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RESIDUAL BASED A POSTERIORI ERROR ESTIMATORS FOR EDDY CURRENT COMPUTATION R. BECK , R. HIPTMAIRy , R.H.W. HOPPEz , AND B. WOHLMUTHx

Abstract. We consider H (curl; )-elliptic problems that have been discretized by means of Nedelec's edge elements on tetrahedral meshes. Such problems occur in the numerical compuation of eddy currents. From the defect equation we derive localized expressions that can be used as a posteriori error estimators to control adaptive re nement. Under certain assumptions on material parameters and computational domains, we derive local lower bounds and a global upper bound for the total error measured in the energy norm. The fundamental tool in the numerical analysis is a Helmholtz-type decomposition of the error into an irrotational part and a weakly solenoidal part. Key words. residual based a posteriori error estimation, Nedelec's edge elements, Helmholtz decomposition, eddy currents AMS subject classi cations. 65 N15, 65 N30, 65 N50.

1. Introduction. The computation of quasistatic electromagnetic elds in conductors usually employs the eddy current model [2,5,23,33]. For the transient case, if we use common formulations based on the electric eld, we end up with the degenerate parabolic initial-boundary value problem @t(E) + curl curlE = ?@tj in

En = 0 on ? := @

(1.1) E(:; 0) = E0 in

Here the unknown quantity is the electric eld E : [0; T ] 7! R3 and R3 stands for a connected bounded polyhedral computational domain. Though the equations are initially posed on the entire space R3, we can switch to a bounded domain by introducing an arti cial boundary suciently removed from the region of interest. This is commonplace in engineering simulations [29]. Further, 2 L1 ( ) denotes the bounded uniformly positive inverse of the magnetic permeability. We con ne ourselves to linear isotropic media, i.e. is a scalar function of the spatial variable x 2 only. Hence, for some ; > 0 holds 0 < a.e. in . We rule out anisotropy also for the conductivity 2 L1 ( ), for which holds 0 a.e. in for some bound > 0. Usually, there is a crisp distinction between conducting regions, where is bounded away from zero, and insulating regions, where = 0. We will take for granted that > 0, whereever 6= 0. The right hand side j = j(x; t) is a time-dependent vector eld in L2 ( ), which represents the source current. For physical reasons div j(t; :) = 0 a.e. in and for all times. We remark that in many applications the exciting current, for instance the current in a coil, is provided through an analytic expression. A typical arrangement is depicted in gure 1.1. We remark that (1.1) is an ungauged formulation, as we have already dropped the divergence constraint div E = 0. Obviously, this forfeits uniqueness of the solution in parts of the domain where = 0. However, the only relevant quantity there is curlE, which remains unique. Where E is of interest, inside the conductor, we have > 0. ZIB-Berlin, e-mail:[email protected] y SFB 382, Universitat Tubingen, e-mail: [email protected] z Mathematisches Institut, Universitat Augsburg, e-mail: [email protected] x Mathematisches Institut, Universitat Augsburg, e-mail: [email protected] 1

2

Beck, Hiptmair, Hoppe, Wohlmuth

j

C = 0

Fig. 1.1. A model problem for eddy current computation (cf. [24, Ch. 8])

There E is unique and div E = 0 is satis ed due to the solenoidality of the right hand side. For the sake of stability, timestepping schemes for (1.1) have to be L-stable [38]. This requirement can only be met by implicit schemes like SDIRK-methods. In each timestep they entail the solution of a degenerate elliptic boundary value problem of the form curl curlu + u = f in

(1.2) u n = 0 on @ : In this context, u denotes the new approximation to E to be computed in the current timestep, and f depends on j and the approximation of E in the previous timestep. Note that we can still assume div f = 0. The coecient agrees with except for a scaling by the length of the current timestep; accordingly, 0 < a.e. in C . Problem (1.2) cast in weak form yields a variational problem in the Hilbert space H (curl; ) of L2 ( )-vector elds whose curl is square integrable: Find u 2 H 0(curl; ) such that ( curlu; curlq)L2 ( ) + ( u; q)L2 ( ) = (f ; q)L2 ( ) ; 8q 2 H 0 (curl; ) : (1.3) A subscript 0 indicates that vanishing tangential traces on @ are imposed on the elds (for details on traces see, e.g. [4,37]). For uniformly positive a.e. in the Lax-Milgram lemma guarantees existence and uniqueness of a solution of (1.3). If = 0 on sets of positive measure, we can still expect a unique solution in the quotient space H 0(curl; )=Ker(curl). It is now generally accepted that an appropriate nite element discretization of (1.3) should rely on genuine H (curl; )-conforming schemes that merely enforce the typical tangential continuity of the eld across interelement boundaries [3,21,22]. For simplicial meshes H (curl; )-conforming nite elements of arbitrary polynomial order were rst introduced by Nedelec [49], generalizing the lowest order Whitney elements [56]. Similar schemes are also known for other shapes of elements [34,49]. For all of them standard a priori error bounds can be established [47,48,52]. Ultimately, the discretization of (1.3) by these so-called edge elements leads to a large sparse system of equations for the degrees of freedom of the nite element space. Usually, an approximate solution can only be obtained by iterative methods [28,39]. Denoting by uh the exact solution of the discretized problem and by u~ h some iterative approximation, we are interested in an ecient and reliable residual based a posteriori error estimator for the total error e := u ? u~ h with respect to relevant

Residual based error estimators

3

norms. The most signi cant is the energy (semi)norm kkE; related to problem (1.1) de ned by kuk2E; := ( curlu; curlu)L2 ( ) + ( u; u)L2 ( ) : In the current context, local a posteriori error estimation serves two purposes: Firstly, the error estimator can be used for adaptive local re nement and coarsening of the underlying triangulation. Since the elds feature strong singularities at reentrant corners [31] and at irregular material interfaces [32], a higher resolution of the mesh in these zones is desirable. Precisely how much can only be concluded on the basis of information about the local error. Secondly, information about the accuracy of the nite element solution is also required to balance the spatial and temporal errors in the context of adaptive timestepping for the original parabolic problem [17,18]. We note that a posteriori error estimators for adaptive local grid re nement are well established tools in the ecient numerical solution of elliptic boundary value problems. In the framework of standard conforming nite element approaches we acknowledge the pioneering work due to Babuska and Rheinboldt [8,9] and the more recent articles [11,12,35,36,54,57]. Further references can be found in the survey article by Bornemann et al. [19] and in the excellent monography by Verfurth [55]. In the context of nonconforming techniques we mention [41,43]. For mixed nite element methods involving Raviart-Thomas elements we refer to [1,25,26,42,44,45]. However, as far as nite element approximations based on Nedelec's curl-conforming edge elements are concerned, to the authors' knowledge no work on a posteriori error estimation has been done so far. The paper is organized as follows: In section 2, we will introduce the curlconforming approximation of (1.3) by Nedelec's edge elements. In addition we are going to supply a few technical devices required for the proofs. Then, in section 3, we consider the variational problem satis ed by the total error e and state the main result of this paper in terms of a cheaply computable, ecient and reliable a posteriori error estimator for kekE; . As the main tool we will use a Helmholtz type decomposition of e into a curl-free part e0 and a \ {weakly solenoidal" part e? . In particular, section 4 contains the estimation of the irrotational part e0 whereas section 5 is devoted to the weakly solenoidal part e? . In both cases, the estimates result from an evaluation of the residuum with respect to a dual norm. In the nal section we report on numerical experiments that examine the performance of the error estimator for a wide range of model problems. 2. Finite element spaces. We consider the nite element approximation of (1.3) by means of Nedelec's edge elements with respect to a hierarchy Thk , k 2 N0, of simplicial triangulations of generated by successive local re nement of an initial coarse triangulation Th0 . We use the standard re nement process developed by Bank et al. [10,11] in the 2D case which has been extended to the 3D setting in [16] and [50] (cf. also [19]). Alternative schemes are also available [7,13,46]. For a description of the re nement strategies we refer to the literature cited above. We demand that the coarsest mesh Th0 can resolve the boundaries of the conductors. This means that any element either entirely belongs to the conducting region or to the nonconducting region. We x some Thk within the hierarchy and, for notational convenience, omit the lower index k, i.e., Th := Thk . For D , the sets of vertices, edges and faces in D are denoted by Nh (D); Eh (D) and Fh (D), respectively. If D = , we will simply write Nh ; Eh; Fh and refer to Nhint ; Ehint; Fhint and Nh? ; Eh?; Fh? as the sets of vertices, edges

4


and faces located in the interior of and on the boundary ?, respectively. Those interior edges that belong to C constitute the set FhC and for the set of elements in

C we write ThC . All the edges have to be endowed with a xed internal oerientation (direction). We denote by hT and hF the maximal diameter of an element T 2 Th and a face F 2 Fh . Since the re nement rules imply regularity and local quasiuniformity of the hierarchy of triangulations (cf. [16]), there exist constants 1 > 0 and 2 > 0 depending only on the local geometry of the initial triangulation T0 := Th0 such that

hT 0 1 hT for T; T 0 2 Th ; T \ T 0 6= ; (2.1) hF 2 hT for F 2 Fh (T ) : Following Nedelec's construction of simplicial edge elements in [49], we denote by Pk (D), k 0, the linear space of multivariate polynomials of degree k on D, and refer to P~k (D), k 0, as the subspace of homogeneous polynomials of degree k. We de ne

Sk (D) := fp 2 P~k (D)3 ; hx; pi :=

3 X i=1

xipi = 0g; k 1 :

Then, for T 2 Th and k 1, the local space for the Nedelec element is given by NDk(T ) := Pk?1(D)3 Sk(D) : In the special case of lowest order edge elements, k = 1, we nd the representation ND1(T ) := fx 7! a + b x ; a; b 2 R3g : (2.2) Appropriate degrees of freedom are provided by linear functionals on NDk (T ) of the form (cf. e.g. [49])

R

p 2 Pk?1(E ); E 2 Eh (T ) ; (i) q 7! E hq; ti p ds ; R (ii) q 7! F hq n; pi d ; p 2 Pk?2(F )2 ; F 2 Fh (T ) ; R (ii) q 7! T hq; pi dx ; p 2 Pk?3(T )3 : Here, polynomialspaces with negative degree are supposed to be empty. This speci cation of the degrees of freedom ensures that the global nite element space NDk ( ; Th) is contained in H (curl; ). Then, setting NDk;0( ; Th) := NDk ( ; Th) \ H 0(curl; ) ; the curl-conforming nite element approximation of (1.3) is as follows: Find uh 2 NDk;0( ; Th) such that (uh ; qh )L2 ( ) + ( uh ; qh )L2 ( ) = (f ; qh)L2 ( ) 8qh 2 ND k;0( ; Th ) : (2.3) We recall that Nedelec's nite elements provide ane equivalent families of nite elements in the sense of [27], if the vector elds are subjected to a covariant transformation: For any T 2 Th write : Tb 7! T for the ane mapping of a xed reference tetrahedron Tb to T and de ne F(v)(xb ) := DT (xb) v((xb)) ; xb 2 Tb : (2.4)


5

Then it turns out that ND k (Tb) = F(ND k (T )) and the degrees of freedom are invariant under the transformation (2.4). Edge elements provide one specimen of discrete dierential forms [20,40]. This accounts for the exceptional property that the result that every curl{free vector eld on a contractible domain is a gradient is preserved in a purely discrete context. The role of the potentials is played by Lagrangian nite element functions Sk;0 ( ; Th) := fh 2 C 0( ); h j@ = 0; h jT 2 Pk (T )g : Lemma 2.1 (Discrete potentials). If the boundary @ is connected, then for any qh 2 ND k;0( ; Th ), k 1, with curlqh = 0 there exists a unique h 2 Sk;0 ( ; Th) such that qh = grad h . 3. Residual based error estimator. We assume that u~ h 2 NDk;0( ; Th ) is some iterative approximation of the unique solution uh of the curl-conforming nite element solution of (2.3). It can be obtained, for instance, by the multigrid iterative solution process as developed in [39]. Denoting the total error by e := u ? u~ h , it is easy to see that e 2 H 0 (curl; ) satis es the defect equation ( curle; curlq)L2 ( ) + ( e; q)L2 ( ) = r(q) 8q 2 H 0 (curl; ) ; (3.1)

where r() stands for the residual r(q) := (f ; q)L2 ( ) ? ( curl u~ h ; curlq)L2 ( ) ? ( u~ h ; q)L2 ( ) ; q 2 H 0 (curl; ) : (3.2) The construction of the error estimator will be based on a direct splitting of the function space H 0(curl; ) (3.3) H 0 (curl; ) = H 00 (curl; ) H ? 0 (curl; ) : It may be labelled a \ {orthogonal" Helmholtz type decomposition, since we require Both H 00 (curl; ) and H ? 0 (curl; ) are closed subspaces of H 0 (curl; ). 0 H 0 (curl; ) := fq 2 H 0 (curl; ) ; curlq = 0g is the kernel of the curl operator. ? q? ; q0 L2 ( ) = 0 for all q0 2 H 00(curl; ), q? 2 H ?0 (curl; ). Evidently, a decomposition complying with these requirement is also orthogonal with respect to the energy seminorm. The following procedure furnishes a splitting of q 2 H 0(curl; ) according to First decompose qj C = u0 ue ?, where u0; ue ? 2 H (curl; C ) and ? u(3.3): 0; u e ? L2( C ) = 0, curlu0 = 0. If meas(@ \ @ C ) > 0, we also require that u0, ue? have vanishing trace on @ . Write u? for the H (curl; )-extension (cf. [4]) of ue ? to . Then, let v? be the unique vector eld in H 0 (curl; = C ) such that curlv? = curlq ? curlu? and v? ?Ker(curl) in H 0(curl; = C ). Finally set ue ? in C ? q := v? + u? in

= C : The functions q? thus constructed form a closed subspace of H 0(curl; ). This can be seen by completeness arguments. Unfortunately, mere existence of such a decomposition is not enough; our theoretical examinations hinge on the following assumption:

6


Assumption 3.1. We assume that a splitting (3.3) with the above features can be found such that H ?0 (curl; ) is continously embedded in H 1( ) \ H 0 (curl; ) and, moreover,

q? ? ? ? H 1 ( ) C ( ) curlq L2 ( ) 8q 2 H 0 (curl; ) :

Thus far, the statement of this assumption can be shown only if C = and ; are continuously dierentiable and uniformly bounded away from zero [6,37]. In this case we simply use the true Helmholtz-decomposition. A characterization of the kernel of the curl-operator is provided by the continuous version of lemma 2.1 (cf. [37]): Lemma 3.2. For any q 2 H 00 (curl; ) there exists a unique 2 H01( ) such that q = grad , provided that the boundary ? of is connected. If ? is not connected, i.e. has embedded cavities, the entire kernel of curl is not provided by grad H01 ( ). This is only true modulo a space of small dimension [6, Prop. 3.12]. To avoid technical diculties we do not allow cavities in . By means of the decomposition (3.3) we may split the total error according to e = e0 + e? , where e0 2 H 00(curl; ) and e? 2 H ?0 (curl; ). We note that e0 represents the curl-free part of the total error whereas e? stands for a \ -weakly solenoidal" part. As a matter of course, e0 is only meaningful in C . It readily follows from (3.1) that e0 and e? are the unique solutions of the variational equations ? e0 ; q 0 0 2 H 00 (curl; ) ; 8 q 2 ( ) = r(q) L ? curle?; curlq? + ? e?; q? ? ? L2 ( ) L2 ( ) = r(q) 8q 2 H 0 (curl; ) : (3.4) The irrotational and the {weakly solenoidal part of the error will be estimated separately. For simplicity, throughout the rest of this paper we assume the functions and to be elementwise constant. As far as the irrotational part e0 is concerned, the estimate is based on the evaluation of the residual r() restricted to H 00(curl; ) = grad H01 ( ). In particular, exploiting that f is solenoidal, Green's formula yields X (f ? u~ h ; grad v )L2 (T) r(grad v) = =

T 2Th

X

T 2ThC

(div u~ h ; v)L2 (T) ?

X

F 2FhC

([hn; u~ h i]J ; v)L2 (F) ;

where [hn; u~ hi]J denotes the jump of the normal component of the vector eld u~ h across the interelement face F 2 Fhint. It is de ned as follows: If F 2 Fhint is the common face of two adjacent elements Tin; Tout 2 Th and n denotes the unit normal vector on F directed towards the interior of Tin , then [hn; qi]J := hn; qijF Tout ? hn; qijF Tin : Note that []J does not depend on the speci cation of Tin and Tout . As will be shown in section 4, the upper and lower bounds for e0 L2 ( ) involve the error terms

0 11=2 0 11=2 X X (0) := @ (0T )2 A + @ (0F )2 A ; T 2ThC

F 2FhC

(3.5)

7


whose local contributions 0T , 0F are given by

p u~

0T := hT div h L2 (T) ;

0F := hF1=2

p1 A [hn; u~ h i]J

L2 (F) ;

T 2 ThC ; F 2 FhC ;

where A is de ned as the average on F , A := 0:5( jTout + jTin ). The scaling in the dierent terms of the error estimator corresponds to the fact that we measure the error in the energy norm. For elements and faces outside C , we formally set the contributions 0T and 0F to zero. The upper bound also involves the iteration error

p it(0) :=

(uh ? u~ h )

L2 ( )

:

(3.6)

On the other hand, concerning bounds for e? , for q 2 H ?0 (curl; ) the residual r(q) can be written as P n(f ? u~ ; q) 2 ? ( curl u~ ; curlq) 2 o r(q) = h L (T) h L (T) TP 2Th = (f ? curl curl u~ h ? u~ h ; q)L2 (T) T 2ThP ([n curl u~ h ]J ; q)L2 (F) : ? F 2Fhint

We note that a localization of the residual is not feasible due to the global character of the space H ?0 (curl; ). Instead, we will use a localization by means of an interpolation with respect to the entire discrete space NDk;0( ; Th ). As we shall see in Section 5, this is at the expense of a coupling between e0 and e? . However, this will not thwart the primary goal of obtaining an ecient and reliable estimate of the total error in

?

the energy norm. In particular, the bounds for e E; comprise the error terms

11=2 !1=2 0 X T )2 +@ (1F )2 A ; (1;1 1(1) := T 2Th F 2Fhint !1=2 X X

2 :=

T )2 (1;2

(3.7)

;

(3.8)

T 2Th T , 1 2, and 1F given by with the local contributions 1;

T := hT

p1 (h f ? curl curl u~ h ? u~ h )

1;1 2 (T) ; L

T := hT

p1 (f ? h f )

1;2 L2 (T) ;

1F := hF1=2

p1 A [n curl u~ h ]J L2 (F) ;

T 2 Th ; T 2 Th ; F 2 Fhint ;

where A is de ned as the average on F , A := 0:5(jTout + jTin ). Here, h f denotes Q 2 the L -projection of f onto T 2Th Pk (T )3 . Again, the iteration error

it(1) := kuh ? u~ h kE;

(3.9)

8


will enter the upper bound. Be aware that using a fast aymptotically optimal iteration scheme, we can quickly tell the truncation error from the size of the correction to the current iterate. Good bounds for it(0) and it(1) are at our disposal, thus. As the main result of this paper we state the following a posteriori estimate for the total error e measured in the energy norm. Theorem 3.3. Let 1 := (0) + 1(1), it := it(1) + it(0) with (0) , 1(1) , it(1) , it(0) given by (3.6), (3.5), (3.7), and (3.9), respectively. If assumption 3.1 holds true, then there exist constants ; ? > 0, 1 2, depending only on , , , , and on the local geometry of Th0 such that

1 1 ? 2 2 kekE; ?1(1 + 2 ) + ?2 it :

Eventually, we need an estimate for the energy of the error on each element. Such an element oriented error estimator can be constructed by assigning half of the contribution of a face to either adjacent element. To oset the impact of jumps in the coecients it is advisable to resort to additional scaling. As the actual error estimator we then get for each T 2 Th X jT F 2 jT F 2 T )2 + (0 ) + 2 (1 ) : (3.10) bT2 := (0T )2 + (1;1 2 A A int F 2F (T)\F h

Since only local information is needed, the evaluation of bT is cheap. Low order T is elusive, numerical quadrature is sucient to compute the local norms. Of course 1;2 but h has been chosen such that for smooth f this quantity can be expected to decrease faster than the other contributions to the error estimator. For lowest order edge elements and locally constant coecients, we can capitalize on the simple local ansatz space (2.2). First note that it contains only piecewise linear, divergence-free vector elds. Therefore, curl u~ h is locally constant and we end up with the simpli ed local error estimator 2

^T2 = hjTT kh f ? u~ hk2L2 (T) + P 1 h jT k[hn; u~ i] k2 + jT k[n curl u~ ] k2 2 : h J L2 (F) A h J L (F) 2 F A F 2Fh (T)\Fhint (3.11) Here, h can be a suitable interpolation onto the space of piecewise linear vector elds. Moreover, Gauian quadrature formulas that are exact for quadratic polynomials on T and F , respectively, can be used to evaluate all the norms. Thus, only the values of degrees of freedom in a neighborhood of T and the local geometry of the mesh will show up in an explicit expression for ^T2 . 4. Estimator of the irrotational part of the error. Here,? we will consider the irrotational part e0 of the error e. Upper and lower bounds for e0 ; e0 L2 ( ) will be established by means of (0) and the iteration error. The starting point for the error analysis is the variational problem (3.4). The defect problem (3.1) restricted to the curl-free subspace of H 0 (curl; ) gives rise to the following uniformely positive de nite variational problem on H?1 ( C ): Find 2 H?1 ( C ) such that ( grad ; grad )L2 ( C ) = r(grad ) =: r~() 8 2 H?1 ( C ): (4.1)

9


Here, the space H?1 ( C ) is de ned as follows: fv 2 H 1( )j (1; v) 2 = 0g; if meas (@ \ @ ) = 0 1 H?( C ) := fv 2 H 1( C )j v L ( C=) 0g; if meas (@ \C @ ) 6= 0; C j@ C \@

C and the residual r() is de ned in (3.2). Thanks to Lemma 3.2 we have e0 = grad in C . We will also write for the harmonic extension to a function in H01( ). Following the same lines as in the H 1( )-elliptic setting [55], the dual norm of 1 ( C ) provides bounds for ? e0 ; e0 2 . The upper bound for r?~() restricted on H ? L ( ) 0 0 e ; e L2 ( ) is obtained by applying Green's formula. Observing Lemma 2.1 and Galerkin orthogonality r~( h ) = ( (~uh ? uh ); grad h )L2 ( C ) ; h 2 Sk ( C ; Th ) :

Here, Sk ( C ; Th) H?1 ( C ) denotes the Pk conforming nite element space. We nd for h 2 Sk ( C ; Th) ? e0; e0 = r~( ? ) + ( (u ? u~ ); grad ) 2 : (4.2) h h h h L ( C ) L2 ( )

In particular, r~( h ) = 0 if the iteration error uh ? u~ h is zero. Upper bounds for the right side are obtained by a suitable choice of h . We set h := Phk , where Phk : H?1 ( C ) ?! Sk ( C ; Th ) is a locally de ned projection-like operator satisfying approximation and stability properties k h ; h 2 Sk ( C ; Th);

? P k

Ph h = Ch ; (4.3) T kgrad k 2 2 h

? P k

L (T) C ph kgrad Lk(DT ) ; (4.4) 2 F

grad Phk

L2(F) C kgrad k L :(DF ) (4.5) h L2 (T) L2 (DT ) Here, DT and DF contain all elements in Th sharing at least one vertex with T and F , respectively. Such operators can be de ned preserving boundary conditions by the use of dual basis functions. We refer to [51,53] for more details. In the case of a posteriori error estimates, the interpolation operator of Clement is very often used [30]. However, the Clement operator restricted on Sk ( C ; Th) is not the identity. Using Green's formula, the approximation properties (4.3), (4.4) of Phk , and div f = 0, we get

X? X? div u~ h ; ? Phk L2 (T) ? [hn; u~ hi]J ; ? Phk L2 (F) T 2ThC F 2FhC 1 12 0

X p 0

2 p

2 A @ C hT div u~ h L2 (T) e L2 ( ) C C T 2Th 0 1 21 X p phF k[hn; u~ hi]J k2L2 (F) A

e0

2 : + C@

r~( ? Phk ) =

F 2FhC

A

L ( C )

Here, we set [hn; u~ hi]J jF := 0 if F @ C \ @ and [hn; u~ hi]J jF := u~ h nT if F @ C n @ , F @T . An upper bound for the second term on the right side of

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(4.2) is obtained by means of (4.5). The stability of the projection-like operator Phk yields

p 0

p ? (~u ? u~ ); grad P k ~ ( u ? u ) C

h h L2 ( ) e L2 ( ) h h h L2 ( C ) C

and thus

C

p 0

e L2 ( C ) C (0) + it(0) :

p We point out that in the proof for the upper bound of e0 L2 ( C ) , the stability

and approximation properties of a suitable quasi-interpolation operator from H?1 ( C ) onto Sk ( C ; Th ) play an important role. We establish the lower bound exploiting properties of local bubble functions: Two dierent types of bubble functions T and F de ned on T are used. We set

T := 256

Y4 l=1

l;T ; F := 27

Y3 l=1

Fl ;T ;

where l;T , 1 l 4 are the barycentric coordinates of T associated with the vertices pl . The face F is spanned by the vertices pFl , 1 l 3 of T . We set out from the following norm equivalences

1=2

1=2

T h L2 (T) kh kL2 (T) C T h L2 (T) 8h 2 Pk (T )

1=2 (4.6)

F h

L2(F) khkL2(F) C

F1=2h

L2(F) 8h 2 Pk(F ): The independence of the constants of the elements and faces can be seen by an ane equivalence argument. The two local components of the error estimator are estimated seperately. In a rst step, we consider the element oriented contribution and show local upper bounds for 0T . Using (4.6), an inverse inequality and taking into account that T = 0 on the boundary of T , we nd (0T )2 =

div p u~

2 C Z 1 (div u~ )2 dx h T h L2 (T) h2T

Z

T

?C hu~ h ; grad(T div u~ h )i dx = C r(T div u~ h ) T

p 0

p

? 0 ?C e ; grad(T div u~ h ) L2 (T) C e L2 (T) grad(T div u~ h ) L2 (T)

p 0

p

C h e L2 (T) div u~ h L2 (T) : T

Local upper bounds for 0F are obtained in a similar fashion. The basic tools to establish the bounds are (4.6), Green's formula and an inverse estimate. Let T1 and T2 be the elements such that @T1 \ @T2 = F . Then we can transform the L2 -norm on F into an integral on T1 and T2 :

(0F )2 =

p1 [hn; u~ i]

2 C Z 1 [hn; u~ i]2 d h J F h J hF A A L2 (F) F

2 Z D E X C u~ h ; grad([hn; u~ hi]J;Ti F ) + div u~ h [hn; u~ hi]J;Ti F dx A i=1 Ti

11


Here, the function [hn; u~ h i]J on F is extended by a continuous piecewise polynomial function [hn; u~ h i]J;Ti to T1 [ T2 such that

[hn; u~ hi]J;Ti

L2(Ti) ChT1=2i k[hn; u~ hi]J kL2(F) :

Now, we are in a position to give an upper bound for 0F by means of 0T and e0

2

0F : (0F )2 C X 1=2

div p u~

?1=2

p e0

+ h h h L2 (T ) 1=2 Ti hF L2 (Ti) Ti i hF i=1 Using for 0T , we nally obtain an upper bound for 0F in terms of

upper ,bound

p e0the L2 (T1 [T2 ) and thus

0T +

X

p

0F C

e0

L2 ( ) ; T F @T

where T is the union of all elements T 0 sharing at least one face with T . Keep in mind that C > 0 does not depend on the meshsize. This is the desired local lower bound for the curl-free part of the error. 5. Estimation of the -weakly solenoidal part of the error. To establish upper bounds for the \ -weakly solenoidal" part e? of the error, we basically follow the same ideas as before. Now, we have to apply Green's formula for the curloperator. Furthermore, the scalar locally de ned quasi-interpolation operator Phk will be replaced by a vector-valued counterpart Pkh . Again, nodal interpolation is not suitable, since the degrees of freedom located on edges cannot be extended to continuous functionals on H 1 ( ) [6]. Lemma 5.1. For T 2 Th , F 2 Fhint and E 2 Ehint, let DT1 , DF1 , and DE be given by

DE := [fT 2 Th ; E 2 Eh (T )g ; DT1 := [fDE ; E 2 Eh (T )g ; DF1 := [fDE ; E 2 Eh (F )g :

There exists a linear projection Pkh : H 1 ( ) \ H 0 (curl; ) 7! NDk;0( ; Th ) such that for all q 2 H 1( )

Pk q

d kqk 1 1 ;

curl Phk q

L2(T) d1 jqj H (DT ); 1 1

q ? Phk q

L2(T) d2h Hjqj(DF ) ; 3 T H 1 (DT1 ) h L2 (T)

q ? Pk q

d p 4 hF jqjH 1 (DF1 ) ; h L2 (F)

(5.1) (5.2) (5.3) (5.4)

where the constants d1; d2; d3; d4 > 0 do neither depend on q nor on T , but only on the shape regularity of the mesh Th . Proof. Details will only be given for the lowest order case k = 1, where the

degrees of freedom are plain path integrals along the edges of the elements. We adopt the notation wE , E 2 Eh , for the canonical basis function of ND 1( ; Th ) attached to edge E .

12


Pick any F 2 Fh with vertices fa1 ; a2; a3 g and edges fE1; E2; E3g = Eh (F ). With ei , i = 1; 2; 3, we abbreviate the length of the edge Ei and set s2 := e21 + e22 + e23 . Then de ne A := (aij ) 2 R3;3 by 03=4s2 ? e2 1=4s2 ? e2 1=4s2 ? e21?1 0e?1 0 0 1 1 1 3 2 A = 6 @1=4s2 ? e23 3=4s2 ? e22 1=4s2 ? e21 A @ 0 e?2 1 0 A (5.5) ? 1 2 2 2 2 2 2 1=4s ? e2 1=4s ? e1 3=4s ? e3 0 0 e3 and set F j (x) :=

3 X

k=1

akj (x ? ak ) ; x 2 T :

(5.6)

The sign takes into account the orientation of the edges, which will be irrelevant for our considerations. Straightforward compuations show

ZD

F

wEi () n; Fj ()

E

d = ij ; i; j = 1; 2; 3 :

(5.7)

Following this procedure, for each face F we construct three functions FE , E 2 Eh (F ), indexed by the edges of the face, so that they satisfy relations like (5.7). Next, to each E 2 Eh we assign one of its adjacent faces and call it FE 2 Fh . We have to comply with the restriction that for E 2 Eh? also FE 2 Fh?. Then we can de ne Pkh (q) :=

XZ D

E 2EhFE

q() n; FE ()

E

d wE :

(5.8)

By virtue of (5.7) this de nes a projection. Obviously boundary conditions are respected. From the formulas (5.5) and (5.6) we conclude that

F

F

? 2 ) Ch

E L1(F)

E L2(F) Ch?F 1 : F

Here C > 0 represents a generic constant that depends on the angles of the face only. Thanks to shape regularity of Th , this means that the constants can be chosen independently of F . As all FE , E 2 Eh (F ), are linear polynomials, the inverse estimate

F

F

? 1=2

E H? 21 (F) ChF E L2(F) ; E 2 Eh(F ) ;

(5.9)

follows from shape regularity. It yields for all E 2 Eh (F ), F 2 Fh ,

ZD

F

E

q() n; FE () d kqkH 21 (F)

FE

H? 21 (F)

ChF?1=2 kqkH 21 (F) Ch?1=2 kqkH 1 (DF1 ) :

Now, consider T 2 Th and recall that, again as a consequence of shape regularity, for vh 2 ND1( ; Th) we have (cf. [39])

2 X Z kvh L (T) ChT E hvh ; tE i ds : k2 2

E 2Eh (T)

13


Combining this with estimate (5.9) and the de nition (5.8) we end up with the estimate (5.1):

D E 2 R P F ChT q() n; E () d E 2Eh (T) FE Ch P h?1 kqk2 1 1

kPh qk2L2 (T)

T

F E 2Eh (T) E C kqk2H 1 (DT1 ) :

H (DF )

The nal estimate is possible, as shape regularity ensures that the number of elements sharing an edge is uniformly bounded. To get the remaining estimates (5.2){(5.4), we have to resort to ane equivalence techniques, mapping T to a reference simplex, where a Bramble-Hilbert argument (cf. [27]) is available. We will skip the technicalities and refer to [52] for an application of those tricks to edge elements. When we compare the properties of Phk and Pkh , we see that both operators are stable and posses the same approximations properties. The variational problem (3.4) yields an expression for the the energy norm of e? :

e?

2 = r(e?) = r(e? ? Pk e?) + r(Pk e?) h h E;

We apply Green's formula on the rst term on the right side and nd

r(e? ? Pkh e? ) =

X?

T 2Th

?

f ? u~ h ? curl( curl u~ h ); e? ? Pkh e? L2(T)

X ? [n curl u~ h ]J ; e? ? Pkh e? L2 (F) :

F 2Fhint

By means of the approximation properties (5.3), (5.4) of Pkh , it is easy to obtain upper bounds for r(e? ? Pkh e? )

2 !1=2 p

h X e? H1( )

pT (f ? u~ h ? curl( curl u~ h)

r(e? ? Pkh e? ) C L2 (T) T 2Th 0 11=2 X +C@ k[n curl u~ h ]J k2L2 (F) A pe? H1 ( ) : F 2Fhint

Due to the Galerkin orthogonality, the second part r(Pkh e? ) is equal to zero, if the iteration error is zero. The estimate for this term involves the stability (5.1), (5.2) of the operator Pkh ? ? r(Pkh e? ) = (uh ? u~ h ); Pkh e? L2 ( ) + curl(uh ? u~ h ); curl Pkh e? L2 ( )

kuh ? u~ h kE; Pkh e? E; C kuh ? u~ h kE; e? H 1 ( ) :

At this point we have to resort to the regularity assumption 3.1 to switch from H 1( )norms back to the relevant energy norm. Taking assumption 3.1 for granted we can the upper estimates for r(e? ? Pkh e? ), r(Pkh e? ). Thus, an upper bound for

combine

e? E; is provided by the sum of 1(1) , 2 and it(1) .

14


To obtain a lower bound for e? E; , we consider the local contributions of 1(1) separately. The global error is localized by means of the bubble functions T T and set and F introduced in Section 4. We start with an upper bound for 1;1 jh := h f ? u~ h ? curl( curl u~ h ) T )2 jT (1;1 22 ~ ~ = k f ? u ? curl ( curl u ) k h h h L (T) 2 hT Z C hh f ? u~ h ? curl( curl u~ h ); jh T i dx : T

Observing that (T jh )j@T = 0 and applying Green's formula yields

Z T )2 jT (1;1 h2T C r(T jh ) ? T hf ? h f ; T jh i dx C kekE;T kT jh kE;T + kf ? h f kL2 (T) kT jh kL2 (T) ! 1 C h?T 1 kekE;T + p kf ? h f kL2 (T) kpjh kL2 (T) ; jT

and we get T C 1;1

T kekE;T + 1;2

:

We remark, that in general T jh is not an element of H ?0 (curl; ). As a consequence, T involves not only the solenoidal part of the error e? but the the upper bound for 1;1 total error e. The estimate for an upper bound for 1F follows the same lines. Here, we use the bubble function F . The face contribution 1F involves the two adjacent elements T1 and T2 , @T1 \ @T2 . We extend the jump [n curl u~ h ]J de ned on the face F to a polynomial function de ned on T1 and T2 such that for 1 i 2

~

[n curl uh]J;Ti L2(Ti) ChT1=2i k[n curl u~ h]J kL2 (F) :

Setting jh jTi := [n curl u~ h ]J;Ti and using the norm equivalence (4.6), we nd

A (1F )2 = k[n curl u~ ] k2 C Z h[n curl u~ ] ; j i d : (5.10) h J L2 (F) hJ h F hF F Green's formula applied on (5.10) yields cA (1F )2 Z (h? curl curl u~ ; j i + h curl u~ ; curl( j )i ) dx h h F h F h hF T1 [T2 = ?r(F jh ) + (f ? u~ h ? curl( curl u~ h ; F jh )L2 (T1 [T2 ) C kekE;T1 [T2 kT jh kE;T1 [T2 T1 T1 ?1 T2 T2 + 1;2) + hT2 (1;1 + 1;2) + C kpT jh kL2 (T1 [T2 ) h?T11 (1;1 ?1 T1 + T1 ) + h?1((kek T2 + T2 ) A F : C hT1 (kekE;T1 + 1;1 + 1;2 1;1 1;2 1 T2 E;T2

15


T with the last inequality provides an upper bound for Combining the bound for 1;1 T1 , T2 1F in terms of the local energy norm of the total error and 1;2 1;2

T1 + T2 : 1F C kekE;T1 + kekE;T2 + 1;2 1;2

6. Numerical experiments. In order to demonstrate the performance of the

proposed error estimator for a variety of settings, we provide several numerical examples. Throughout we use lowest order edge elements on an unstructured tetrahedral grid. The stiness matrix and load vector corresponding to (2.3) are computed using Gaussian quadrature of order 5. Interpolation of boundary values is of the same order. The linear systems of equations are approximately solved by means of a multigrid preconditioned conjugate gradient method [14,15]. The iterations are terminated, when the Euclidean norm of the algebraic residual for the current iterate is less than 10?10 times the Euclidean norm of the vector on the right hand side. Thus, the truncation error it can be neglected. In all cases the local error estimator (3.10) in the form of (3.11) was used to provide the local and global error estimates. Most of the examples were chosen so that the exact solution and, hence, the energy norm T of the true error were available for each element T . To gauge the quality of the error estimator in particular settings we evaluate dierent functionals: The eectivity index " := ^=, which gives the ratio between the estimated P 2 and the true discretization error. Here, := kekE; and ^ := T ^T2 . This quantity merely re ects the quality of the global estimate. For a good error estimator, the eectivity index is to approach a constant rapidly as re nement proceeds. We point out that, since we can only expect equivalence of the estimated energy of the error and its true energy, the eectivity index may be far o the ideal value 1. The proportion (1) of \incorrect decisions", measuring how much re nement controlled by the actual estimator diers from re nement based upon an \ideal" estimator. Consider the set of elements marked for re nement by the error estimator

(

)

X 2 (6.1) ^ ; A^ := T > n1 T T 2Th T where = 0:95 and nT = #Th , and the set of elements that should have been marked 2 Th : ^T2

(

)

X 2 A := T 2 Th : T2 > n1 : T T 2Th T

Then we de ne

n o 1 (1) ^ ^ := n # (A \ C A) [ (CA \ A) : T If the estimator performs satisfactorily, we expect (1) to stay bounded well below 1 as re nement proceeds.

16


A measure (2) for the \severity of incorrect decisions", which gives crude

information how much smaller the discretization error might have been, if an \ideal" estimator had steered local re nement. Since lowest order edge elements provide a rst order approximation in the energy norm, we expect a reduction of the error on a single element by local re nement roughly like T2 ! 14 T2 . Thus the total error on the adaptively re ned mesh can be expressed as 2 = 1 X 2 + X 2 : new 4 T 2A^ T T 2C A^ T If A is substituted for A^, a case we regard as \optimal" local re nement, we 2 . end up with an error opt 2 ? 2 j jnew opt (2) := 2 opt

:

If (2) stays neatly bounded, the error estimator performs satisfactorily. To be able to zero in on singularities, the error estimator must detect local errors. We de ne the quantity (3) by

v u 2 2 X ^T2 u (3) T t j T j k^2k 1 ? k2k 1 : := L L T 2TH

with

k2 kL1 =

X T 2TH

T2 j T j ; k^2kL1 =

X T 2TH

^T2 j T j :

This quantity will be big, if the estimator fails to tell the approximate spatial distribution of the discretization error. The gain from adaptive re nement is illustrated by plotting the discretization error versus total number of degrees of freedom both for uniform and adaptive re nement. All numerical experiments are conducted on uniformly re ned meshes, some of them also on meshes generated by adaptive re nement. The latter relies on an averaging strategy, which singles out the elements in the set A^ from (6.1) for re nement. When we do so, we also monitor the decrease of the true error against the degrees of freedom for both uniform and adaptive cases in order to assess the gain of adaptivity. The numerical experiments 1 to 5 are carried out on the unit cube := ]0; 1[3. Dirichlet boundary conditions are applied on @ . In order to be able to evaluate the true discretization errors, we choose boundary data and right hand sides such that an analytical expressions for the solutions of the continuous problems are available. In each case we start with a coarse grid (level 0) consisting of 6 tetrahedrons, which is re ned uniformely up to level 5. In our rst experiment the coecients and are kept constant all over the domain; is always set to 1. In this situation the regularity assumption 3.1 is ful lled. Dierent values for are taken into account, because in the case of implicit timestepping will be scaled by the size of the timestep. Therefore, it is essential that the error estimator is robust with respect to the relative scaling of and . The

17


solution is rather smooth and is given by u = (0; 0; sin(x1)). Consequently there are no particular local features to be detected. The results are reported in table 6.1 and bear out a decent performance of the estimator for this benign setting. We also observe that the error estimator is not severely aected by dierent values for . Level Eectivity = 10?4 = 10?2 = 1:0 = 102 = 104 (1) = 10?4 = 10?2 = 1:0 = 102 = 104 (2) = 10?4 = 10?2 = 1:0 = 102 = 104 (3) = 10?4 = 10?2 = 1:0 = 102 = 104

0

1

index 4.05 8.05 4.05 8.05 3.01 7.64 2.29 4.27 2.33 4.23

2

3

4

5

8.18 8.17 7.78 4.70 4.66

8.24 8.23 7.84 4.95 4.86

8.27 8.27 7.87 5.20 4.95

8.29 8.29 7.89 5.26 5.00

0.33 0.33 0.33 0 0

0.17 0.17 0.25 0.42 0.44

0.12 0.12 0.18 0.088 0.11

0.1 0.1 0.14 0.14 0.15

0.1 0.1 0.13 0.15 0.16

0.085 0.086 0.13 0.16 0.16

0.28 0.28 0.29 0 0

0.41 0.41 0.056 0.091 0.26

0.057 0.056 0.0039 0.014 0.078

0.033 0.033 0.013 0.0085 0.074

0.014 0.014 0.034 0.0039 0.065

0.0062 0.0062 0.038 0.0034 0.047

0.2 0.2 0.2 0.22 0.21

0.078 0.078 0.084 0.081 0.08

0.026 0.026 0.025 0.024 0.024

0.0079 0.0079 0.0077 0.0086 0.0082

0.0026 0.0026 0.0026 0.0032 0.0029

0.00089 0.00089 0.00089 0.0012 0.0011

Table 6.1

Quality measures for the residual based error estimator on the unit cube (Exp. 1).

For our second experiment boundary data and right hand sides are chosen such that the solenoidal solution u = curl(sin(x2 x3); cos(x1x3 ); sin(x1 x2)) is generated. We included this experiment to study how the error estimator responds to a smooth divergence-free solution. We refer to table 6.2 for the results. Little dierence compared to the previous experiment can be seen. In the third experiment we generate a smooth irrotational solution u = grad(xyz ). The coecients and are chosen like in the previous experiments. In this case the irrotational part of the error is the only remaining component. Values for the dierent quality measures are given in table 6.3. They show that the error estimator is insensitive to irrotational solutions. Our fourth experiment deals with constant = 1, whereas the other coecient is varying on the domain: (x) = 1:5 + sin(2x1 ) sin(2x2) sin(2x3 ). We choose the smooth solution = (0; 0; sin(x1)). In this case the energy norm features a certain anisotropy, but the coecients are still smooth.

18

Beck, Hiptmair, Hoppe, Wohlmuth Level Eectivity = 10?4 = 10?2 = 1:0 = 102 = 104 (1) = 10?4 = 10?2 = 1:0 = 102 = 104 (2) = 10?4 = 10?2 = 1:0 = 102 = 104 (3) = 10?4 = 10?2 = 1:0 = 102 = 104

0

1

2

3

4

5

index 5.26 5.25 3.82 1.86 1.70

5.94 6.23 6.08 3.14 3.17

7.07 7.08 6.92 3.99 3.98

7.51 7.50 7.34 4.46 4.34

7.72 7.71 7.55 4.96 4.49

7.84 7.84 7.66 5.09 4.57

0.33 0.33 0.5 0.83 0.83

0.19 0.15 0.17 0.42 0.4

0.13 0.13 0.14 0.25 0.25

0.1 0.1 0.1 0.17 0.16

0.1 0.1 0.1 0.17 0.13

0.1 0.1 0.1 0.17 0.12

0.31 0.31 0.055 0.74 0.9

0.043 0.062 0.084 0.39 0.36

0.054 0.059 0.048 0.11 0.18

0.037 0.036 0.029 0.068 0.1

0.041 0.041 0.039 0.015 0.069

0.044 0.044 0.042 0.01 0.057

0.1 0.1 0.12 0.32 0.35

0.06 0.065 0.065 0.11 0.12

0.033 0.033 0.032 0.037 0.04

0.013 0.013 0.012 0.01 0.011

0.0044 0.0044 0.0043 0.0032 0.0034

0.0015 0.0015 0.0015 0.001 0.001

Table 6.2

Measures for quality of residual based error estimator for Exp. 2

How the error estimator behaves can be seen from table 6.4. All the observations made in the previous experiments remain true. The fth experiment exchanges the roles of the coecients. Now is set to 1 throughout the domain and is given by (x) = 1:5 + sin(2x1) sin(2x2) sin(2x3 ). Again the solution is u = (0; 0; sin(x1)). See table 6.5 for information about the performance of the error estimator. We remark that the performance remains satisfactory. Our sixth experiment is again carried out on the unit cube := ]0; 1[3 with 1, but we enforce a vanishing zero{order term on part of the domain. Thus we depart from the situation where the regularity assumption holds. As far as the coecients are concerned, this experiment comes fairly close to the arrangements in realistic eddy current computations. In particular, we choose as follows:

1 : max fjx ? 0:5j; jx ? 0:5j; jx ? 0:5jg 0:25 1 2 3 (x) = 0 : elsewhere

Boundary data and right hand side are again adjusted to produce the smooth solution u = (0; 0; sin(x1)). Of course, in this respect we fail to capture the usual singular behavior of the electric eld at the edges of the conductor.

19

Residual based error estimators Level

0

Eectivity = 10?4 = 10?2 = 1:0 = 102 = 104 (1) = 10?4 = 10?2 = 1:0 = 102 = 104 (2) = 10?4 = 10?2 = 1:0 = 102 = 104 (3) = 10?4 = 10?2 = 1:0 = 102 = 104

1

index 4.21 4.70 4.21 4.70 4.21 4.70 4.22 4.68 4.22 4.66

2

3

4

5

4.90 4.90 4.89 4.87 4.84

4.98 4.98 4.98 4.96 4.92

5.01 5.01 5.01 5.01 4.96

5.03 5.03 5.03 5.03 4.99

0.5 0.5 0.5 0.33 0.33

0.042 0.042 0.042 0.1 0.12

0.13 0.13 0.13 0.13 0.14

0.14 0.14 0.14 0.14 0.13

0.14 0.14 0.14 0.14 0.14

0.14 0.14 0.14 0.14 0.14

0.46 0.46 0.46 0.32 0.32

0.054 0.054 0.054 0.057 0.083

0.042 0.042 0.047 0.04 0.051

0.037 0.037 0.037 0.036 0.026

0.034 0.034 0.034 0.032 0.019

0.023 0.023 0.023 0.023 0.016

0.24 0.24 0.24 0.24 0.24

0.095 0.095 0.095 0.082 0.077

0.032 0.032 0.032 0.03 0.028

0.01 0.01 0.01 0.0098 0.0088

0.0034 0.0034 0.0034 0.0033 0.003

0.0011 0.0011 0.0011 0.0011 0.001

Table 6.3

Measures for the quality of the residual based error estimator on the unit cube (Exp. 3). Level

0

1

2

3

4

5

(1) (2) (3)

5.59 0.67 0.32 0.26

9.80 0.31 0.28 0.077

11.20 0.21 0.072 0.028

11.46 0.2 0.1 0.0095

11.57 0.19 0.11 0.0033

11.61 0.19 0.11 0.0011

Table 6.4

Measures for the quality of the residual based error estimator on the unit cube (Exp. 4). Level

0

1

2

3

4

5

(1) (2) (3)

2.92 0.33 0.29 0.22

7.83 0.35 0.071 0.085

8.18 0.19 0.0027 0.025

8.97 0.18 0.025 0.0082

11.47 0.17 0.083 0.0032

18.37 0.18 0.15 0.0014

Table 6.5

Measures for the quality of the residual based error estimator on the unit cube (Exp. 5).

20


The results of the computations are recorded in table 6.6. All measures for the quality of the error estimator reveal a awless performance. Level

0

1

2

3

4

(1) (2) (3)

5.18 0.17 0.099 0.12

7.22 0.081 0.027 0.05

7.46 0.066 0.011 0.017

7.52 0.051 0.0026 0.0058

7.53 0.047 0.0031 0.0021

Table 6.6


Owing to the smooth solution adaptive re nement does not really pay o in this situation (cf. gure 6.1). Nevertheless, we report the quality measures in table 6.7 to show that non-uniform meshes do not make a dierence. Level

0

1

2

3

4

5

6

(1) (2) (3)

5.18 0.17 0.099 0.12

7.29 0.11 0.083 0.048

7.89 0.15 0.024 0.032

7.89 0.17 0.1 0.014

7.81 0.14 0.11 0.0054

7.73 0.11 0.039 0.0036

7.88 0.17 0.076 0.0022

Table 6.7

Measures for the quality of the residual based error estimator on the unit cube in the adaptive case (Exp. 6).

true error

uniform adaptive 0.5

0.05 102

103

104

105

106

degrees of freedom Fig. 6.1. True error for uniform and adaptive mesh re nement on the unit cube (Exp. 6).

For the seventh experiment we employ the domain and coecients of the previous

21


one. But now we enforce homogeneous Dirichlet boundary conditions u n = 0 on the boundary and a smooth right hand side f = (1; 1; 1). Thus the vector eld will be non-smooth and cannot be described analytically. Now, all the features of an actual eddy current problem are present. To estimate the true errors, we carried out two re nement steps more than reported in tables 6.8 and 6.9, respectively, and compare the discrete solutions to those obtained on the nest levels. The results are collected in table 6.8 for uniform re nement and in table 6.9 for adaptive re nement. Evidently, the additional singularities in the solution hardly aect the error estimator in either case. Level

0

1

2

3

(1) (2) (3)

15.19 0.1 0.24 0.076

10.70 0.094 0.016 0.046

9.89 0.093 0.051 0.021

10.34 0.1 0.079 0.0083

Table 6.8


Level

0

1

2

3

4

5

(1) (2) (3)

15.15 0.1 0.24 0.076

11.03 0.29 0.046 0.04

10.41 0.22 0.23 0.024

10.81 0.12 0.072 0.015

10.52 0.18 0.079 0.0075

11.86 0.14 0.14 0.0041

Table 6.9

Measures for the quality of the residual based error estimator on the unit cube in the adaptive case (Exp. 7).

Surprisingly, the singularities do hardly reward adaptive re nement, as can be seen from gure 6.2. However, as the right tails of the curves indicate, adaptivity might pay o on higher levels. But due to lacking computer resources we could not proceed with grid re nement in our experiment. The eighth experiment deals with an edge singularity of the eld. We use a non{ convex \L-shaped" domain :=] ? 1; 1[3 n [0; 1]2 [?1; 1]. The coarsest grid comprises 52 tetrahedra. We set and to 1 and employ such boundary conditions and right hand side 2 3 that the solution is given (in polar coordinates) by u = grad(r sin( 32 ). The gradient eld u is both irrotational and divergence{free and does not even belong to H 1 ( ). How this aects the error estimator is conveyed by table 6.10 for the case of uniform re nement and table 6.11 for local grid adaptation. The numbers illustrate that the error estimator can be relied upon even under these extreme circumstances. Now that the vector eld is singular along the concave edge of the L-shaped domain, local mesh re nement should provide favourable grids. Indeed, g. 6.3 reveals a clearly superior performance in the adaptive case. Figure 6.4 gives a view of both the initial and an adaptively re ned triangulation; observe how the grid adaption concentrates on the concave edge of the domain.

22


uniform adaptive true error

0.1

0.01

102

103

104

105

degrees of freedom Fig. 6.2. True error for uniform and adaptive mesh re nement on the unit cube (Exp. 7).

Level

0

1

2

3

4

(1) (2) (3)

3.39 0.27 0.17 0.11

3.73 0.1 0.079 0.07

3.85 0.047 0.07 0.047

3.94 0.029 0.018 0.032

3.99 0.019 0.013 0.022

Table 6.10

Measures for the quality of the residual based error estimator on the L-shaped domain (Exp. 8). Level

0

1

2

3

4

5

6

(1) (2) (3)

3.39 0.27 0.17 0.11

3.67 0.18 0.11 0.065

3.82 0.12 0.08 0.11

4.05 0.12 0.018 0.037

4.18 0.11 0.012 0.02

4.28 0.096 0.021 0.0056

4.37 0.088 0.025 0.0027

Table 6.11

Measures for the quality of the residual based error estimator on the L-shaped domain in the adaptive case (Exp. 8).

7. Conclusion. In the present paper we have designed a local a posteriori error estimator for H (curl; ){elliptic problems by taking into account the dual norm of

the residual. We could show that this estimator is ecient, i.e. apart from scaling it provides a local lower bound for the energy norm of the error. `Under additional assumptions we could establish that it is also reliable in the sense that we can also obtain a global upper bound. However, the assumptions are hardly ever met in realistic settings. Nevertheless, the numerical results oer strong evidence that the error

23


uniform adaptive

true error

0.5

0.05

102

103

104

105

degrees of freedom Fig. 6.3. True error for uniform and adaptive mesh re nement on the L-shaped domain (Exp. 8).

Fig. 6.4. The L-shaped domain of Exp. 8. On the left hand side the initial triangulation is shown; the other gure displays a cross section of the grid after ve adaptive re nement steps.

estimator performs excellently beyond the scope of the theoretical analysis. Hence, it would be desirable to extend the rigorous treatment to the case of discontinuous coecients. REFERENCES [1] B. Achchab, A.Agouzal, J. Baranger, and J. Maitre, Estimateur d'erreur a posteriori

24 [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28]

Beck, Hiptmair, Hoppe, Wohlmuth hierarchique. Application aux elements nis mixtes, IMPACT Comput. Sci. Engrg., 1 (1995), pp. 3{35. R. Albanese and G. Rubinacci, Fomulation of the eddy{current problem, IEE Proc. A, 137 (1990), pp. 16{22. , Analysis of three dimensional electromagnetic leds using edge elements, J. Comp. Phys., 108 (1993), pp. 236{245. A. Alonso and A. Valli, Some remarks on the characterization of the space of tangential traces of H (rot; ) and the construction of an extension operator, Manuscripta mathematica, 89 (1996), pp. 159{178. H. Ammari, A. Buffa, and J.-C. Nedelec, A justi cation of eddy currents model for the Maxwell equations, tech. rep., IAN, University of Pavia, Pavia, Italy, 1998. C. Amrouche, C. Bernardi, M. Dauge, and V. Girault, Vector potential in three{ dimensional nonsmooth domains, Tech. Rep. 96{04, IRMAR, Rennes, France, 1996. D. Arnold, A. Mukherjee, and L. Pouly, Locally adapted tetrahedral meshes using bisection, SIAM Journal on Scienti c Computing, (1998). submitted. I. Babuska and W. Rheinboldt, Error estimates for adaptive nite element computations, SIAM J. Numer. Anal., 15 (1978), pp. 736{754. , A posteriori error estimates for the nite element method, Int. J. Numer. Methods Eng., 12 (1978), pp. 1597{1615. R. Bank, PLTMG: A Software Package for Solving Elliptic Partial Dierential Equations, User's Guide 6.0, SIAM, Philadelphia, 1990. R. Bank, A. Sherman, and A. Weiser, Re nement algorithm and data structures for regular local mesh re nement, in Scienti c Computing, R. Stepleman et al., ed., vol. 44, IMACS North{Holland, Amsterdam, 1983, pp. 3{17. R. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial dierential equations, Math. Comp., 44 (1985), pp. 283{301. E. Bansch, Local mesh re nement in 2 and 3 dimensions, IMPACT Comput. Sci. Engrg., 3 (1991), pp. 181{191. R. Beck, P. Deuflhard, R. Hiptmair, R. Hoppe, and B. Wohlmuth, Adaptive multilevel methods for edge element discretizations of Maxwell's equations, Tech. Rep. SC 97{66, ZIB Berlin, 1997. To appear in Surveys for Mathematics in Industry. R. Beck and R. Hiptmair, Multilevel solution of the time-harmonic Maxwell equations based on edge elements, Tech. Rep. SC-96-51, ZIB Berlin, 1996. To appear in Int. J. Num. Meth. Eng. J. Bey, Tetrahedral grid re nement, Computing, 55 (1995), pp. 355{378. F. Bornemann, An adaptive multilevel approach to parabolic equations I. General theory and 1D-implementation, IMPACT Comput. Sci. Engrg., 2 (1990), pp. 279{317. , An adaptive multilevel approach to parabolic equations II. Variable-order time discretization based on a multiplicative error correction, IMPACT Comput. Sci. Engrg., 3 (1991), pp. 93{122. F. Bornemann, B. Erdmann, and R. Kornhuber, A posteriori error estimates for elliptic problems in two and three spaces dimensions, SIAM J. Numer. Anal, 33 (1996), pp. 1188{ 1204. A. Bossavit, Mixed nite elements and the complex of Whitney forms, in The Mathematics of Finite Elements and Applications VI, J. Whiteman, ed., Academic Press, London, 1988, pp. 137{144. , A rationale for edge elements in 3D eld computations, IEEE Trans. Mag., 24 (1988), pp. 74{79. , Solving Maxwell's equations in a closed cavity and the question of spurious modes, IEEE Trans. Mag., 26 (1990), pp. 702{705. , Electromagn etisme, en vue de la modelisation, Springer{Verlag, Paris, 1993. , Computational Electromagnetism. Variational Formulation, Complementarity, Edge Elements, no. 2 in Academic Press Electromagnetism Series, Academic Press, San Diego, 1998. D. Braess and R. Verfurth, A posteriori error estimators for the Raviart-Thomas element, SIAM J. Numer. Anal., 33 (1996), pp. 2431{2445. C. Carstensen, A posteriori error estimate for the mixed nite element method, Math. Comp., 66 (1997), pp. 465{476. P. Ciarlet, The Finite Element Method for Elliptic Problems, vol. 4 of Studies in Mathematics and its Applications, North-Holland, Amsterdam, 1978. M. Clemens, R. Schuhmann, U. van Rienen, and T. Weiland, Modern Krylov subspace methods in electromagnetic eld computation using the nite integration theory, ACES J.


25

Appl. Math., 11 (1996), pp. 70{84. [29] M. Clemens and T. Weiland, Transient eddy current calculation with the FI{method, in Proc. CEFC '98, IEEE, 1998. submitted to IEEE Trans. Mag. [30] P. Clement, Approximation by nite element functions using local regularization, RAIRO Anal. Numer., 2 (1975), pp. 77{84. [31] M. Costabel and M. Dauge, Singularities of electromagnetic elds in polyhedral domains, Tech. Rep. 97{19, IRMAR, Rennes, September 1997. [32] M. Costabel, M.Dauge, and S. Nicaise, Singularities of Maxwell interface problems, Tech. Rep. 98-24, IRMAR, Rennes, France, May 1998. [33] H. Dirks, Quasi{stationary elds for microelectronic applications, Electrical Engineering, 79 (1996), pp. 145{155. [34] P. Dular, J.-Y. Hody, A. Nicolet, A. Genon, and W. Legros, Mixed nite elements associated with a collection of tetrahedra, hexahedra and prisms, IEEE Trans Magnetics, MAG-30 (1994), pp. 2980{2983. [35] K. Erikson, D. Estep, P. Hansbo, and C. Johnson, Introduction to adaptive methods for dierential equations, Acta Numerica, 4 (1995), pp. 105{158. [36] K. Eriksson and C. Johnson, An adaptive nite element method for linear elliptic problems, Math. Comput., 50 (1988), pp. 361{383. [37] V. Girault and P. Raviart, Finite element methods for Navier{Stokes equations, Springer{ Verlag, Berlin, 1986. [38] E. Hairer and G. Wanner, Solving Ordinary Dierential Equations II. Sti and DierentialAlgebraic Problems, Springer-Verlag, Berlin, Heidelberg, New York, 1991. [39] R. Hiptmair, Multigrid method for Maxwell's equations, Tech. Rep. 374, Institut fur Mathematik, Universitat Augsburg, 1997. To appear in SIAM J. Numer. Anal. [40] , Canonical construction of nite elements, Math. Comp., (1999). To appear. [41] R. Hoppe and B. Wohlmuth, Adaptive multilevel iterative techniques for nonconforming nite element discretizations, East-West J. Numer. Math., 3 (1995), pp. 179{197. , A comparison of a posteriori error estimators for mixed nite elements, Tech. Rep. [42] 350, Math.{Nat. Fakultat, Universitat Augsburg, 1996. To appear in Math. Comp. [43] , Element-oriented and edge-oriented local error estimators for nonconforming nite element methods, M 2 AN Math. Modelling and Numer. Anal., 30 (1996), pp. 237{263. [44] , Adaptive multilevel techniques for mixed nite element discretizations of elliptic boundary value problems, SIAM J. Numer. Anal., 34 (1997), pp. 1658{1687. [45] , Hierarchical basis error estimators for Raviart-Thomas discretizations of arbitrary order, in Finite Element Methods: Superconvergence, Post-processing and A Posteriori Estimates, P. et al., ed., Marcel Dekker, New York, 1997, pp. 155{167. [46] J. Maubach, Local bisection re nement for n-simplicial grids generated by re ection, SIAM J. Sci. Comp., 16 (1995), pp. 210{227. [47] P. Monk, A mixed method for approximating Maxwell's equations, SIAM J. Numer. Anal., 28 (1991), pp. 1610{1634. [48] , Analysis of a nite element method for Maxwell's equations, SIAM J. Numer. Anal., 29 (1992), pp. 714{729. [49] J. Nedelec, Mixed nite elements in R3 , Numer. Math., 35 (1980), pp. 315{341. [50] E. Ong, Hierarchical basis preconditioners for second order elliptic problems in three dimensions, PhD thesis, Dept. of Math., UCLA, Los Angeles, CA, 1990. [51] P. Oswald, Multilevel nite element approximation, Teubner Skripten zur Numerik, B.G. Teubner, Stuttgart, 1994. [52] J. P. Ciarlet and J. Zou, Fully discrete nite element approaches for time{dependent Maxwell equations, Tech. Rep. TR MATH{96-31 (105), Department of Mathematics, The Chinese University of Hong Kong, 1996. To appear in Num. Math. [53] L. R. Scott and Z. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Math. Comp., 54 (1990), pp. 483{493. [54] R. Verfurth, A posteriori error estimation and adaptive mesh-re nement techniques, J. Comp. Appl. Math., 50 (1994), pp. 67{83. [55] , A Review of A Posteriori Error Estimation and Adaptive Mesh{Re nement Techniques, Wiley{Teubner, Chichester, Stuttgart, 1996. [56] H. Whitney, Geometric Integration Theory, Princeton Univ. Press, Princeton, 1957. [57] J. Zhu and O. Zienkiewicz, Adaptive techniques in the nite element method, Commun. Appl. Numer. Methods, 4 (1988), pp. 197{204.