potential V to exist; see [5] and the formal argument at the end of this section. Notice that, from (2.1),

Moreover, since H [member of] H (curl; [OMEGA]), (2.1) and (2.3) also imply that

(i[omega][sigma]A + i[omega][sigma] grad V) * [n.sub.C] = 0 on [[GAMMA].sub.C].

These last two equations will be also collected in the potential formulation.

Equation (3.5) determines the electric potential V on each connected component of [[OMEGA].sub.C] up to an additive constant. Thus, if [[OMEGA].sub.C] has me connected components [[OMEGA].sup.j.sub.C], then the natural space for V is

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

endowed with the norm [parallel]grad V[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Finally, a magnetic scalar potential [psi] is defined in

[[OMEGA].sub.[psi]] := [OMEGA] \ [[OMEGA].sub.A]

(see Fig. 3.1). To do this, notice that since [[OMEGA].sub.A] is a disjoint union of convex sets with [[OMEGA].sub.A] [subset] [OMEGA] and [OMEGA] is simply connected, it turns out that [[OMEGA].sub.[psi]] is simply connected too. Therefore, from (2.3) and (3.1) we know that there exists [psi] [member of] [H.sup.1]([[OMEGA].sub.[psi]]) (unique up to an additive constant) such that

H = [omega] grad [psi] in [[OMEGA].sub.[psi]].

Thus, we are lead to the following formulation of problem (2.1)-(2.6) in terms of the potentials A [member of] X, V [member of] M and [psi] [member of] [H.sup.1]([[OMEGA].sub.[psi]])/C:

(3.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.12) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.13) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.14) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.15) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(3.16) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Let us remark that (3.9) and (3.15) are consequences of the fact that H [member of] H(curl; [OMEGA]), whereas (3.14) follows from the fact that [micro]H [member of] H(div; [OMEGA]), which in its turn is a consequence of (2.4)

To end this section we show that any solution of the above equations yields a solution of the eddy current problem (2.1)-(2.6). In fact, let (A, V, [psi]) [member of] X x M x [H.sup.1] ([[OMEGA].sub.[psi]])/C satisfying (3.6)-(3.16). Let

(3.17) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

and E be defined by (3.5). It is immediate to show that H [member of] H(curl; [OMEGA]), [member of] [member of] H (curl; [[OMEGA].sub.C]), and they satisfy (2.1)-(2.5). There only remains to prove that (2.6) also holds true. To do this, notice that (2.4) implies that there exists a vector potential B [member of] H(curl; [OMEGA]) such that

(3.18) [micro]H = curl B in [OMEGA].

Taking into account that the sets [[OMEGA].sup.j.sub.A] are simply connected and mutually disjoint, from (3.17) there follows that there exists [xi] [member of] [H.sup.1] ([[OMEGA].sub.A]) such that

A = B + grad [xi] in [[OMEGA].sub.A].

Consequently, if we define V := V + [xi]/[[OMEGA].sub.C] we obtain from (3.5) that

(3.19) E = - i[omega](B + grad V) in [[OMEGA].sub.C].

Equations (3.18) and (3.19) fall in the framework analyzed by Alonso et al. [5, Section 6 (ii)], where it is shown that (2.6) holds true. This can be formally verified by using (3.17), the fact that grad [[phi].sub.j] [member of] [H.sub.[micro]] ([GAMMA], [[GAMMA].sub.C]) and (3.19), as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where for the last equality we have used that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with V * [member of] [H.sup.1] ([OMEGA]) being an extension of V to the whole [OMEGA].

4. Variational formulation. Existence and uniqueness of solution. The aim of this section is to give a variational formulation of problem (3.6)-(3.16) and to prove its well-posedness.

First, we recall some results settled in [20] for Lipschitz domains. We write these results for [[OMEGA].sub.A], as will be used in the sequel. The tangential trace operator [[gamma].sub.[tau]](u) := u|[[GAMMA].sub.A] x [n.sub.A] is a bounded linear operator from H (curl; [OMEGA] A) onto [H.sup.-1/2] (dive; I7A). The tangential projection [[pi].sub.[tau]],(v) := [n.sub.A] x (v|[[GAMMA].sub.A] x [n.sub.A]) is a bounded linear operator from H(curl; [[OMEGA].sub.A]) onto [H.sup.-1/2] (curl[GAMMA]; [[GAMMA].sub.A]). Thus, the duality pairing between [H.sup.-1/2] ([div.sub.[GAMMA]]; [[GAMMA].sub.A]) and [H.sup.-1/2] (curl[GAMMA]; [[GAMMA].sub.A]) is well defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

For any w [member of] H(curl; [[OMEGA].sub.[psi]]), its tangential trace on [[GAMMA].sub.A] also belongs to [H.sup.-1/2] ([div.sub.[GAMMA]]; [[GAMMA].sub.A]) and, consequently, (w x [n.sub.A], [[pi].sub.[tau]](v))[[GAMMA].sub.A] is also well defined.

To obtain a variational formulation of problem (3.6)-(3.16), notice that by virtue of (3.6), (3.8) and (3.9) we have that 1/[micro] curl A [member of] H (curl; [[OMEGA].sub.A]), and for all Z [member of] X

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Integrating by parts the left-hand side above and using (3.11) and (3.15), there follows

(4.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the other hand, from (3.7), by integrating by parts and using (3.16) we have for all U [member of] [H.sup.1]([[OMEGA].sub.C])

(4.2) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Finally, for any [psi] [member of] [H.sup.1.sub.[GAMMA]] ([[OMEGA].sub.[psi]]), from (3.10), by integrating by parts and using (3.14), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where the last integral must be understood as the duality pairing between [H.sup.-1/2] ([[GAMMA].sub.A]) and [H.sup.1/2] ([[GAMMA].sub.A]). Now, let [psi]* [member of] [H.sup.1] ([OMEGA]) be an extension of [psi] to the whole [OMEGA]. Hence,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore, we obtain

(4.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Equations (4.1)-(4.3) together with the essential condition (3.13), provide the following variational formulation of problem (3.6)-(3.16):

Find A [member of] X, V [member of] M and [psi] [member of] [H.sup.1] ([[OMEGA].sub.[psi]])/C such that:

(4.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(4.5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

(4.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(4.7) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Our next goal is to prove that this variational problem has a unique solution. For this purpose, first of all notice that (4.4) can be satisfied only if [f.sub.d] is the tangential trace on [GAMMA] of a gradient. Thus, this additional hypothesis turns out necessary for the problem to have a solution. So, we make the following assumption:

(4.8) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, let A be the bilinear form defined on X x M x [H.sup.1] ([[OMEGA].sub.[psi]])/C by A((A, V, [psi]), (Z, U [psi]))

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Clearly, (4.4)-(4.7) can be equivalently written as follows:

Find (A, V, [psi]) [member of] X x M x [H.sup.1] ([[OMEGA].sub.[psi]])/C such that:

(4.9) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

(4.10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

THEOREM 4.1. Under assumption (4.8), the variational problem (4.9)-(4.10) has a unique solution.

Proof. It is enough to show that A is elliptic, since, in such a case, the theorem follows from Lax-Milgram's Lemma.

To prove the ellipticity, for (Z, U [psi]) [member of] X x M x [H.sup.1] ([[OMEGA].sub.[psi]])/C we write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

Thus,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Next, we proceed as in [16] and use the elementary inequality

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

to obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, since (1)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

with [KAPPA] > 0 independent of Z, we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

To estimate the last term in the right-hand side above, notice first that, for all [epsilon] > 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

On the other hand, [there exists]C > 0 independent of [psi] and Z such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore, by combining the last three inequalities and taking [epsilon] and [rho] small enough, we obtain that [there exists][alpha] > [psi] such that, [for all] (Z, U [psi]) [member of] X x M x [H.sup.1]([[OMEGA].sub.[psi]])/C,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

which allows us to conclude the ellipticity of A.

To end this section, we prove that the unique solution of the variational problem (4.9)-(4.10) is actually a solution of the strong form of the problem given by equations (3.6)-(3.16).

THEOREM 4.2. The solution (A, V, [psi]) of (4.9)-(4.10) satisfies (3.6)-(3.16).

Proof. First, let [xi] [member of] [H.sup.1] ([[OMEGA].sub.A]) be a solution of the compatible Neumann problem [DELTA][xi] = div A in [[OMEGA].sub.A], [delta][xi]/[delta][n.sub.A] = 0 on [[GAMMA].sub.A]. By testing (4.5) with Z = grad [xi] [member of] X, we obtain (3.11) by using (4.6) (since [xi]|[[OMEGA].sub.C] [member of] M) and (grad [psi] x [n.sub.A], [[pi].sub.[tau]], (grad [xi]))[[GAMMA].sub.A] = 0 (which is a consequence of the definition of the duality pairing).

Second, by testing (4.5)-(4.7) with smooth functions supported in adequate domains and proceeding in the standard way, it is easy to verify equations (3.6)-(3.10), (3.14) and (3.16). Since (3.12) is imposed in the definition of the space X and (3.13) coincides with (4.9), there only remains to prove (3.15) in [H.sup.-1/2] ([div.sub.[GAMMA]];[[GAMMA].sub.A]); namely, that for all [zeta] [member of] H(curl; [[OMEGA].sub.A]),

(4.11) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

To do this, notice first that by substituting (3.11) in (4.5), integrating by parts and having into account (3.6) and (3.8), we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Next, for [zeta] [member of] H(curl; [[OMEGA].sub.A]), let [psi] be a solution of the following auxiliary problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Hence, div([zeta] - grad [psi]) = 0 in [[OMEGA].sub.A] and ([zeta] - grad [psi]) * [n.sub.A] = 0 on [[GAMMA].sub.A]. Consequently, Z := [zeta] - grad [psi] [member of] X, and using it as a test function in the equation above, we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Now, from (3.6) and (3.8), we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where, for the last step, we have used integration by parts, (3.7), (3.16), the assumption that [J.sub.d] is divergence-free and (3.1).

Thus, using again that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.] vanishes, (4.11) follows from the last two equations, and we conclude the proof.

5. Numerical approximation. In this section we describe and analyze a finite element method to approximate the solution of problem (4.9)-(4.10). To do this, first notice that (4.9) implies that the surface gradient of [psi] can be written as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Therefore, if we take an arbitrary but fixed point [x.sub.0] [member of] [GAMMA] and if the data [f.sub.d] is sufficiently smooth (for instance, it is enough that [f.sub.d] [member of] [H.sup.1/2+[delta]][([GAMMA]).sup.3] with [delta] > 0), then we can compute in advance the values of [psi] on P as follows:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [alpha] (x) is any simple curve lying on [GAMMA] and joining [x.sub.0] with x, and [t.sub.[alpha](x)] is its unit tangent vector. Notice that the computed value of [psi](x) is independent of the particular curve [alpha] (x). Thus, if we define

(5.1) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

then problem (4.9)-(4.10) is equivalent to the following one:

Find (A, V, [psi]) [member of] X x M x [H.sup.1] ([[OMEGA].sub.[psi]]) such that:

(5.2) [psi] = [g.sub.d] on [GAMMA],

(5.3) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

To obtain a discrete formulation of this problem, we further assume that all the domains are Lipschitz polyhedra. Let {[T.sub.h]} be a family of tetrahedral meshes of [OMEGA] such that, for each mesh, all the elements T [member of] [T.sub.h] are completely included in one of the three subdomains [[OMEGA].sub.A], [[OMEGA].sub.C] or [[OMEGA].sub.[psi]].

Consider the following finite element spaces:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

where [P.sub.m], m [greater than or equal to] 1, is the set of polynomials of degree not greater than m.

For the boundary condition, we choose the following discrete approximation of [g.sub.d]:

(5.4) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

where [[PI].sup.[GAMMA].sub.h] is the Lagrange interpolant on the triangular mesh on [GAMMA] which consists of the faces of tetrahedra of T [member of] [[tau].sub.h] lying on [GAMMA], that we denote [[tau].sub.h]. Notice that the definition of [g.sub.d,h] makes sense because [g.sub.d], as defined by (5.1), is continuous. Let us remark that [g.sub.d,h] is completely determined by its values at the vertices of the triangulation [[tau].sup.[GAMMA].sub.h], which can be conveniently computed from the data [f.sub.d] by means of (5.1), with [alpha] (x) being a curve formed by edges of [[tau].sup.[GAMMA].sub.h].

Thus, we are lead to the following discrete problem:

Find ([A.sub.h], [V.sub.h], [[psi].sub.h]) [member of] X h x [M.sub.h] x [Q.sub.h] such that:

(5.5) [[psi].sub.h] = [g.sub.d,h] on [GAMMA],

(5.6) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The existence and uniqueness of the solution of this discrete problem is again an immediate consequence of the ellipticity of d, proved in the proof of Theorem 4.2, and Lax-Milgram's Lemma. Moreover, if the solution of the continuous problem is smooth enough, the standard finite element error analysis techniques yield the following result:

THEOREM 5.1. Let [g.sub.d] [member of] C([GAMMA]) and [g.sub.d,h] be defined by (5.4). Let (A, V, [psi]) and ([A.sub.h], [V.sub.h], [[psi].sub.h]) be the solutions of problems (5.2)-(5.3) and (5.5)-(5.6), respectively.

If A [member of] [H.sup.1+s][([[OMEGA].sub.A]).sup.3], V [member of] [H.sup.1+s]([[OMEGA].sub.C]) and [psi] [member of] [H.sup.1+s]([[OMEGA].sub.[psi]]) with s > 0, then there exists a strictly positive constant C, independent of h, A, V and [psi], such that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

with r := min {m, s}.

Proof. Let [[PI].sub.h] be the Lagrange interpolant on [Q.sub.h]. Since

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

we have that [[psi].sub.h] - [[PI].sub.h][psi] [member of] [Q.sub.[GAMMA],h]. Therefore, the theorem is a direct consequence of the ellipticity of A, Cea's lemma and the approximation properties of the Lagrange interpolant; see, for instance, Ciarlet [21].

To end the paper we discuss the convenience of choosing the domain [[OMEGA].sub.A] of the vector potential so that its connected components be convex polyhedra. For simplicity, we take [[OMEGA].sub.A] connected in what follows, but all the statements hold true for each of its connected components. So let [[OMEGA].sub.A] be simply connected with a connected boundary.

According to [25, Theorem 1.3.4], since div([micro]H) = 0 in [OMEGA], there exists [PHI] [member of] [H.sup.1][([OMEGA]).sup.3] satisfying:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

Moreover, according to Remark 1.3.12 of the same reference, if [micro]H [member of] [H.sup.p][([OMEGA]).sup.3] with 0 < p [less than or equal to] 1, then [PHI] [member of] [H.sup.1+P][([OMEGA]).sup.3].

Therefore, by virtue of (3.2)-(3.4), there holds:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.]

The first equation above and the simple-connectedness of [[OMEGA].sub.A] implies that there exists a unique X [member of] [H.sup.1] ([[OMEGA].sub.A])/(C such that A - [PHI] = grad [chi] in [[OMEGA].sub.A], whereas the remaining equations imply that X is the solution of the following compatible Neumann problem:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

The Neumann data of this problem will be in general smooth on each polygonal face F of [[GAMMA].sub.A], since [[GAMMA].sub.A] is an arbitrary polyhedral surface within the dielectric domain. In fact, if [micro]H [member of] [H.sup.p] [([OMEGA]).sup.3] with 0 < p [less than or equal to] 1, then [PHI]|F * [n.sub.A] [member of] [H.sup.1/2+p](F) for all faces F.

Therefore, if [[OMEGA].sub.A] is a convex polyhedron, then there exists q > 0 such that [chi] [member of] [H.sup.2+q] ([[OMEGA].sub.A]); see [23]. Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.],

with s := min {p, q} > 0. Conversely, if [[OMEGA].sub.A] were a non-convex polyhedron, then, in general, [chi] [??] [H.sup.2] ([[OMEGA].sub.A]) and, consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII.].

In such a case, Theorem 5.1 would become meaningless.

Moreover, Y := {Z [member of] [H.sup.1]([[OMEGA].sub.A])3 : Z * [n.sub.A] = 0 on [[GAMMA].sub.A]} is a closed subspace of X; see [22]. When [[OMEGA].sub.A] is a polyhedron, it is well-known that Y = X if and only if [[OMEGA].sub.A] is convex; see [25, Theorem 1.3.9] and [22].

The finite element space [X.sub.h] is clearly a subspace of Y. Therefore, when [[OMEGA].sub.A] is a convex polyhedron, it makes sense to approximate A [member of] X by finite elements from [X.sub.h].

Instead, if [[OMEGA].sub.A] were not convex, then there would be no hope of approximating A by finite elements from [X.sub.h]. Indeed, as stated above, in general A [??] [H.sup.1][([[OMEGA].sub.A]).sup.3] in such a case. Hence, A would not belong to the closed set Y containing the finite element spaces [X.sub.h] for all meshes. So, there could not exist [A.sub.h] such that [parallel]A - [A.sub.h][[parallel].sub.x] [right arrow] 0 as h goes to zero.

6. Conclusions. We have proved that the A, V - A - [psi] formulation of the eddy current problem is well posed and that its discretization by standard nodal finite elements leads to an optimal-order numerical approximation. This gives mathematical support to the well-known efficiency of this approach in applications.

However, for the convergence of the numerical method, the connected components of the domain of the vector potential A must be chosen as convex polyhedra. Since this domain can be chosen freely (as far as it contains the conductors and the source current), this is not a severe restriction in practice.

Acknowledgments. The authors want to express their gratitude to Monique Dauge for helpful discussions.

* Received August 30, 2006. Accepted for publication January 22, 2007. Recommended by O. Widlund.

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(1) For the first inequality, see, for instance, [25, Lemma l.3.6].

RAMIRO ACEVEDO ([dagger]) AND RODOLFO RODRIGUEZ ([double dagger])

([dagger]) Departamento de Ingenieria Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile (racevedo@ing-mat.udec.cl). Permanent address: Departamento de Matematicas, Universidad del Cauca, Popayan, Colombia. The research of this author was partially supported by MECESUP U000406 (Chile) and Universidad del Cauca (Colombia).

([double dagger]) G[I.sup.2]MA, Departamento de Ingenieria Matematica, Universidad de Concepcion, Casilla 160-C, Concepcion, Chile (rodolfo@ing-mat.udec.c1). The research of this author was partially supported by FONDAP in Applied Mathematics (Chile).
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