Preconditioners for higher order edge finite element discretizations of Maxwell s equations

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1 Science in Cina Series A: Matematics Aug., 2008, Vol. 51, No. 8, mat.scicina.com Preconditioners for iger order edge finite element discretizations of Maxwell s equations ZHONG LiuQiang 1,2,SHUSi 1,2, SUN DuDu 3 & TAN Lin 4 1 Scool of Matematical and Computational Sciences, Xiangtan University, Xiangtan , Cina 2 Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Xiangtan , Cina 3 Institute of Computational Matematics and Scientific/Engineering Computing, Academy of Matematics and Systems Science, Graduate University of Cinese Academy of Sciences, Cinese Academy Sciences, P.O. Box 2719, Beijing , Cina 4 Department of Mat-Pysics, Nanua University, Hengyang , Cina ( zonglq@xtu.edu.cn, susi@xtu.edu.cn, sundudu@lsec.cc.ac.cn, tanlinboy@yaoo.com.cn) Abstract In tis paper, we are concerned wit te fast solvers for iger order edge finite element discretizations of Maxwell s equations. We present te preconditioners for te first family and second family of iger order Nédélec element equations, respectively. By combining te stable decompositions of two kinds of edge finite element spaces wit te abstract teory of auxiliary space preconditioning, we prove tat te corresponding condition numbers of our preconditioners are uniformly bounded on quasi-uniform grids. We also present some numerical experiments to demonstrate te teoretical results. Keywords: preconditioner, iger order edge finite element, stable decomposition MSC(2000): 65F10, 65N22 1 Introduction Nédélec edge finite elements in [1, 2] are most popular coices for discretizations of Maxwell s equations, but te resulting discrete systems are usually large and iger ill-condition, ence constructing te fast algoritms for te corresponding numerical solutions is necessary for realistic computational electromagnetism. Some literature concerning te fast solvers of te discrete Maxwell s variational problems can be found in [3 9]. Exploiting available efficient MG metods on auxiliary mes for te same bilinear form leading efficient auxiliary mes preconditioners to unstructured problems was sown in [6]. Bot te teory of auxiliary space preconditioning and te preconditioners of te first family were constructed in [7]. Te preconditioners of te second family of Nédélec edge finite element equations were presented in [8]. However, te researces mentioned above deal wit te discrete systems resulting from te low-order Nédélec edge finite elements, wereas, te ig-order edge finite element metods are superior and necessary under certain conditions over te low-order ones, for example, it can reduce te Received Marc 7, 2008; accepted May 8, 2008 DOI: /s Corresponding autor Tis work was partially supported by te National Natural Science Foundation of Cina (Grant Nos , ), te National Key Basic Researc Program of Cina (973 Program) (Grant No. 2005CB321702), te Key Project of Cinese Ministry of Education and Scientific Researc Fund of Hunan Provincial Education Department (Grant Nos , 07A068)

2 1538 ZHONG LiuQiang et al. numerical dispersion error, moreover, te construction of ig order finite element metods is a very active area nowadays in computational electromagnetism. Te construction of te base functions of H(curl)-conforming spaces on tetraedral can be found in [10]. In tis paper, we are concerned wit te preconditioned conjugate gradient (PCG) metods for iger order edge finite element equations, te key to wic is ow to construct ig efficient preconditioners. We will design and analyze two kinds of preconditioners for two families of te iger order edge finite element equations, respectively. Te main idea of te construction of preconditioners is to recursively translate te computation of te preconditioners for te iger order edge finite element equations into te one for oter edge or H 1 -conforming finite element equations wic possess less degrees of freedom witout costing too muc computational efforts. For te preconditioner for k +1 ordernédélec element equations of first kind, by using a Jacobi smooting (or Gauss-Seild smooting), we recursively translate te construction of its preconditioner into te one of te k order Nédélec element equations of second kind. For te preconditioner for k order Nédélec element equations of second kind, by solving an H 1 - conforming k + 1 order Lagrange finite element equations, we can translate it into te one for k order Nédélec element equations of first kind. Tere ave existed many works about teeffectivesolversforteh 1 -conforming ig order Lagrange finite element equations, for example, see [11], ence, we essentially translate te computation of iger order edge finite element equations into te one of te linear edge element equations of first kind by tis recursive metod. Furtermore, by using te abstract teory in [7], we prove tat te above condition numbers are uniformly bounded on quasi-uniform meses. We also present some numerical experiments to demonstrate te teoretical results. Te rest of te paper is organized as follows. In te next section, we introduce two kinds of ig order edge finite element equations, and present te corresponding frame of constructing preconditioner. We construct te preconditioners for two kinds of ig order Nédélec element equations, and prove tat teir corresponding condition number is uniformly bounded in Sections 3 and 4, respectively. We also present some numerical experiments in Section 5. 2 Edge finite element equations and preconditioner Let Ω be a simply connected polyedron in R 3 wit boundary Γ and unit outward normal ν 1). We now define some Sobolev functional spaces (see [12]): H0 1 (Ω) = {q L 2 (Ω) q (L 2 (Ω)) 3,u=0onΓ}, (2.1) H 0 (curl;ω)={u (L 2 (Ω)) 3 u (L 2 (Ω)) 3, ν u = 0 on Γ}, (2.2) H 0 (div; Ω) = {u (L 2 (Ω)) 3 u L 2 (Ω), ν u =0onΓ}. (2.3) For all u H 0 (curl;ω) and u H 0 (div; Ω), we define te norms u H(curl;Ω) = ( u /2 0) u 2, u H(div;Ω) = ( u /2 0) u 2, were 0 denotes te norm in (L 2 (Ω)) 3 or L 2 (Ω). In tis paper, we consider te following variational problem: Find u H 0 (curl;ω) suc tat a(u, v) =(f, v), v H 0 (curl;ω), (2.4) 1) In tis paper, vectors are distinguised from scalars by te use of bold type.

3 Preconditioners for iger order edge finite element discretizations of Maxwell s equations 1539 were f (L 2 (Ω)) 3 is a given data and a(u, v) = [( u) ( v)+τu v] dx, (f, v) = f vdx, (2.5) Ω Ω wit te constant τ>0. Te bilinear form a(, ) induces te energy norm v 2 A = a(v, v), v H 0(curl;Ω). (2.6) Variational problem of te form (2.4) arises in many simulations of electromagnetic fields, for instance, it can describe te eddy current model, and it is also a core task in te time-domain simulation of electrimagnetic fields if implicit timestepping is employed (see [3, 13]). Let T be a quasi-uniform tetraedron mes of Ω (see Figures 1 and 2), were is te maximum diameter of te tetraedra in T. Figure 1 Structured grids Figure 2 Unstructured grids For eac k>0, define polynomial spaces P k = {polynomials of maximum total degree k in x 1,x 2,x 3 }, P k = {omogeneous polynomials of maximum total degree exactly k in x 1,x 2,x 3 }. Now, we present te following two families of edge finite element spaces (see [12 14]): (i) k order Nédélec element of first kind V k,1 = {u k,1 H 0 (curl;ω) u k,1 K R k for all K T }, (2.7) were R k =(P k 1 ) 3 {p ( P k ) 3 p(x) x =0}. (ii) k order Nédélec element of second kind u k,l V k,2 = {u k,2 H 0(curl;Ω) u k,2 K (P k ) 3 for all K T }. (2.8) We are interested in te following finite element equations of variational problem (2.4): Find V k,l (k 1,l=1, 2) suc tat a(u k,l, vk,l )=(f, vk,l ), Teir algebraic systems can be described as vk,l V k,l. (2.9) A k,l U k,l = F k,l. (2.10)

4 1540 ZHONG LiuQiang et al. Since A k,l is symmetric positive definite, we use PCG metods to solve algebraic systems (2.10). In tis paper, we will construct te preconditioners for te cases of ig order edge finite equations, and present some estimates of te corresponding condition numbers. For tis purpose, we need to introduce some auxiliary spaces and corresponding operators. Let V = V k,l wit inner product a(, ) given by (2.5). Let V 1,..., V J,J N, be Hilbert spaces endowed wit inner products ā j (, ),j =1,...,J. Te operators Āj : V j V j are isomorpisms induced by ā j(, ), namely ā j (ū j, v j )= Ājū j, v j, ū j, v j V j, ere we tag dual spaces by and use angle brackets for duality pairings. For eac V j, tere exist continuous transfer operators Π j : V j V. Ten we can construct te preconditioner for operator A k,l as follows: B = J Π j Bj Π j, (2.11) j=1 were B j : V j V j are given preconditioners for Āj, andπ j are adjoint operators of Π j. Te following teorem of an estimate for te spectral condition number of te preconditioner given by (2.11) was presented in [7]. Teorem 2.1. Assume tat tere exist constants c j, suc tat Π j ū j A c j ū j Āj, ū j V j, 1 j J, (2.12) and for u V,tereexistū j V j suc tat u = J j=1 Π jū j and ( J ) 1/2 ū j 2 Ā j c 0 u A, (2.13) j=1 ten for te preconditioner B given by (2.11), we ave te following estimate for te spectral condition number K(BA k,l ) max K( B j Ā j )c j J J c 2 j. (2.14) Te principal callenge confronted in te development of preconditioners by applying Teorem 2.1 is to construct some spaces and operators wic satisfy (2.12) and (2.13). In te following two sections, we present te corresponding spaces and operators for two kinds of Nédélec edge finite element spaces, respectively. 3 Preconditioner for edge element equations of second kind Let V = V k,2 and k order Lagrangian finite element space is j=1 S k = {p H 1 0 (Ω) p K P k for all K T }. We coose te following two auxiliary spaces and te corresponding transfer operator: (i) V 1 = V k,1 wit inner product ā 1 (, ) =a(, ) in te sense tat ā 1 (ū 1, v 1 ):= Ā1ū 1, v 1 = a(ū 1, v 1 ), ū 1, v 1 V 1,

5 Preconditioners for iger order edge finite element discretizations of Maxwell s equations 1541 wic concludes tat Ā1 = A k,1. Te corresponding transfer operator is Π 1 = Id. (ii) V 2 = S k+1 wit inner product ā 2 (ū 2, v 2 ):= Ā2ū 2, v 2 = τ( ū 2, v 2 ), ū 2, v 2 V 2. (3.1) Te corresponding transfer operator is Π 2 = grad. Ten by using (2.11), we obtain te auxiliary space preconditioner for A k,2 as follows: B k,2 = B k,1 + grad B 2 grad, (3.2) were B k,1 is te preconditioner of A k,1,and B 2 is te preconditioners of Ā2 given by (3.1). Especially, we adopt te new algebraic multigrid metods for te H 1 -conforming ig order Lagrange finite element equations as te preconditioner B 2, tis coice fulfils te following estimate (see [11]): K( B 2 Ā 2 ) C 1, (3.3) were te constant C 1 is independent of te mes parameters. It is easy to prove tat te above transfer operators satisfy te conditions (2.12). In fact, using te definitions of inner products and transfer operators in spaces V l (l =1, 2), we ave Π 1 v 1 A = v 1 A = v 1 Ā1, v 1 V 1, (3.4) Π 2 v 2 2 A = v 2 2 A = τ v = v 2 2 Ā 2, v 2 V 2, (3.5) namely, te conditions (2.12) of Teorem 2.1 old wit te constants c 1 = c 2 =1. In order to give te corresponding decomposition wic satisfies (2.13), we need to present some preliminary materials. We first introduce te following k order divergence conforming finite elements of first kind (R-T elements): W k,1 = {v k,1 H 0 (div; Ω) v k,1 K D k for all K T }, (3.6) were D k =(P k 1 ) 3 P k 1 x. Te Sobolev spaces (2.1) (2.3) and te corresponding finite element spaces possess te exceptional exact sequence properties (see [12, 14]): H 0 (curl0;ω):={u H 0 (curl;ω): u = 0} = H0 1 (Ω), (3.7) V k,l (curl0) :={vk,l V k,l : vk,l = 0} = Sk+l 1, l =1, 2, (3.8) W k,1 (div0) :={wk,1 W k,1 : w k,1 =0} = V k,1 = V k,2. (3.9) We suppose tat te bases B(k, 1) of V k,1 are L 2 stable in te sense tat 2) v = v b, v b span{b}, v b 2 0 v 2 0, vk,1 V k,1. (3.10) b B(k,1) b B(k,1) Assuming tat u as te necessary smootness, we can define two kinds of interpolants: Π k,1,curl and Π k,1,div, suc tat Πk,1,curl u V k,1 and Π k,1,div u W k,1 (for more details, refer to [1, 2, 12]). Especially, te interpolation Π k,1,curl is not defined for a general function in H 0 (curl; Ω). Here let us quote a sligtly simplified version (see Lemma 5.38 of [12]). 2) Trougout tis paper, a b is abbreviated to a Cb wit a mes-size being independent, generic constant C>0. Finally, a b is abbreviated to a b a.

6 1542 ZHONG LiuQiang et al. Lemma 3.1. Suppose tat tere are constants δ>0 and p>2 suc tat u (H 1/2+δ (K)) 3 and u (L p (K)) 3 for eac K in T.TenΠ k,1,curlu is well-defined and bounded. In te following, we present te error estimate wic is known for te interpolation of Π k,1,curl (see Teorem 5.41 of [12] and Lemma 4.6 of [14], respectively). Lemma 3.2. If u (H 1/2+δ (K)) 3, 0 <δ 1/2 and u K D k, ten we ave (Id Π k,1,curl )u 0,K 1/2+δ K u (H 1/2+δ (K)) 3 + K u 0,K, (3.11) wit a constant only depending on te sape regularity of T. Lemma 3.3. Te interpolation operator Π k,1,curl is bounded on {v (H0 1 (Ω)) 3, v W k,1 } (H1 0 (Ω))3 and satisfies 1 (Id Π k,1,curl )ψ 0 ψ (H 1 (Ω)) 3, ψ (H1 0 (Ω)) 3, ψ W k,1, (3.12) wit a constant only depending on te sape regularity of T. Furtermore, all above operators possess te following commuting diagram property (see [12]): curl Π k,1,curl =Πk,1,div curl. (3.13) We may apply te quasi-interpolation operators for Lagrangian finite element space introduced in [15] to te components of vector fields separately. Tis gives rise to te projectors Q :(H0 1 (Ω)) 3 (S 1)3, wic inerits te continuity Q Ψ (H 1 (Ω)) 3 Ψ (H 1 (Ω)) 3, Ψ (H1 0 (Ω)) 3, (3.14) and satisfies te local projection error esitmate 1 (Id Q )Ψ 0 Ψ (H1 (Ω)) 3, Ψ (H1 0 (Ω))3. (3.15) Now, we present te stable decomposition of V k,2. Lemma 3.4. For any u k,2 V k,2, tere are uk,1 V k,1 and p S k+1 suc tat and u k,2 = u k,1 + p, (3.16) ( u k,1 2 A + p 2 A) 1/2 c 0 u k,2 A, (3.17) were te constant c 0 only depends on Ω and te sape regularity of T. Proof. For any u k,2 V k,2, we can interpolate uk,2 by Lemma 3.1. Tus, using (3.13), we ave In view of (3.9), we ave Π k,1,curl uk,2 =Π k,1,div uk,2. (3.18) u k,2 W k,1. (3.19)

7 Preconditioners for iger order edge finite element discretizations of Maxwell s equations 1543 Making use of (3.19) and noting tat Π k,1,div W k,1 = Id in (3.18), we get Π k,1,curl uk,2 = u k,2, namely Note tat u k,2 tat (u k,2 Πk,1,curl uk,2 V k,2 Πk,1,curl uk,2 )=0. (3.20), ten by (3.8) and (3.20), tere exists p S k+1, suc u k,2 = u k,1 + p, (3.21) were u k,1 =Πk,1,curl uk,2, wic completes te proof of (3.16). Using (3.21), (3.11) wit δ =1/2, and te inverse estimate, we obtain p 0,K = u k,2 Πk,1,curl uk,2 0,K u k,2 (H 1 (K)) 3 uk,2 0,K. Squaring and summing over all te elements, we get p 2 0 = p 2 0,K u k,2 2 0,K = uk, (3.22) K T K T In view of (2.6) and (3.22), we find p 2 A = τ p 2 0 τ u k,2 2 0 u k,2 2 A. (3.23) Making use of (3.21), triangular inequality and (3.22), we ave A direct manipulation of (3.21) gives tat u k,1 0 u k,2 0 + p 0 u k, (3.24) u k,1 0 = u k,2 0. (3.25) A combination of (3.23), (3.24) and (3.25) concludes (3.17). As a direct consequence of Teorem 2.1, (3.4), (3.5) and Lemma 3.4, we ave Teorem 3.5. For B k,2 is given by (3.2), and B 2 satisfies te condition (3.3), we ave K(B k,2 Ak,2 ) K(Bk,1 Ak,1 wit a constant only depending on te constants c 0 and C 1. ), (3.26) 4 Preconditioner for edge element equations of first kind In tis case, we take V = V k+1,1 and coose anoter two auxiliary spaces and te corresponding transfer operator as follows. (i) V 1 = V k+1,1 wit inner product ā 1 (, ) wic is defined by ā 1 (ū 1, ū 1 ):= Ā 1 ū 1, v 1 = a(v b, v b ), were ū 1 = v b, v b span{b}. Te transfer operator is Π 1 = Id.

8 1544 ZHONG LiuQiang et al. (ii) V 2 = V k,2 wit inner product ā 1 (, ) =a(, ) in te sense tat ā 1 (ū 1, v 1 ):= Ā2ū 2, v 2 = a(ū 1, v 1 ), ū 1, v 1 V 1, wic concludes tat Ā2 = A k,2. Te transfer operator is Π 2 = Id. Making use of (2.11), te auxiliary space preconditioner for A k+1,1 reads were B k,2 is te preconditioner of A k,2 Noting tat Ā1 denotes te diagonal matrix of A k+1,1 B k+1,1 = B 1 + B k,2, (4.1), B1 is te preconditioners of Ā1., in te practical application, we will. Obviously, tis special take B 1 as te Jacobi (or Gauss-Seidel) smooting operator for A k+1,1 coose satisfies K( B 1 Ā 1 ) C 1, (4.2) were te constant C 1 is independent of te mes parameters. First, we prove tat te above transfer operators satisfy te condition (2.12). Due to te definitions of inner product and transfer operator in space V 1, for any given ū 1 = α bb V 1, were α b R, weave Π 1 ū 1 2 A = ū 1 2 A = M K T α b b 2 A = M α b b K T j=1 2 A,K α b 2 b 2 A,K = M ū 1 2 Ā 1, (4.3) were te constant M bounds te number of basis functions wose support overlaps wit a single element K. For any given ū 2 V 2, it is easy to obain Π 2 ū 2 A = ū 2 A = ū 2 Ā2. (4.4) Combining (4.3) wit (4.4), we conclude tat (2.12) olds wit te constants c 1 = M and c 2 =1. In te sequel, we present anoter lemma wic is devoted to a decomposition of V k+1,1. Here, we ave to use anoter metod wic is different from te proof of Lemma 3.4, since (Id Π k,2,curl )uk+1,1 does not belong to te space V k+1,1 (curl 0). In fact, it is te igfrequency contribution of u k+1,1. Lemma 4.1. For any u k+1,1 u k,2 V k,2, suc tat V k+1,1 u k+1,1 =,tereexist v b V k+1,1, v b Span{b}, v b + u k,2, (4.5) and ( ) 1/2 v b 2 A + u k,2 2 A c 0 u k+1,1 A, (4.6)

9 Preconditioners for iger order edge finite element discretizations of Maxwell s equations 1545 were te constant c 0 only depends on Ω and te sape regularity of T. Proof. For any given u k+1,1 V k+1,1, using te continuous Helmoltz decomposition, tere exist Ψ (H0 1(Ω))3,p H0 1 (Ω) suc tat and Ψ (H1 (Ω)) 3 u k+1,1 = Ψ + p, (4.7) uk+1,1 0, p H1 (Ω) u k+1,1 H(curl;Ω), (4.8) wit constants only depending on Ω. Taking te curl of bot sides of (4.7) and using (3.9), we get Ψ = u k+1,1 W k+1,1. OwingtoLemma3.1,Π k+1,1,curlψ is well defined. Furtermore, te commuting diagram property (3.13) implies Π k+1,1,curl Ψ =Πk+1,1,div Tis confirms tat te tird term in te splitting Ψ = Ψ (Id Πk+1,1,curl )Ψ =0. Ψ =Π k+1,1,curl (Id Q )Ψ +Π k+1,1,curl Q Ψ +(Id Π k+1,1,curl )Ψ (4.9) actually belongs to te kernel of curl. By (3.7), ten tere exists q H0 1 (Ω) suc tat Note tat Q Ψ (S 1)3 V k+1,1,wicleadsto Substituting (4.9), (4.10) and (4.11) into (4.7), we ave u k+1,1 (Id Π k+1,1,curl )Ψ = q. (4.10) Π k+1,1,curl Q Ψ = Q Ψ. (4.11) =Π k+1,1,curl (Id Q )Ψ + Q Ψ + (q + p). (4.12) Since u k+1,1, Π k+1,1,curl (Id Q )Ψ,Q Ψ V k+1,1,weobtain (q + p) V k+1,1 (4.12), ten observing (3.8), tere exists q S k+1, suc tat by using q = (q + p). (4.13) Let ũ k+1,1 =Π k+1,1,curl (Id Q )Ψ = v b, v b Span{b}, (4.14) u k,2 = Q Ψ + q. (4.15) It is easy to obtain u k,2 V k,2 by noting tat Q Ψ (S 1)3 V k,2 and q S k+1. Substituting (4.13), (4.14) and (4.15) into (4.12), we conclude V k,2 u k+1,1 = v b + u k,2, (4.16)

10 1546 ZHONG LiuQiang et al. wic completes te proof of (4.5). Using (4.14), triangular inequality, Lemma 3.3, (3.15) and (4.8), we ave wic leads to 1 ũ k+1,1 0 = 1 Π k+1,1,curl (Id Q )Ψ 0 1 (Id Π k+1,1,curl )(Id Q )Ψ (Id Q )Ψ 0 (Id Q )Ψ (H1 (Ω)) 3 + Ψ (H 1 (Ω)) 3 Ψ (H 1 (Ω)) 3 uk+1,1 0, ũ k+1,1 0 u k+1,1 0. (4.17) It follows readily from inverse estimate and (3.10) tat v b 2 A = ( v b τ v b 2 0) Using inverse estimate again yields ũ k+1,1 2 A b B(curl) ( 1 v b τ v b 2 0) ( 2 + τ) ũ k+1, (4.18) = ũk+1,1 By means of (4.17) and inverse estimate, we get ( 2 + τ ) ũ k+1, τ ũk+1,1 2 0 ( 2 + τ ) ũ k+1, (4.19) 2 0 ( 2 + τ ) 2 u k+1,1 2 0 u k+1, τ u k+1,1 2 0 = u k+1,1 2 A. (4.20) In view of (4.16), triangular inequality (4.18), (4.19) and (4.20), we ave v b 2 A + u k,2 2 A v b 2 A + ( u k+1,1 A + ũ k+1,1 ) 2 A ( 2 + τ ) ũ k+1,1 u k+1,1 2 A, uk+1,1 2 A wic completes te proof of (4.6). As a direct consequence of Teorem 2.1, (4.3), (4.4) and Lemma 4.1, we ave Teorem 4.2. For B k+1,1 is given by (4.1), and B 1 satisfies te condition (4.2), we ave K(B k+1,1 A k+1,1 ) K(B k,2 Ak,2 ), (4.21) wit a constant only depending on te constants c 0, C 1 and M. Combining Teorems 4.2 and 3.5, by using a Jacobi (or Guass-Seild) smooting, we can translate te construction of preconditioner for k + 1orderNédélec element equations of first kind into te one of te k order Nédélec element equations of second kind. Furtermore, by

11 Preconditioners for iger order edge finite element discretizations of Maxwell s equations 1547 solving an H 1 -conforming k + 1 order Lagrange finite element equations, we can translate te preconditioner for k order Nédélec element equations of second kind into te one for k order Nédélec element equations of first kind. Since Hiptmair and Xu [7] ave constructed an efficient preconditioner B 1,1 for A 1,1, we can prove tat te spectral condition numbers K(B k,l Ak,l )(k>1,l =1, 2) are uniformly bounded and independent of mes size and te parameter τ by tis recursive form. In te next section, we will give some numerical experiments to verify te correctness of teories and sow te efficiency of te preconditioners. 5 Numerical experiments For variational problem (2.9), we construct two examples as follows: Example 5.1. Te computational domain is Ω = [0, 1] [0, 1] [0, 1] and te corresponding structured grids can be seen in Figure 1. For te convenience of computing te exact errors, we construct an exact solution u =(u 1,u 2,u 3 )as u 1 = xyz(x 1)(y 1)(z 1), u 2 =sin(πx)sin(πy)sin(πz), u 3 =(1 e x )(1 e x 1 )(1 e y )(1 e y 1 )(1 e z )(1 e z 1 ). Example 5.2. Te computational domain is te speres of radius 1, and te corresponding unstructured grids can be seen in Figure 2, te exact solution u =(u 1,u 2,u 3 )is u 1 = x 2 + y 2 + z 2 1, u 2 = x 2 + y 2 + z 2 1, u 3 = x 2 + y 2 + z 2 1. In te following, we will keep track of te number of PCG-iterations required to solve te discrete variational problems (2.9). A relative reduction of te Euklidean norm of te residual vector by a factor of 10 6 wasusedastermination criterion. Te numerical results for Example 5.1 in V 2,1 and V 2,2 are reported in Tables 1 and 2, respectively. Te numerical results for Example 5.2 in V 2,1 and V 2,2 are also presented in Tables 3 and 4, respectively. Table 1 No. of PCG-iterations for Example 5.1 in V 2,1 Level Cells τ Table 2 No. of PCG-iterations for Example 5.2 in V 2,1 τ Level Cells

12 1548 ZHONG LiuQiang et al. Table 3 No. of PCG-iterations for Example 5.1 in V 2,2 τ Level Cells Table 4 No. of PCG-iterations for Example 5.2 in V 2,2 τ Level Cells By te observation of te numerical results, we know tat te condition numbers do ardly deteriorate on successively finite structured grids and unstructured grids, and tey are almost independent of te parameter τ. Acknowledgements Te autors wis to tank Xu J, Department of Matematics, Pennsylvania State University, USA, for providing many constructive suggestions. References 1 Nédélec J C. Mixed finite elements in R 3. Numer Mat, 35: (1980) 2 Nédélec J C. A new family of mixed finite elements in R 3. Numer Mat, 50: (1986) 3 Hiptmair R. Multigrid metod for Maxwell s equations. SIAM J Numer Anal, 36: (1998) 4 Cen Z, Wang L, Zeng W. An adaptive multilevel metod for time-armonic Maxwell equations wit singularities. SIAM J Sci Comput, 29: (2007) 5 Arnold D, Falk R, Winter R. Multigrid in H(div) and H(curl). Numer Mat, 85: (2000) 6 Hiptmair R, Widmer G, Zou J. Auxiliary space preconditioning in H 0 (curl;ω). Numer Mat, 103(3): (2006) 7 Hiptmair R, Xu J. Nodal auxiliary space preconditioning in H(curl) andh(div) spaces. SIAM J Numer Anal, 45(6): (2007) 8 Zong L, Tan L, Wang J, Su S. A fast algoritm for a second family of Nédélec edge finite elements equations (in Cinese). Mat Numer Sin, (in press) 9 Tan L, Zong L, Su S. Fast iterative metod for solving first family quadratic edge finite element equations (in Cinese). Natur Sci J Xiangtan Univ, 30(1): (2008) 10 Ainswort M, Coyle J. Hierarcic finite element bases on unstructured tetraedral meses. Int J Num Met Eng, 58: (2003) 11 Su S, Sun D, Xu J. An algebraic multigrid metod for iger-order finite element discretizations, Computing, 77(4): (2006) 12 Monk P. Finite element metods for Maxwell s equations. Numerical Matematics and Scientific Computation. Oxford: Oxford University Press, Bossavit A. Computational Electromagnetizm. Variational Formulation, Complementarity, Edge Elments. Vol. 2 of Electromagnetizm Series. San Diego: Academic Press, Hiptmair R. Finite elements in computational electromagnetism. Acta Numerical, 11: (2002) 15 Scott L R, Zang S. Finite element interpolation of nonsmoot functions satisfying boundary conditions. Mat Comp, 54: (1990)