AN OPTIMAL DOMAIN DECOMPOSITION PRECONDITIONER FOR LOW-FREQUENCY TIME-HARMONIC MAXWELL EQUATIONS

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1 MATHEMATICS OF COMPUTATION Volume 68, Number 226, April 1999, Pages S (99) AN OPTIMAL DOMAIN DECOMPOSITION PRECONDITIONER FOR LOW-FREQUENCY TIME-HARMONIC MAXWELL EQUATIONS ANA ALONSO AND ALBERTO VALLI Abstract. The time-harmonic Maxwell equations are considered in the lowfrequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields. 1. Introduction The Maxwell equations read D = rot H J, t B = rot E, t where E and H are the electric and magnetic field, D and B the electric and magnetic induction, respectively, and J is the density of the electric current. The following constitutive relations D = εe, B = µh (where ε and µ are the dielectric and magnetic permeability coefficients, respectively) are assumed to hold, as well as the Ohm s law J = σe (where σ is the electric conductivity). The quantities ε, µ and σ are in general symmetric matrices, depending on the space variable x; ε and µ are assumed to be positive definite, whereas σ is positive definite in a conductor and vanishing in an insulator. Received by the editor December 2, 1996 and, in revised form, July 30, Mathematics Subject Classification. Primary 65N55, 65N30; Secondary 35Q60. ey words and phrases. Domain decomposition methods, Maxwell equations. Partially supported by H.C.M. contract CHRX c 1999 American Mathematical Society

2 608 ANA ALONSO AND ALBERTO VALLI Writing the Maxwell equations in terms of E and H only, we find ε E = rot H σe, t µ H = rot E. t We are interested in the so-called time-harmonic case, i.e., we assume that E and H are given by E(t, x) =Re[E(x)exp(iαt)], H(t, x) =Re[H(x)exp(iαt)], where E and H are three-dimensional complex-valued vector fields, and α 0 is a given angular frequency. Therefore, the equations become iαεe = rot H σe, (1.1) iαµh = rot E, and eliminating H we find (1.2) rot(µ 1 rot E) α 2 (ε iα 1 σ)e = 0. If we are considering the low-frequency case, i.e., the parameter α is small, by checking the effective values of the dielectric coefficient ε, the magnetic permeability µ and the conductivity σ for general media, it can be seen that the parameter α 2 ε is much smaller than µ 1 and ασ. Therefore, in this case the term α 2 εe can be dropped out, and one is left with (1.3) rot(µ 1 rot E)+iασE = 0. Formally speaking, the low-frequency model is thus obtained from the general equation (1.2) by setting ε = 0. Afterwards we will refer to the low-frequency case as to the case where ε =0. Considering (1.1) or (1.2) in a bounded domain Ω R 3,wehavetoimposethe boundary condition (1.4) n E = Ψ on Ω, where n is the unit outward normal vector on Ω andψis a tangential vector on Ω. Most often, it is assumed that a vector function Ê is known, satisfying n Ê = Ψ on Ω. Then the resulting boundary value problem reads rot(µ 1 rot u) α 2 (ε iα 1 σ)u = F in Ω, (1.5) (n u) Ω = 0 on Ω, where u = E Ê and F = rot(µ 1 rot Ê)+α2 1 (ε iα σ)ê. Let us now make precise some notation. As usual, we indicate by H k (Ω), k 0, the Sobolev space of (classes of equivalence of) real or complex functions belonging to L 2 (Ω) together with all their distributional derivatives of order less than or equal to k. In particular, L 2 (Ω) = H 0 (Ω). We also consider the Sobolev space H s (Ω) for s R, whose definition can be found in Adams [1]. It is well known that the trace space of H 1 (Ω) over Ω is given by the Sobolev space H 1/2 ( Ω); more generally, if Σ is a proper (non-empty) subset of Ω, the trace space of H 1 (Ω) over Σ is given by H 1/2 (Σ). The spaces H 1/2 ( Ω) and

3 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 609 H 1/2 (Σ) are the dual spaces of H 1/2 ( Ω) and H 1/2 (Σ), respectively. The duality pairing between these spaces will be denoted by, Ω and, Σ. The norm in the Sobolev space H s (Ξ) will be denoted by s,ξ,wheres RandΞcanbe either the whole domain Ω, or the boundary Ω, or else a suitable surface. The space H(rot; Ω) (respectively, H(div; Ω)) indicates the set of the real or complex (vector) functions v (L 2 (Ω)) 3 such that rot v (L 2 (Ω)) 3 (respectively, div v L 2 (Ω)). We also need the definition of the tangential divergence of a tangential vector field η. Beinggivenη (H 1/2 ( Ω)) 3 with (η n) Ω =0,wedefinethetangential divergence div τ η of η as the distribution in H 3/2 ( Ω) which satisfies div τ η,ψ Ω := η, ( ψ 2 ) Ω Ω ψ H 3/2 ( Ω), where ψ2 H 2 (Ω) is any extension of ψ in Ω, and we have denoted by, Ω the duality pairing between H 3/2 ( Ω) and H 3/2 ( Ω). Notice that, due to the condition (η n) Ω = 0, the right hand side indeed depends only on the value of ψ on Ω. We can now introduce the Hilbert spaces X Ω and X Σ, where Σ is a proper (non-empty) subset of Ω. The former one is defined as X Ω := {η (H 1/2 ( Ω)) 3 (η n) Ω =0anddiv τ η H 1/2 ( Ω)}, with the norm η X Ω := η 1/2, Ω + div τ η 1/2, Ω. Denoting by γ (H 1/2 ( Ω)) 3 the extension of γ by 0 on Ω \ Σ, the space X Σ is X Σ := {γ (H 1/2 (Σ)) 3 (γ n) Σ =0anddiv τ γ H 1/2 ( Ω)}, endowed with the norm γ XΣ := γ 1/2,Σ + div τ γ 1/2, Ω. In Alonso and Valli [2] it has been proven that, if either Ω C 1,1 or Ω is a convex polyhedron, the space X Ω is equal, algebraically and topologically, to the space of tangential traces of H(rot; Ω). Similarly, X Σ is the space of tangential traces of H Ω\Σ (rot; Ω) := {v H(rot; Ω) (n v) Ω\Σ = 0}. Furthermore, in [2] it has been shown that there exist two linear and continuous extension operators satisfying for each η X Ω and γ X Σ. R Ω : X Ω H(rot; Ω), R Σ : X Σ H Ω\Σ (rot; Ω) (n R Ω η) Ω =η, (n R Σ γ) Σ =γ

4 610 ANA ALONSO AND ALBERTO VALLI 2. Weak formulation of the problem and finite element approximation We are going to make precise the variational formulation. First of all, we need the following notation: H 0 (rot; Ω) := {v H(rot; Ω) (n v) Ω = 0}, H 0 (rot; Ω) := {v H(rot; Ω) rot v = 0}, H 0 (div; Ω) := {v H(div; Ω) (n v) Ω =0}, H 0 (div; Ω) := {v H(div; Ω) div v =0}. We also assume that the coefficients µ =(µ ij (x)) 1 i,j 3, ε =(ε ij (x)) 1 i,j 3 and σ = (σ ij (x)) 1 i,j 3 are symmetric matrices with real coefficients belonging to L (Ω). The magnetic permeability is uniformly positive definite (UPD from now on); namely, there exists a constant µ 0 > 0 such that 3 µ lm (x)ξ l ξ m µ 0 ξ 2 for almost all x Ωandforallξ C 3. l,m=1 The dielectric coefficient ε is assumed to be UPD in the high-frequency case and 0 in the low-frequency case. The conductivity σ can be UPD (when Ω is a conductor), or else given by σ = σχ Ω\Ω0,whereΩ 0 is a (non-empty) subset of Ω (representing an insulator), χ Ω\Ω0 is the characteristic function of Ω \ Ω 0,and σis UPD. In particular, the case Ω 0 = Ω corresponds to the case of a perfect insulator. We introduce in H(rot; Ω) the following bilinear form: a ε (w, v) :=(µ 1 rot w, rot v) α 2 ([ε iα 1 σ]w, v), where (, ) denotes the (L 2 (Ω)) 3 -scalar product (for complex-valued vector functions), and we set L(v) :=(F,v). Definition 2.1. A weak solution of (1.5) is a function u H 0 (rot; Ω) such that (2.1) a ε (u, v) =L(v) v H 0 (rot; Ω). The high-frequency case (ε is assumed to be UPD) has been considered by Leis [9]. First of all, the bilinear form a ε (, ) has been proven to be coercive in H(rot; Ω) when σ is UPD. Moreover, the Fredholm alternative theorem holds for problem (2.1) when σ = 0 (i.e., Ω 0 = Ω) (see [9]). A unique solvability result has been proven by Alonso and Valli [4] for the conductivity given by σ = σχ Ω\Ω0,Ω 0 Ω. We are mainly interested in the sequel in the low-frequency case (ε is taken to be 0) for a conductor (σ is assumed to be UPD). In that case we can verify at once that the bilinear form a 0 (, ) is continuous and coercive in H(rot; Ω); therefore, the Lax-Milgram lemma yields Theorem 2.2. Let Ω be a bounded domain, and assume that ε =0and σ is UPD. Then there exists a unique solution of (2.1). A different approach is needed in the low-frequency case when the conductivity is given by σ = σχ Ω\Ω0. In this case the problem rot(µ 1 rot u)+iασu = F in Ω, (2.2) (n u) Ω = 0 on Ω,

5 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 611 does not have a unique solution, as we can always add the gradient of a harmonic function supported in Ω 0 to a solution. Therefore, we have to complete the differential model by adding suitable equations. Alonso and Valli [3], by means of a perturbation argument, have proposed the following problem: rot(µ 1 rot u)+iασu = F in Ω, div(u Ω0 )=0 inω 0, (2.3) (n u) Ω =0 on Ω, (u Ω0 n) Γ0,j, 1 Γ0,j =0 j=1,..., p, where Γ 0,j are the internal connected components of Ω 0. In [3] it has been proven that (2.3) has a unique solution when Ω 0 Ω, and the interface Ω 0 (Ω \ Ω 0 )is either a C 1,1 surface or a convex polyhedral portion of Ω 0. Finally, in the case Ω 0 = Ω (i.e., σ = 0) problem (2.3) reduces to a coercive problem in H 0 (rot; Ω) H(div; Ω) H(e),havingset H(e):={ω H 0 (rot; Ω) H 0 (div; Ω) (n ω) Ω = 0} (see Saranen [15], Valli [17]). We are now going to present some approximation results that have been obtained for problem (2.1) via the finite element method. In the low-frequency case for a conductor, the bilinear form a 0 (, ) is coercive in H(rot; Ω); therefore, the problem is rather standard, and one only needs to construct a suitable internal finite dimensional approximation of the space H(rot; Ω). To this end, the so-called Nédélec finite elements (see Nédélec [12], [13]) can be used, as they are conforming in H(rot; Ω) (their tangential components are continuous across the faces of the finite elements). An optimal order error estimate can be obtained straightforwardly. In the same case, by means of a different approach, řížek and Neittaanmäki [8] proposed a finite element space given by standard Lagrangian piecewise-linear vector functions satisfying suitable conditions on the interfaces. In particular, when σ is a constant, these conditions reduce to the continuity across the interfaces, thus furnishing a finite dimensional subspace of (H 1 (Ω)) 3. Also in the high-frequency case Monk [10] has used the Nédélec finite elements, both for the case where σ is assumed to be UPD and for σ = 0, yielding an optimal order error estimate. The low-frequency heterogeneous problem (2.3), in which σ = σχ Ω\Ω0,Ω 0 Ω, has been considered in [3]. At first the problem has been rewritten in an equivalent two-domain formulation, and then the Nédélec finite elements are employed in Ω \ Ω 0, whereas Lagrangian piecewise-polynomial finite elements are used in Ω 0 for approximating a scalar potential of the magnetic field. Due to the heterogeneous nature of the problem, a natural domain decomposition algorithm can be devised, solving the problem iteratively in Ω 0 andinω\ω 0.The convergence of this iterative procedure is proven in [3], where the rate of convergence is also shown to be independent of the mesh size h. Clearly, it is also interesting to use a domain decomposition technique for solving the two subproblems in Ω 0 and in Ω\Ω 0. For what is concerned with the problem in

6 612 ANA ALONSO AND ALBERTO VALLI the perfect insulator Ω 0, is equivalent to the construction of an extension operator from X Ω0 into H(rot; Ω 0 ) (see Alonso and Valli [2]). Moreover, this last problem can be reduced to a non-homogeneous Neumann boundary value problem for the Laplace operator, and domain decomposition techniques for its finite element approximation are well known. In the next section we are going to consider the domain decomposition approach to the finite element approximation of the low-frequency conductor problem, namely the case where it is assumed that ε = 0 and that σis UPD. 3. The domain decomposition procedure We consider the low-frequency conductor problem rot(µ 1 rot u)+iασu = F in Ω, (3.1) (n u) Ω = 0 on Ω. The bilinear form associated to (3.1) is given by (3.2) a 0 (w, v) := (µ 1 rot w rot v + iασw v), Ω and the weak formulation reads as in Definition 2.1. Let the bounded domain Ω be decomposed in two subdomains Ω 1 and Ω 2 such that Ω=Ω 1 Ω 2 and Ω 1 Ω 2 =. We will set Γ := Ω 1 Ω 2. In each subdomain we want to solve rot(µ 1 rot u j )+iασu j = F in Ω j, (n u j ) Ωj \Γ = 0 on Ω j \ Γ with the interface conditions (3.3) (n Γ u 1 ) Γ =(n Γ u 2 ) Γ, (3.4) (n Γ µ 1 rot u 1 ) Γ =(n Γ µ 1 rot u 2 ) Γ. Set (3.5) (3.6) (3.7) V j := {v j H(rot; Ω j ) (n v j ) Ωj\Γ = 0}, a j (w j, v j ):= (µ 1 rot w j rot v j + iασw j v j ), Ω j w j, v j V j, L j (v j ):= F v j. Ω j

7 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 613 The bilinear forms a j are clearly continuous and coercive in V j. The variational formulation of the two-domain problem reads find, (u 1, u 2 ) V 1 V 2 : (3.8) a 1 (u 1, v 1 )=L 1 (v 1 ) v 1 H 0 (rot; Ω 1 ) (n Γ u 1 ) Γ =(n Γ u 2 ) Γ a 2 (u 2,v 2 )=L 2 (v 2 )+L 1 (R 1 (n Γ v 2 ) Γ ) a 1 (u 1,R 1 (n Γ v 2 ) Γ ) v 2 V 2, where R 1 : X Γ V 1 is any extension operator. The equivalence of the formulations (2.1) and (3.8) can be easily proven (see, for instance, Alonso and Valli [3], where a similar situation is considered). For the numerical approximation, we will use the Nédélec finite elements of the first kind (see Nédélec [12]). However, the same results could be proven also for the Nédélec finite elements of the second kind (see Nédélec [13]). For the reader s convenience, we present here the precise definitions of the former elements. Let us assume that Ω, Ω 1 and Ω 2 are a Lipschitz polyhedrons. Let {T h } h>0 be a family of triangulations composed by tetrahedrons, where h is their maximum diameter. Moreover, assume that each element of T h only intersects either Ω 1 or Ω 2. Let P k, k 1, be the space of polynomials of degree less than or equal to k, and denote by P k the space of homogeneous polynomials of degree k. Weset S k := {p (P k )3 p(x) x =0}, R k := (P k 1 ) 3 S k. Notice that (P k 1 ) 3 R k (P k ) 3. We will employ the finite element space and we define N k j,h := {v h H(rot; Ω j ) v h R k T j,h } (3.9) V j,h := N k j,h V j, (3.10) (3.11) V 0 j,h := N k j,h H 0(rot; Ω j ), X Γ,h := {(n Γ v 1,h ) Γ v 1,h V 1,h } = {(n Γ v 2,h ) Γ v 2,h V 2,h }. The finite dimensional approximation problem reads find, (u 1,h, u 2,h ) V 1,h V 2,h : (3.12) a 1 (u 1,h, v 1,h )=L 1 (v 1,h ) v 1,h V 0 1,h (n Γ u 1,h ) Γ =(n Γ u 2,h ) Γ a 2 (u 2,h, v 2,h )=L 2 (v 2,h )+L 1 (R 1,h (n Γ v 2,h ) Γ ) a 1 (u 1,h, R 1,h (n Γ v 2,h ) Γ ) v 2,h V 2,h, where R 1,h is any extension operator from X h,γ to V 1,h.

8 614 ANA ALONSO AND ALBERTO VALLI Let us introduce now for each γ h X h,γ the solution E h j,γ γ h of the problem E h j,γ γ h V j,h : (3.13) a j (E h j,γ γ h, v j,h )=0 v j,h Vj,h 0 (n Γ E h j,γ γ h) Γ = γ h, and also the solution û j,h V 0 j,h of (3.14) û j,h Vj,h 0 : a j(û j,h, v j,h )=L j (v j,h ) v j,h Vj,h 0, whose existence and uniqueness is a consequence of Lax-Milgram Lemma. Then the couple (E h 1,Γ λ h + û 1,h, E h 2,Γ λ h + û 2,h ) is a solution to (3.12) if and only if a 2 (E h 2,Γ λ h + û 2,h, v 2,h )=L 2 (v 2,h )+L 1 (E h 1,Γ (n Γ v 2,h ) Γ ) (3.15) a 1 (E h 1,Γ λ h + û 1,h, E h 1,Γ (n Γ v 2,h ) Γ ) v 2,h V 2,h is satisfied. Due to (3.13) and (3.14), this is equivalent to a 2 (E h 2,Γλ h, E h 2,Γη h )+a 2 (û 2,h, E h 2,Γη h )=L 2 (E h 2,Γη h )+L 1 (E h 1,Γη h ) (3.16) a 1 (E h 1,Γλ h,e h 1,Γη h ) a 1 (û 1,h, E h 1,Γη h ) η h X Γ,h. We define the Steklov-Poincaré operators S j,h, j =1,2, in the following way: (3.17) S j,h γ h, η h h := a j (E h j,γ γ h, Eh j,γ η h ) γ h, η h X Γ,h, where, h denotes the duality pairing between (X Γ,h ) and X Γ,h. Define moreover Φ h, η h h := L 1 (E h 1,Γ η h) a 1 (û 1,h, E h 1,Γ η h) + L 2 (E h 2,Γ η h) a 2 (û 2,h, E h 2,Γ η h) η h X Γ,h. Problem (3.12) is therefore reduced to finding (3.18) λ h X Γ,h : (S 1,h + S 2,h )λ h, η h h = Φ h, η h h η h X Γ,h. We will see that the operators S j,h are continuous and coercive in X Γ,h ; hence, for solving (3.18) we can apply the Richardson method with one of these operators (say, S 2,h ) as a preconditioner. In other words, given λ 0 h X Γ,h, foreachm 0solve (3.19) λ m+1 h = λ m h + θs 1 2,h [Φ h (S 1,h + S 2,h )λ m h ] =(1 θ)λ m h +θs 1 2,h (Φ h S 1,h λ m h ). By proceeding in a standard way (see, for instance, Alonso and Valli [3], Section 5, for a similar computation), it can be seen that (3.19) is equivalent to the following iteration-by-subdomain algorithm: being given λ 0 h X Γ,h,foreachm 0solve u m+1 1,h V 1,h : (3.20) a 1 (u m+1 1,h, v 1,h) =L 1 (v 1,h ) v 1,h V 0 1,h (n Γ u m+1 1,h ) Γ = λ m h,

9 (3.21) and pose A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 615 u m+1 2,h V 2,h : a 2 (u m+1 2,h, v 2,h) =L 2 (v 2,h )+L 1 (E h 1,Γ(n Γ v 2,h ) Γ ) a 1 (u m+1 1,h, Eh 1,Γ (n Γ v 2,h ) Γ ) v 2,h V 2,h, (3.22) The convergence of the sequence λ m h the following abstract theorem. λ m+1 h := (1 θ)λ m h + θ(n Γ u m+1 2,h ) Γ. constructed in (3.19) is a consequence of Theorem 3.1. Let X be a complex Hilbert space and let S 1,h and S 2,h be two linear operators from a finite dimensional space X h X into its dual X h.letχ s, s=1,..., M h,abasisofx h. Define the matrices S j,h associated to the operators S j,h as (S j,h γ, η) h := S j,h γ h, η h h γ, η C M h, j =1,2, where (, ) h denotes the euclidean scalar product in C M h and M h M h (3.23) γ h := γ s χ s, η h := η s χ s. s=1 Let us assume that there exist two constants C 1 > 0 and C 2 > 0, independent of h, such that (3.24) S 1,h γ h, η h h C 1 γ h X η h X γ h, η h X h, s=1 (3.25) S 2,h γ h, γ h h C 2 γ h 2 X γ h X h, Re S 1,h γ h, γ h h Re S 2,h γ h, γ h h (3.26) +Im S 1,h γ h, γ h h, Im S 2,h γ h, γ h h 0 γ h X h. Then each eigenvalue ν s of S 1 2,h (S 1,h + S 2,h ) satisfies where C 2 Re ν s ν s 2 s =1,..., M h, ( C := min 1, Therefore, for any θ (0,C ) one has 2C 2 2 C C2 2 ). ν s 2 < 2 θ Re ν s s =1,..., M h, and the preconditioned Richardson iterations converge with a rate independent of h. Proof. The proof is similar to that of Theorem 7.2 in [3]. However, for the reader s convenience we will give it in complete detail. If ν is an eigenvalue of S 1 2,h (S 1,h +S 2,h )=I+S 1 2,h S 1,h, we can write ν =1+κ, where κ is an eigenvalue of S 1 2,h S 1,h. The corresponding eigenvector γ C M h, γ 0, satisfiess 1,h γ = κs 2,h γ; therefore, S 1,h γ h, γ h h = κ S 2,h γ h, γ h h, where γ h X h is the function defined in (3.23).

10 616 ANA ALONSO AND ALBERTO VALLI Writing κ = κ 1 + iκ 2, κ 1,κ 2 R,wehave Re S 1,h γ h, γ h h = κ 1 Re S 2,h γ h, γ h h κ 2 Im S 2,h γ h, γ h h, Im S 1,h γ h, γ h h = κ 1 Im S 2,h γ h, γ h h + κ 2 Re S 2,h γ h, γ h h. Multiply now these equations by Re S 2,h γ h, γ h h and Im S 2,h γ h, γ h h, respectively: by adding the results we find κ 1 [(Re S 2,h γ h, γ h h ) 2 +(Im S 2,h γ h, γ h h ) 2 ] =Re S 1,h γ h, γ h h Re S 2,h γ h, γ h h +Im S 1,h γ h, γ h h Im S 2,h γ h, γ h h. From (3.25) we have that S 2,h γ h, γ h h 0; therefore, (3.26) yields κ 1 0. On the other hand, from (3.24), (3.25) it follows κ 2 = S 1,hγ h, γ h h 2 ( ) 2 S 2,h γ h, γ h h 2 C1 ; C 2 therefore, 2 Re ν ν 2 =2 1+κ 1 1+2κ 1 + κ κ 1 1+2κ 1 +( C1 C 2 ) 2. Notice now that the function 1+ξ F (ξ) :=2 1+2ξ+( C1 C 2 ) 2 is strictly increasing when C 1 >C 2, strictly decreasing when C 1 <C 2 and constantly equal to 1 when C 1 = C 2. Moreover, F (0) = 2C2 2 C1 2 +, lim F (ξ) =1; C2 2 ξ hence, C is the infimum of F for ξ 0. The proof of the convergence of the iterations (3.19) reduces now to verify that the operators S 1,h and S 2,h satisfy the assumptions of Theorem 3.1, i.e., (3.24) (3.26). Noting that Re S j,h γ h, γ h h = µ 1 rot E h j,γ γ h 2, Ω j Im S j,h γ h, γ h h = α σ E h j,γ γ h 2, Ω j estimate (3.26) is trivially satisfied. By using the coerciveness of a 2 (, ) and the following tangential trace inequality (see Alonso and Valli [2]) we have (n Γ v) Γ 2 X Γ C v 2 H(rot;Ω 2) S 2,h γ h, γ h h C( E h 2,Γγ h 2 0,Ω 2 + rot E h 2,Γγ h 2 0,Ω 2 ) C C γ h 2 X Γ ; hence, (3.25) holds.

11 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 617 The proof of (3.24) is more difficult. The crucial point is the proof of the continuity of the extension operator E h 1,Γ uniformly with respect to h. Let us start by introducing for each δ>0thespace X Ω:= δ {η (H δ ( Ω)) 3 (η n) Ω =0anddiv τ η H δ ( Ω)}, and for each r>0thespace H r (rot; Ω) := {v (H r (Ω)) 3 rot v (H r (Ω)) 3 }, endowed with the following norms, respectively: η X δ Ω := ( η 2 δ, Ω + div τ η 2 1/2 δ, Ω), v Hr (rot ;Ω) := ( v 2 r,ω + rot 1/2 r,ω) v 2. Let us denote by F 1,Γ : X Γ V 1 the extension operator which at each γ X Γ associates F 1,Γ γ such that F 1,Γ γ V 1 : ((F 1,Γ γ, v 1 )) Ω1 =0 v 1 H 0 (rot; Ω 1 ) (n Γ F 1,Γ γ) Γ = γ, where ((w 1, v 1 )) Ω1 := (rot w 1 rot v 1 + w 1 v 1 ). Ω 1 The existence of such an operator is guaranteed provided that we can characterize X Γ as the space of tangential traces on Γ of V 1. In that case, as a consequence of the closed graph theorem, it easily follows that F 1,Γ is a continuous operator, i.e., (3.27) F 1,Γ γ H(rot ;Ω1) C 0 γ XΓ γ X Γ. The needed characterization result on X Γ was proved in [2], under the assumption that Γ is either a C 1,1 surface or a convex polyhedral portion of Ω 1. Finally, introduce the extension operator F h 1,Γ : X h,γ V 1,h, which is the finite dimensional counterpart of F 1,Γ : F h 1,Γ γ h V 1,h : ((F h 1,Γ γ h, v 1,h )) Ω1 =0 v 1,h V1,h 0 (n Γ F h 1,Γ γ h) Γ = γ h. We need the following regularity result, which is Proposition 3.7 in Amrouche, Bernardi, Dauge and Girault [5]. Let us set X T := H(rot; Ω) H 0 (div; Ω), X N := H 0 (rot; Ω) H(div; Ω), both endowed with the norm v 0,Ω + div v 0,Ω + rot v 0,Ω. Theorem 3.2. Let Ω be a Lipschitz polyhedron. Then there exists s Ω (1/2, 1) such that the spaces X T and X N are both continuously imbedded in (H sω (Ω)) 3.

12 618 ANA ALONSO AND ALBERTO VALLI We notice that s Ω only depends on the geometry of Ω. It is related to the exponent of maximal regularity of the solutions to the Laplace operator with L 2 (Ω) on the right-hand side and homogeneous Dirichlet or Neumann boundary datum (see Amrouche, Bernardi, Dauge and Girault [5], Remark 3.8). The proof of (3.24) is based on the following three theorems, which will be proven in the Sections 4, 5 and 6. From now on the subdomain Ω 1 is always assumed to be a Lipschitz polyhedron. Finally, set κ Ω := s Ω 1/2, where s Ω is as in Theorem 3.2. Theorem A. Assume that Γ is a convex portion of Ω 1. Given δ (0,κ Ω1 ], there exists 1 > 0 such that for all γ X Γ with γ X Ω δ 1 one has F 1,Γ γ H 1/2+δ (rot; Ω 1 ) and F 1,Γ γ 1 H +δ 2 (rot ;Ω 1) 1 γ X δ Ω1. Here, as usual, γ denotes the extension of γ by 0 on Ω 1 \ Γ. Theorem B. Let T h be a regular family of triangulations. Assume that γ h X Γ,h and that F 1,Γ γ h H r (rot; Ω 1 ) for a certain r (1/2, 1). Then there exists a constant 2 > 0, independent of h, suchthat F 1,Γ γ h F h 1,Γ γ h H(rot ;Ω 1) 2 h r F 1,Γ γ h Hr (rot ;Ω 1) γ h X Γ,h. Theorem C. Let M h be the family of triangulations of Ω 1 induced by T h. Assume that M h is quasi-uniform. Then for each ɛ (0, 1/2) there exists a constant 3 > 0, independent of h, suchthat γ h X ε Ω1 3 h 1 2 ε γ h XΓ γ h X Γ,h. Once we have established these results, we are in a condition to prove the following Proposition 3.3. Assume that Γ is convex portion of Ω 1 and that M h is a quasiuniform family of triangulations of Ω 1. Then there exists a constant 4 > 0, independent of h, such that S 1,h γ h, η h h 4 γ h XΓ η h XΓ γ h, η h X Γ,h, which is estimate (3.24). Proof. From the definition of the Steklov-Poincaré operator S 1,h we have S 1,h γ h, η h h = a 1 (E h 1,Γ γ h, E h 1,Γ η h) β 1 E h 1,Γ γ h H(rot ;Ω1) E h 1,Γ η h H(rot ;Ω1), where β 1 > 0 is the continuity constant of a 1 (, ). Therefore, the proof is complete if we show that there exists a constant C>0, independent of h, such that E h 1,Γ γ h H(rot ;Ω 1) C γ h XΓ γ h X Γ,h. Taking in (3.13) the test function v 1,h = E h 1,Γ γ h F h 1,Γ γ h we have Hence a 1 (E h 1,Γ γ h, Eh 1,Γ γ h )=a 1(E h 1,Γ γ h,fh 1,Γ γ h ). E h 1,Γ γ h H(rot ;Ω 1) β 1 (3.28) F h 1,Γ α γ h H(rot ;Ω 1), 1 where α 1 is the coerciveness constant of the bilinear form a 1 (, ).

13 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 619 Moreover (3.29) F h 1,Γγ h H(rot ;Ω1) F h 1,Γγ h F 1,Γ γ h H(rot ;Ω1) + F 1,Γ γ h H(rot ;Ω1); therefore, from (3.27) we only have to estimate the first term in (3.29). At first, remark that γ h X Γ,h yields γ h X Ω δ 1 for each δ (0, 1/2), as both γ h and div τ γ h are (discontinuous) piecewise-polynomial. Therefore, from Theorem A we have that F 1,Γ γ h H 1/2+δ (rot; Ω 1 ) for any δ (0,κ Ω1 ]. Using also Theorem B (where r =1/2+δ)wehave F h 1,Γγ h F 1,Γ γ h H(rot ;Ω1) 2 h 1 2 +δ F 1,Γ γ h 1 H +δ 2 (rot ;Ω 1) 1 2 h 1 2 +δ γ h X δ Ω1. Now we can apply Theorem C (for ɛ = δ) and we find (3.30) F h 1,Γ γ h F 1,Γ γ h H(rot ;Ω1) γ h XΓ. The proof follows from (3.27) (3.30). Remark 3.4. In Theorem A and in Proposition 3.3 the assumption on Γ is only needed to assure that X Γ is the space of tangential traces on Γ of V Proof of Theorem A Let us introduce the finite dimensional spaces H(e) :={ω (L 2 (Ω)) 3 rot ω = 0, div ω =0, (n ω) Ω =0}, H(m):={ϱ (L 2 (Ω)) 3 rot ϱ = 0, div ϱ =0, (ϱ n) Ω =0}. We start recalling the following theorems, whose proof can be essentially found in Saranen [16] (see also Valli [17]). Theorem 4.1. Let Ω R 3 be a bounded domain with Lipschitz boundary Ω. Each function w (L 2 (Ω)) 3 can be written as n (4.1) w = rot p + q + α k ϱ k, where p satisfies rot rot p = rot w k=1 in Ω (4.2) q satisfies (4.3) div p =0 in Ω (n p) Ω = 0 on Ω (p, ω) =0 q =divw ( ) q =(w n) Ω n Ω q =0, Ω ω H(e), in Ω on Ω

14 620 ANA ALONSO AND ALBERTO VALLI the functions {ϱ k } n k=1 are an orthonormal basis of H(m), and the coefficients α k are given by α k =(w,ϱ k ),k=1,..., n. Theorem 4.2. Let Ω R 3 be a bounded domain with Lipschitz boundary Ω. Then there exists a constant C Ω > 0 such that w 0,Ω C Ω ( rot w 0,Ω + div w 0,Ω ) w X T H(m). We are now in a position to prove some auxiliary results, which are interesting on their own as they are regularity results for harmonic fields. In the particular case in which the parameter δ is equal to 0, Costabel [6] proved the same regularity result for a simply connected Lipschitz domain with connected boundary. Theorem 4.3 (Regularity for Dirichlet harmonic fields). Let Ω be a Lipschitz polyhedron. Then for each δ (0, 1/2) the space W := {w H(rot; Ω) H(div; Ω) (n w) Ω (H δ ( Ω)) 3 } is continuously imbedded in (H 1/2+ɛ (Ω)) 3,whereɛ := min(δ, κ Ω ). Proof. From Theorem 4.1 each function w W can be written as n (4.4) w = rot p + q + α k ϱ k. Since rot rot p = rot w (L 2 (Ω)) 3,and(n p) Ω =0yields (rot p n) Ω = div τ (n p) Ω = 0, we have that rot p X T. Moreover, ϱ k H(m) X T ; hence, from Theorem 3.2 we find that rot p and each ϱ k belong to (H 1/2+κΩ (Ω)) 3. From α k =(w,ϱ k ) it follows at once that n (4.5) α k ϱ k 1 2 +κω,ω C w 0,Ω. k=1 Moreover, it is easily verified that rot p H(m) ; hence, from Theorems 3.2 and 4.2 we have rot p 1 2 +κω,ω C rot p XT (4.6) C(1 + C Ω )( rot rot p 0,Ω + div rot p 0,Ω ) = C(1 + C Ω ) rot w 0,Ω. On the other hand, from q =( q n) Ω n n (n q) Ω we have τ q Ω = n (n q) Ω, and therefore ( ) n τ q Ω = n (n w) Ω (n rot p) Ω (n α k ϱ k ) Ω. The unit normal vector n is piecewise constant, as Ω is a polyhedron; hence, τ q Ω (H ɛ (Ω)) 3. From q H 1 (Ω) we also have that q Ω H 1/2 ( Ω) L 2 ( Ω); therefore, we conclude that q Ω H 1+ɛ ( Ω) k=1 k=1

15 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 621 and we find the estimate q Ω 1+ε, Ω C( q 0, Ω + τ q Ω ε, Ω) C( q 1,Ω + τ q Ω ε, Ω) C( w 0,Ω + (n w) Ω δ, Ω + rot p 1 2 +κω,ω + n k=1 α k ϱ k 1 2 +κω,ω ). Since q =divw L 2 (Ω), from the regularity results for the Dirichlet boundary value problem for the Laplace operator it follows that q H 3/2+ɛ (Ω) (see Dauge [7], Corollary 18.13), and (4.7) q 1 2 +ε,ω q 3 2 +ε,ω C( div w 0,Ω + q Ω 1+ε, Ω). From the representation formula (4.1) we finally have w (H 3/2+ɛ (Ω)) 3 and from (4.5) (4.7) (4.8) w 1 2 +ε,ω rot p 1 2 +κω,ω + q 1 2 +ε,ω + which concludes the proof. n k=1 α k ϱ k 1 2 +κω,ω C( w 0,Ω + rot w 0,Ω + div w 0,Ω + (n w) Ω δ, Ω ), A similar result is the following Theorem 4.4 (Regularity for Neumann harmonic fields). Let Ω be a Lipschitz polyhedron. Then for each δ (0, 1/2) the space V := {w H(rot; Ω) H(div; Ω) (n w) Ω H δ ( Ω)} is continuously imbedded in (H 1/2+ɛ (Ω)) 3,whereɛ := min(δ, κ Ω ). Proof. As in the proof of Theorem 4.3, we use the representation formula (4.1). The first and the third term can be treated as done there; hence, we have only to check the regularity of q. It is the solution of a Neumann boundary value problem for the Laplace operator with L 2 (Ω) right-hand side, and H δ ( Ω) Neumann datum. As a consequence of Corollary 23.5 in Dauge [7] we have that q H 3/2+ɛ (Ω) and q 1 2 +ε,ω C( div w 0,Ω + (n w) Ω δ, Ω ). Using (4.5), (4.6) we finally have (4.9) n w 1 2 +ε,ω rot p 1 2 +κω,ω + q 1 2 +ε,ω + α k ϱ k 1 2 +κω,ω and the proof is concluded. k=1 C( w 0,Ω + rot w 0,Ω + div w 0,Ω + (n w) Ω δ, Ω ) We are now in a position to give the proof of Theorem A. Proof of Theorem A. F 1,Γ γ H(rot; Ω 1 ) satisfies rot rot F 1,Γ γ + F 1,Γ γ = 0 in Ω 1 ; therefore, div F 1,Γ γ = 0 in Ω 1. We can apply Theorem 4.3 and we find that F 1,Γ γ H 1/2+δ (Ω 1 )and F 1,Γ γ 1 C( F 1,Γγ 2 +δ,ω1 H(rot ;Ω1) + γ δ, Ω1 ).

16 622 ANA ALONSO AND ALBERTO VALLI Assuming that Γ is convex portion of Ω 1, from (3.27) we finally have (4.10) F 1,Γ γ 1 C( γ 2 +δ,ω1 δ, Ω 1 + div τ γ 1/2, Ω1 ). Let us denote by G 1,Γ γ := rot F 1,Γ γ. We first notice that (G 1,Γ γ n) Ω1 = div τ γ H δ ( Ω 1 ) and that rot G 1,Γ γ = F 1,Γ γ (L 2 (Ω 1 )) 3. We can apply Theorem 4.4 and we find that G 1,Γ γ H 1/2+δ (Ω 1 )and G 1,Γ γ 1 C( G 1,Γγ 2 +δ,ω1 H(rot ;Ω1) + div τ γ δ, Ω1 ). On the other hand, it is at once verified that G 1,Γ γ H(rot ;Ω1) C F 1,Γ γ H(rot ;Ω1); hence, from (3.27) (4.11) rot F 1,Γ γ 1 C( γ 2 +δ,ω1 1/2,Γ + div τ γ δ, Ω1 ). From (4.10), (4.11) we have (4.12) F 1,Γ γ 1 H +δ 2 (rot ;Ω 1) 1 γ X δ Ω1, which concludes the proof. 5. Proof of Theorem B The proof of Theorem B is based on the estimate for the interpolation error. Let us recall that the finite elements we are going to employ are the Nédélec finite element of first type (however, as we already noticed, the same results hold also for the Nédélec finite element of second type introduced in [13]). They are defined for k 1as (5.1) Nh k := {v h H(rot; Ω) v h R k T h }, where R k := (P k 1 ) 3 S k, S k := {p (P k) 3 p(x) x =0} and P k is the space of homogeneous polynomials of degree k. The degrees of freedom of Nh k are given by { } (5.2) m 1 (v) := v t a qfor all q P k 1 (a) for the six edges a of, a where t a is a unit vector having the same direction as the edge a; whenk 2one has to add (5.3) { } m 2 (v) := (v n) qfor all q (P k 2 (f)) 2 for the four faces f of ; f and finally for k 3 one has to take also { } (5.4) m 3 (v) := v qfor all q (P k 3 ()) 3. Nédélec [12] has proven that these degrees of freedom are curl-conforming and determine a unique element of R k. Let us denote by Π k h the interpolation operator valued in Nh k.

17 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 623 Lemma 5.1. The interpolation operator Π k h is defined in Hr (rot; Ω) for any r> 1/2. Proof. It is only necessary to see that the moments introduced in (5.2) (5.4) are well defined. If v H r (rot; Ω) for r>1/2, then in particular v (H r ()) 3 and from the trace theorem v f (L 2 (f)) 3 ; therefore, the moments m 3 (v) andm 2 (v) are defined. Concerning the moments m 1 (v), let a be one of the edges of a face f. Denote as usual by n the unit outward normal vector on and by ν the unit vector contained in the plane identified by f, pointing outward f and normal to a. The unit vector t a can be written as t a = n ν. Therefore we have (5.5) v t a q = v (n ν) q = (v n) ν q. a a From the assumption on v we know v f n (H r 1/2 (f)) 3 and div τ (v f n) = (rot v) f n H r 1/2 (f); hence, in particular v f n (L p (f)) 3 for a suitable p>2 and div τ (v f n) L 2 (f). This easily yields ((v f n) ν) f W 1/p,p ( f) and then the moments m 1 (v) are defined by means of a duality argument. Now we want to prove that (5.6) v Π k hv H(rot ;Ω) Ch min(r,k) v H r (rot ;Ω) v H r (rot; Ω), where, as before, r>1/2andk 1. This result is already known when r k, as for any v H k (rot; Ω) Nédélec [12] has proven (5.7) v Π k h v 0,Ω Ch k ( v k,ω + rot v k,ω ), and Monk [11] has obtained (5.8) rot(v Π k hv) 0,Ω Ch k rot v k,ω. By following their proofs, it is an easy matter to verify that (5.6) holds also for a positive integer r, r k 1. Therefore, we are left with the proof of (5.6) in the case of a non-integer r, 1/2<r<k. The reference tetrahedron ˆ is the one with vertices P 0 := (0, 0, 0), P 1 := (1, 0, 0), P 2 := (0, 1, 0) and P 3 := (0, 0, 1), and each tetrahedron T h can be obtained from ˆ by means of an invertible affine map F (ˆx) =B ˆx+b. Let us denote the local interpolation operator by Π k. The relation Πk (v ) = (Π k h v) clearly holds. Moreover, as in Nédélec [12], consider the map ˆv = B T v F, which easily yields (Π k v) =Π kˆˆv. Finally, for the sake of convenience we introduce the matrix v 0 2 x 1 v1 v 3 x 2 x 1 v1 x 3 Rot v := v 1 x 2 v2 v x x 2 v2 x 3, v 1 x 3 v3 v 2 x 1 x 3 v3 x 2 0 a

18 624 ANA ALONSO AND ALBERTO VALLI in such a way that We have Rot v(x) =(B T ) 1 Rot ˆv(F 1 (x)) B 1 Lemma 5.2. Let T h be a regular family of triangulations. Then there exists a constant C>0such that v 2 0, Ch ˆv 2 0, ˆ rot v 2 0, Ch 1 rot ˆv 2 0, ˆ for each v H(rot; ) and T h. Proof. The procedure is classic, and we report it here for the sake of completeness. Firstly we have v 2 0, = v(x) 2 dx = (B T 1 ) 1ˆv(F (x)) 2 dx = det B (B T ) 1ˆv(ˆx) 2 dˆx ˆ det B (B T ) 1 2 ˆv 2 0, ˆ. Analogously, rot v 2 0, = 1 Rot v(x) 2 dx = 1 (B T 2 2 ) 1 Rot ˆv(F 1 (x)) B 1 2 dx = 1 2 det B (B T ) 1 Rot ˆv(ˆx) B 1 2 dˆx ˆ C det B (B T ) 1 2 B 1 2 rot ˆv 2 0, ˆ. The proof then follows by noticing that B 1 Ch 1 and det B Ch 3.. Using this Lemma we find (5.9) v Π k v 2 H(rot ;) C(h ˆv Π kˆ ˆv 2 + h 1 0, ˆ rot(ˆv Πkˆ ˆv) 2 ). 0, ˆ Now we want to write the term rot(π kˆ ˆv) in an equivalent form. To start with, let us consider the case k 2. As in Nédélec [13], for l 1 introduce the finite element space Mh l := {v h H(div; Ω) v h (P l ) 3 T h }, with the moments { } m 1 (v) := (v n)qfor all q P l (f) for the four faces f of, f at which one has to add, in the case l 2, also { } m 2 (v) := v qfor all q q R l 1 (). These moments are div-conforming and determine a unique element of (P l ) 3.We will denote by πh l the interpolation operator related to M h l, which is clearly well defined and continuous in (H r (Ω)) 3, r>1/2, and by π l the local interpolation operator. Again, we have π l (v ) =(πh lv). The following lemma was proved by Nédélec in [13], Proposition 2.

19 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 625 Lemma 5.3. For each function ˆv H r (rot; ˆ), r>1/2, and for each k 2 the following relation rot(π kˆ ˆv) =π k 1 (rot ˆv) ˆ holds. Notice that the proof in [13] refers to the curl-conforming Nédélec elements of second type; however, the curl of the interpolant is the same for both types of Nédélec elements. Notice also that (though not explicitly underlined) in [13], Proposition 2, it is assumed that k 1 1, i.e., k 2. Finally, there the result is stated for ˆv H 2 ( ˆ), but this assumption can be weakened, as a consequence of Lemma 5.1. Consider now the case k =1. Wedenotebyf i thefaceof ˆorthogonal to the axis x i, i =1,2,3. The following operator π 0ˆ :(H r (ˆ)) 3 R 3, π 0ˆ ˆv = 2 f 1 ˆv n f 2 ˆv n f 3 ˆv n, is clearly well defined and continuous for each r>1/2. Moreover Lemma 5.4. For each function ˆv H r (rot; ˆ), r>1/2, the following relation holds. rot(π 1ˆ ˆv) =π 0ˆ(rot ˆv) Proof. As k = 1 we only have to deal with the moments of the first type ˆm 1 (ˆv) = aˆv t a. Denote by ˆm s (ˆv) the degree of freedom on ˆ associated to the edge a s, s =1,..., 6, where a 1 = P 0 P 1, a 2 = P 0 P 2, a 3 = P 0 P 3, a 4 = P 3 P 2, a 5 = P 1 P 3 and a 6 = P 2 P 1. It can be easily shown that the basis ˆψ s of R 1 on ˆ satisfying ˆm s ( ˆψ j )=δ sj is given by ˆψ 1 = 1 ŷ ẑ ˆx ˆx ˆψ 4 = Let us notice that rot ˆψ 1 =, ˆψ2 = 0 ẑ ŷ ŷ 1 ˆx ẑ ŷ, ˆψ3 =, ˆψ5 = ẑ 0, ˆψ6 = ˆx ŷ ˆx 0 ẑ ẑ 1 ˆx ŷ., rot ˆψ 2 = 2 0 2, rot ˆψ 3 = 2 2 0, rot ˆψ 4 = 2 0, rot ˆψ 5 = , rot ˆψ 6 = 0 0 2,,

20 626 ANA ALONSO AND ALBERTO VALLI and therefore ˆm 2 (ˆv) ˆm 3 (ˆv) ˆm 4 (ˆv) rot(π 1ˆ ˆv) =2 ˆm 1 (ˆv)+ ˆm 3 (ˆv) ˆm 5 (ˆv). ˆm 1 (ˆv) ˆm 2 (ˆv) ˆm 6 (ˆv) Taking the unit vector t = n ν as in Lemma 5.1, it follows that f 1 ˆv t 1 f 1 rot ˆv n rot(π 1ˆ ˆv) = 2 f 2 ˆv t 2 = 2 f 2 rot ˆv n = π0ˆ(rot ˆv), f 3 ˆv t 3 f 3 rot ˆv n having used the Stokes Theorem on each face f i. We are now in a position to consider the operators I Π kˆ : H r (rot, ˆ) (L 2 ( ˆ)) 3, I π k 1 ˆ :(Hr (ˆ)) 3 (L 2 ( ˆ)) 3, which are clearly linear and continuous. Since we are dealing with a non-integer r, with 1/2 <r<k, it follows that the integral part [r] ofrsatisfies [r] k 1. Hence, the operators above take value zero for each polynomial in P [r], and applying the Bramble-Hilbert Lemma we find (5.10) (I Π kˆ)ˆv 2 0, ˆ C inf ˆv + ˆp (P ˆp 2 [r] ) 3 H r (rot ; ˆ), (5.11) (I π k 1 ˆ ) rot ˆv 2 C inf 0, ˆ rot ˆv + ˆp (P [r] ) ˆp 2 3 r, ˆ. By repeating the proof of the Deny-Lions Lemma we finally have (5.12) inf ˆv + ˆp (P ˆp 2 [r] ) 3 H r (rot ; ˆ) C( ˆv 2 r, ˆ + rot ˆv 2 [r], ˆ + rot ˆv 2 ), r, ˆ (5.13) inf ˆp (P [r] ) 3 rot ˆv + ˆp 2 r, ˆ C rot ˆv 2 r, ˆ. We recall that for an integer k the semi-norm in (H k (Ω)) 3 is defined as ˆv k, ˆ := 1/2 D αˆv 2 0, ˆ, α =k whereas for a non-integer value s it holds ˆv s, ˆ := D αˆv 2 s [s], ˆ where for θ (0, 1) we have set ( ˆv θ, ˆ := ˆ α =[s] ˆ 1/2 ˆv(ˆx) ˆv(ŷ) 2 ) 1/2 ˆx ŷ 3+2θ dˆxdŷ.,

21 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS 627 From (5.9) (5.13) we have thus obtained v Π k v H(rot ;) C(h ˆv Π kˆ ˆv 2 + h 1 0, ˆ rot ˆv πk 1 [ ˆ (rot ˆv) 2 0, ˆ ) ] C h ( ˆv 2 r, ˆ + rot ˆv 2 [r], ˆ + rot ˆv 2 )+h 1 r, ˆ rot ˆv 2 r, ˆ C We need the following result. [ h ( ˆv 2 r, ˆ + rot ˆv 2 )+h 1 [r], ˆ rot ˆv 2 r, ˆ Lemma 5.5. Let T h be a regular family of triangulations. Then there exists a constant C>0such that ˆv 2 Ch 1+2s s, ˆ v 2 s,, for each real number s 0. rot ˆv 2 s, ˆ Ch1+2s rot v 2 s, Proof. We will present the proof only for 0 <s<1, the other cases being standard. We have ˆv 2 s, ˆ = ˆv(ˆx) ˆv(ŷ) 2 ˆ ˆ ˆx ŷ 3+2s dˆxdŷ = det B 2 B T (v(x) v(y)) 2 B 1 (x dxdy. y) 3+2s We can write x y = B B 1 (x y) B B 1 (x y) ; therefore, B 1 (x y) B 1 x y. Hence, ˆv 2 s, ˆ det B 2 B 3+2s B T (v(x) v(y)) 2 x y 3+2s dxdy C det B 2 B 3+2s B T 2 v 2 s, Ch 1+2s v 2 s,. In an analogous way rot ˆv 2 s, ˆ = 1 Rot ˆv(ˆx) Rot ˆv(ŷ) 2 2 ˆ ˆ ˆx ŷ 3+2s dˆxdŷ = 1 2 det B 2 B T (Rot v(x) Rot v(y)) B 2 B 1 dxdy (x y) 3+2s C det B 2 B 5+2s B T 2 rot v 2 s, Ch 1+2s rot v 2 s,, which concludes the proof. We can conclude with the following interpolation result. ]. Proposition 5.6. Let r be a non-integer with 1/2 <r<k. Let T h be a regular family of triangulations. Then there exists a constant C > 0, independent of h, such that (5.14) v Π k h v H(rot ;Ω) Ch r v H r (rot ;Ω) for each v H r (rot; Ω).

22 628 ANA ALONSO AND ALBERTO VALLI Proof. Using the additivity of the integral we can write v Π k h v 2 H(rot ;Ω) = ( v Π k v 2 0, T h T h + rot v πk 1 rot v 2 0, ) [ h ( ˆv 2 r, ˆ + rot ˆv 2 )+h 1 [r], ˆ rot ˆv 2 r, ˆ From Lemma 5.5 we find v Π k hv 2 H(rot ;Ω) [ h (h 1+2r v 2 r, + h 1+2[r] rot v 2 [r], ) T h ] + h 1 h1+2r rot v 2 r, C v 2 r, + h2+2[r] rot v 2 [r], + h2r rot v 2 r, ) and the thesis is proved. T h (h 2r Ch 2r v 2 H r (rot ;Ω), The estimates (5.7), (5.8) and (5.14) yield the interpolation estimate (5.6), and we are now in a position to conclude the proof of Theorem B. Proof of Theorem B. We are going to follow the lines of the proof of Céa Lemma. We have (5.15) F 1,Γ γ h F h 1,Γγ h 2 H(rot ;Ω =((F 1) 1,Γγ h F h 1,Γγ h,f 1,Γ γ h F h 1,Γγ h )) Ω1 ]. =((F 1,Γ γ h F h 1,Γ γ h,f 1,Γ γ h Π k h F 1,Γγ h +Π k h F 1,Γγ h F h 1,Γ γ h)) Ω1. Let us take now γ h X Γ,h.Abasisφ 1,j in X Γ,h is given by (n Γ ψ 1,j ) Γ,where ψ 1,j are basis functions of V 1,h. Moreover, due to relation (5.5), for any tangential element γ (H r 1/2 (Γ)) 3 H(div τ ; Γ) the degrees of freedom related to X Γ,h are given by a γ ν q for each q P k 1(a) andby f γ qfor each q (P k 2(f)) 2, where a and f are any edge and face of Γ, respectively. Therefore, for any v 1 H r (rot; Ω 1 ) the interpolant on Γ of (n Γ v 1 ) Γ is given by (n Γ Π k h v 1) Γ ; hence, (n Γ Π k h F 1,Γγ h ) Γ = γ h. Consequently, we have (Π k h F 1,Γγ h F h 1,Γ γ h ) V 0 1,h and Hence we find ((F 1,Γ γ h F h 1,Γ γ h, Πk h F 1,Γγ h F h 1,Γ γ h )) Ω 1 =0. F 1,Γ γ h F h 1,Γ γ h H(rot ;Ω1) F 1,Γ γ h Π k h F 1,Γγ h H(rot ;Ω1). From the assumption F 1,Γ γ h H r (rot; Ω 1 ) and the interpolation inequality (5.6) we finally find F 1,Γ γ h F h 1,Γ γ h H(rot ;Ω 1) 2 h r F 1,Γ γ h Hr (rot ;Ω 1), and Theorem B is completely proved.

23 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS Proof of Theorem C We have to prove that if M h is a quasi-uniform family of triangulations of Ω, then there exists a constant C>0such that for each η h X Ω,h it holds (6.1) η h ɛ, Ω + div τ η h ɛ, Ω Ch 1 2 ɛ ( η h 1/2, Ω + div τ η h 1/2, Ω ). Noticing that both η h and div τ η h are polynomials in each triangle on Ω, one can apply the inverse inequality for the non-integer exponent ɛ obtaining (6.2) η h ɛ, Ω + div τ η h ɛ, Ω Ch ɛ ( η h 0, Ω + div τ η h 0, Ω ). It remains for us to show that for each real scalar function z h L 2 ( Ω) and such that z h T P k for each triangle T M h we have (6.3) z h 0, Ω Ch 1/2 z h 1/2, Ω. A similar result in the two-dimensional case can be found in Quarteroni, Sacchi Landriani and Valli [14]. We are going to adapt their proof to the case under consideration. Let us set M h := {v H 1 ( Ω) v T P k+3 T M h }. For each φ L 2 ( Ω) denote by φ h the L2 ( Ω)-orthogonal projection of φ onto M h. Moreover, denote by φ h the function belonging to M h whichisdefinedas (6.4) φ h T = φ h T + (φ φ h, [M(x)]α ) T P α,t, α k where M(x) is the rigid motion sending T on the (x, y)-plane with one vertex in (0, 0), and P α,t P k+1 satisfies P α,t T =0and (P α,t, [M(x)] β ) T = { 0 if α β 1 if α = β, α, β k. Clearly, the function z h T can be written as a linear combination of [M(x)] α, α k; therefore, it follows at once that Ω z hφ = Ω z hφ h for each φ L2 (Ω), and we have Ω z h 0, Ω = z hφ = φ 0, Ω sup φ L 2 ( Ω) φ 0 sup φ L 2 ( Ω) φ 0 sup φ L 2 ( Ω) φ 0 Ω z hφ h φ 0, Ω z h 1/2, Ω φ h 1/2, Ω φ 0, Ω. By using the inverse inequality as in (6.2) we obtain hence, φ h 1/2, Ω Ch 1/2 φ h 0, Ω; z h 0, Ω Ch 1/2 z h 1/2, Ω To conclude the proof, we have to show that (6.5) sup φ L 2 ( Ω) φ 0 φ h 0, Ω C φ 0, Ω. φ h 0, Ω φ 0, Ω.

24 630 ANA ALONSO AND ALBERTO VALLI We have φ h 2 0, Ω = φ h 2 0,T = φ h + (φ φ h, [M(x)]α ) T P α,t T M h T M T h α k C φ h 2 + (φ φ h, [M(x)]α ) 2 T P α,t 2 T M h T C φ h 2 0, Ω + T M h α k φ φ h 2 0,T [M(x)] 2α α k T Pα,T 2 T 2. By a straightforward computation it can be shown that max [M(x)] 2α Pα,T 2 C, T M h T T uniformly with respect to h, and (6.5) follows by recalling that φ h is the L2 ( Ω)- orthogonal projection of φ onto M h. References 1. R.A. Adams, Sobolev Spaces, Academic Press, New York, MR 56: A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H(rot;Ω) and the construction of an extension operator, Manuscr. Math. 89 (1996), MR 96k: A. Alonso and A. Valli, A domain decomposition approach for heterogeneous time-harmonic Maxwell equations, Comput. Meth. Appl. Mech. Engrg. 143 (1997), MR 98b: A. Alonso and A. Valli, Unique solvability for high-frequency heterogeneous time-harmonic Maxwell equations via the Fredholm alternative theory, Math. Meth. Appl. Sci., to appear. 5. C. Amrouche, C. Bernardi, M. Dauge and V. Girault, Vector potentials in three-dimensional nonsmooth domains, preprint R 96001, Laboratoire d Analyse Numérique, Université Pierre et Marie Curie, Paris, M. Costabel, A remark on the regularity of solutions of Maxwell s equations on Lipschitz domains, Math. Meth. Appl. Sci. 12 (1990), MR 91c: M. Dauge, Elliptic Boundary Value Problems on Corner Domains, Springer-Verlag, Berlin, MR 91a: M. řížek and P. Neittaanmäki, On time-harmonic Maxwell equations with nonhomogeneous conductivities: solvability and FE-approximation, Apl. Mat. 34 (1989), MR 90m: R. Leis, Exterior boundary-value problems in mathematical physics, in Trends in Applications of Pure Mathematics to Mechanics, Vol. 11, H. Zorski ed., Pitman, London, 1979, pp MR 81d: P. Monk, A finite element method for approximating the time-harmonic Maxwell equations, Numer. Math. 63 (1992), MR 94b: P. Monk, Analysis of a finite element method for Maxwell s equations, SIAM J. Numer. Anal. 29 (1992), MR 93k: J.C. Nédélec, Mixed finite elements in R 3, Numer. Math. 35 (1980), MR 81k: J.C. Nédélec, A new family of mixed finite elements in R 3,Numer.Math.50 (1986), MR 88e: A. Quarteroni, G. Sacchi Landriani and A. Valli, Coupling of viscous and inviscid Stokes equations via a domain decomposition method for finite elements, Numer.Math.59 (1991), MR 93b: J. Saranen, On generalized harmonic fields in domains with anisotropic nonhomogeneous media, J. Math. Anal. Appl. 88 (1982), ; Erratum: J. Math. Anal. Appl. 91 (1983), 300. MR 84d:78004a, MR 84d:78004b

25 A DOMAIN DECOMPOSITION METHOD FOR MAXWELL EQUATIONS J. Saranen, On electric and magnetic problems for vector fields in anisotropic nonhomogeneous media, J. Math. Anal. Appl. 91 (1983), MR 85i: A. Valli, Orthogonal decompositions of (L 2 (Ω)) 3, preprint UTM 493, Dipartimento di Matematica, Università di Trento, Dipartimento di Matematica, Università di Trento, Povo (Trento), Italy address: alonso@science.unitn.it address: valli@science.unitn.it