Hierarchical Error Estimator for Eddy Current ... - Semantic Scholar

Hierarchical Error Estimator for Eddy Current Computation R. Beck

R. Hiptmairy

B. Wohlmuthz

October 27, 1999 Abstract We consider the quasi-magnetostatic eddy current problem discretized by means of lowest order curl-conforming finite elements (edge elements) on tetrahedral meshes. Bounds for the discretization error in the finite element solution are desirable to control adaptive mesh refinement. We propose a local a-posteriori error estimator based on higher order edge elements: The residual equation is approximately solved in the space of p-hierarchical surpluses. Provided that a saturation assumption holds, we show that the estimator is both reliable and efficient. Key words. Edge elements, a posteriori error estimator, hierarchical error estimator MSC 1991. 65N15, 65N30, 65N5

1 Introduction The eddy current model arises from Maxwell’s equations as a magneto-quasistatic approximation by dropping the displacement current [3, 22]. This is reasonable for low-frequency, high-conductivity applications like electrical machines. A wealth of different formulations have been proposed [1, 8]. We single out one that zeros in on the magnetic vector potential = ( ; t) as primary unknown [18]. Then we end up with the degenerate parabolic initial-boundary value problem

A Ax

@t(A) + curl  curlA = ?j

An A(:; 0)

= 0 = A0

in

on ? := @

in :

(1)

Here,  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 artificial boundary sufficiently removed from the region of interest. This is commonplace in engineering simulations [8, 18].  ZIB-Berlin, e-mail:[email protected] y Sonderforschungsbereich 328, Universit¨at

T¨ubingen, e-mail: [email protected]. This work was supported by DFG as part of SFB 382 z Mathematisches Institut, Universit¨at Augsburg, e-mail: [email protected]

1

Further,  2 L1 ( ) denotes the bounded uniformly positive inverse of the magnetic permeability (magnetic susceptibility). We confine ourselves to linear isotropic media, i.e.  is a scalar function of the spatial variable 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 . Usually, there is a crisp distinction between conducting regions C , where  is bounded away from zero, and insulating regions, where  = 0. We will take for granted that    > 0, for some bound  > 0, wherever  6= 0. Often, the material parameters vary only moderately inside and outside the conductor. The right hand side = ( ; t) is a time-dependent vectorfield in 2 ( ), which represents the source current. For physical reasons div (t; :) = 0 a.e. in and for all times t. 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 figure 1.

x

j jx

j

L

j



C

 > 0 > 0

Figure 1: A model problem for eddy current computation (cf. [14, Ch. 8])

A B curlA

It is important to note that the vector potential lacks physical meaning. The . This is why we can really interesting quantity is the magnetic induction = use an ungauged formulation as in (1), which does not impose a constraint on div outside C . Obviously, this forfeits uniqueness of the solution in parts of the domain where  = 0, but the solution for remains unique everywhere. Inside the conductor, where  > 0, we get a unique . For the sake of stability, timestepping schemes for (1) have to be L-stable [26]. 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

A

u

A

B

curl  curlu + u un

= f = 0

in

on @ :

A

(2)

In this context, denotes the new approximation to to be computed in the current timestep, and depends on and the approximation of in the previous timestep. Note that we can still assume div = 0 outside C . The coefficient agrees with 

f

j

f

2

A

except for a scaling by the length of the current timestep; Accordingly, 0 <    a.e. in C . Problem (2) cast in weak form yields a variational problem in the Hilbert space ( ; ) of 2 ( )-vectorfields whose is square integrable: Find 2 ; ) such that for all 2 0 ( ; ) 0(

H curl

H curl

L

curl

q H curl ( curlu; curlq)L ( ) + ( u; q)L ( ) = (f ; q)L ( ) : 2

2

2

u

(3)

A subscript 0 indicates that vanishing tangential traces on @ are imposed on the fields. For uniformly positive a.e. in the Lax-Milgram lemma guarantees existence and uniqueness of a solution of (3). If = 0 on sets of positive measure, we can still expect a unique solution in some quotient space. It is now generally accepted that an appropriate finite element discretization of ; )-conforming schemes that merely enforce the (3) should rely on genuine ( typical tangential continuity of the vector potential across interelement boundaries [2, 12, 13]. For simplicial meshes ( ; )-conforming finite elements of arbitrary polynomial order were first introduced by N´ed´elec [31], generalizing the lowest order Whitney elements [34]. Similar schemes are also known for other shapes of elements [23, 31]. For all of them standard a priori error bounds can be established as was done in [16, 28, 29]. Ultimately, the discretization of (3) by these so-called edge elements leads to a large sparse linear system of equations for the degrees of freedom of the finite element space. Usually, an approximate solution can only be obtained by iterative methods [17, 27]. If stands for the exact solution of (3) and h for the discrete finite element solution, we seek to obtain information about the spatial discretization error := ? ~ h . This is desirable for two reasons: Firstly, knowledge about can be used for adaptive local refinement and coarsening of the underlying triangulation. Since the fields feature strong singularities at reentrant corners [19] and at irregular materia l interfaces [20], a finer the mesh in these zones is recommended. Secondly, information about the accuracy of the finite element solution is also required to balance the spatial and temporal errors in the context of adaptive timestepping for the original parabolic problem [9, 10]. Techniques of a-posteriori error estimation have been thoroughly researched for many types of problems, in particular second order elliptic problems. Among the most widely used approached are residual based error estimators and hierarchical error estimators. We refer to the monograph by Verf¨urth [33] and the survey articles by Bornemann et al. [11] and Johnson et al. [25]. In [6] a residual based a-posteriori error estimator for the very edge element approximation of the eddy current problem is studied. The same was done for high-frequency scattering problems, i.e. the full Maxwell’s equations, in [30]. To the knowledge of the authors, hierarchical error estimator for quasistatic electromagnetics problems have not been investigated before, though they are widely used for ordinary elliptic problems [5, 21, 24, 32]. The paper is organized as follows: In the next section we introduce edge elements of lowest and second order. Then we discuss the principles of hierarchical error estima-

H curl H curl

u

u

e

3

e u u

tion and present the concrete hierarchical error estimator. It turns out that it is essential to pay special attention to -free vectorfields in the process of hierarchical decoupling and localization. Finally, we report on a number of numerical experiments that examine the performance of the hierarchical error estimator for some model proble ms.

curl

2 Edge elements

curl

-conforming finite elements [31] on families of shapeWe employ N´ed´elec’s regular simplicial triangulations (in the sense of [15]). Usually, the meshes are generated by repeated refinement of an initial coarse triangulation T0 . We may use the standard “red-green” refinement process (cf. [7, 11]). We require that the boundary of the conductor C is resolved by all meshes, that is, C is always the union of elements. We denote by k ( ; Th ) the global space of edge elements of order k 2 N built upon a simplicial mesh Th of . Compliance with homogeneous Dirichlet boundary conditions is taken for granted. For any tetrahedron T the local spaces are given by

ND

NDk (T ) := (Pk?1(T ))3 + p 2 (Pk (T ))3 ; hp(x); xi = 0; 8x 2 T ;

where Pk (T ) designates the spaces of polynomials of degree  k over T . For the lowest order case k = 1 this leads to the representation 1 (T ) = f 7! +  ; ; 2 R3g. Then the global space is obtained by prescribing degrees of freedom (d.o.f.) that ensure tangential continuity of the discrete vectorfields. In the lowest order case k = 1 the d.o.f. are given by path integrals along oriented edges of elements [31]:

ND

x ab

x a b

Z

u 7! hu; ti d? ; e edge of Th

(4)

e

There is an explicit representation of the related local basis functions on a tetrahedron

T 2 Th using the barycentric coordinate functions i , i = 1; : : : ; 4, of T [14]: ND1(T ) = Span fb1; : : : ; b6g ; bk (x) = i grad j + j grad i ; where bk is associated with edge k , with endpoints i and j .

The local degrees of freedom for second order edge elements additional integrals over faces of tetrahedra and can be stated as

ND2( ; Th) involve

R u 2 ND2(T ) ! 7 pj hu; ti d? ; e edge of T; fp1; p2g basis of P1(e) ; eR

u 2 ND2(T ) ! 7 u  n; tFj  d ; F face of T; j = 1; 2 ;

t t

(5)

(6)

F

where f F1 ; F2 g is some basis of the tangential space of F . By (6) the local basis functions are not entirely fixed, since they depend on the choice of “test polynomials” pj and of tangential vectors Fj . Based on the degrees of freedom we can introduce conventional nodal projection operators k : C ( )3 7! k ( ; Th ) that can easily be extended to non-continuous

t ND

4

vectorfields, for which the degrees of freedoms are well defined. The exceptional feature of the nodal projectors is that they preserve irrotational vectorfields

curlu = 0 ^ u 2 D(k ) ) curl(k u) = 0 :

(7)

It will be crucial that edge elements form affine equivalent families of finite elements in the sense of [15]. That is, if Tb is the reference tetrahedron and T : Tb 7! T , T (b) := AT b + T , AT 2 R3;3, T 2 R3, the affine mapping taking it onto some T 2 Th , then

x

x t

t

NDk (T ) = FT (NDk (Tb)) ; where FT : H (curl; Tb) 7! H (curl; T ) is the mapping ub 7! A?T T (u  ?1 ) :

(8)

Moreover, if the weighting polynomials are suitably chosen, the local degrees of freedom remain invariant under this transformation. The transformation has the following impact on norms [16]: hT kbk2L2(Tb)  kF(b)k2L2(T ) b (9) k bk2L2(Tb)  hT k F(b)k2L2(T ) ; 8b 2 k (T ) ;

u curl u

u curl u

u ND

with  meaning equivalence up to constants that only depend on k , the shaperegularity measure of the element T (i.e., the ratio of its diameter and the radius of the largest inscribed ball), and the variation of the material parameters in individual elements. We have written hT for the diameter of T . Affine equivalence is the key to establishing the following simultaneous approximation estimates [16] that are valid for quasiuniform, shape-regular meshes of meshwidth h > 0:





ku ? vhkH (curl; )  Ch jujH ( ) + jcurlujH ( ) ; v 2ND ( ;T ) h

inf 1

h



1

1



ku ? vhkH (curl; )  Ch2 kukH ( ) + kcurlukH ( ) ; v 2ND ( ;T ) h

inf 2

2

h

2

(10) (11)

u

where C > 0 stands for a generic constant and the vectorfields are to be sufficiently smooth. The estimates (10) and (11) directly translate into error estimates for the finite element discretization of (3). More precisely, if the solution is sufficiently regular, we get k kL2( ) = O(hk ) for edge elements of order k, k = 1; 2. In addition, the same orders of convergence can be expected for k kL2 ( C ) .

u

curle

e

3 Hierarchical error estimator Hierarchical error estimators invariably target the energy norm of the er ror. However, ; ) in the current context we have to deal with an energy-seminorm on 0 (

H curl

kukA := a(u; u) ; a(u; q) := ( curlu; curlq)L ( ) + ( u; q)L ( ) ; 1 2

2

5

2

(12)

as we dispensed with a gauge condition outside C . As usual, the theoretical analysis of the hierarchical error estimator starts with a saturation assumption, which is to some extent justified by the a-priori error estimates (10) and (11). We assume that there is a sequence (h )h , 0 < h < 1, belonging to the shape-regular family fTh gh of meshes such that

ND u ND u H curl u

ku ? uehkA  h ku ? uh kA :

(13)

ND

Here, e h 2 2 ( ; Th ) is the discrete solution of (3) in 2( ; Th ), whereas h is that in 1( ; Th ). As 1 ( ; Th )  2 ( ; Th ),   1 is guaranteed, but (13) will eventually hinge on extra smoothness of the continuous solution 2 0( ; ). In fact, if the solution is smooth enough, we expect from (10) and (11) that h ! 0 as the meshwidth h tends to 0. Admittedly, the discontinuity of the conductivity  at the edge of the conducting zone C will spawn singularities of the solution [20] and destroy the regularity required for (10) and (11). On the other hand, we merely demand that h is uniformly bounded away from 1, which is much weaker than h ! 0. Using (13) and Galerkin orthogonality, the following lemma is readily established: Lemma 1 [4] If (13) holds, we can conclude

ND

ND

u

kueh ? uh kA  ku ? uh kA  1 ?1  kueh ? uhkA :

h Thus, keh ? h kA can be used to gauge the energy seminorm of the error. Only in theory, of course, because it takes tremendous computational effort to compute eh . To get a cheaper estimator we first observe that for all h 2 2 ( ; Th )

u

u

a(ueh ? uh ; qh) = r(qh) ; where

q ND r(qh) := (f ; qh)L ( ) ? a(uh; qh)

u

(14)

2

is the residual. Then the following lemma comes handily: Lemma 2 Let b : 2 ( ; Th )  2 ( ; Th ) 7! R be a positive semidefinite symmetric bilinear form, which is equivalent to a(
;
) from (12), i.e.

ND

for C; C

ND

C b(qh; qh)  a(qh; qh)  C b(qh; qh) 8qh 2 ND2( ; Th )

> 0. Then

C kwh kA  kwe h kA  C kwh kA ; wh ; w~ h 2 ND2( ; Th ) satisfy a(wh; qh) = b(we h; qh)

q

where for all h 2 2 ( ; Th ). It enables us to replace a(
;
) by an equivalent bilinear form b(
;
), for which the defect equation is much easier to solve. Nevertheless, e will still furnish an efficient and reliable error estimator. In order to construct a viable b(
;
), define the direct phierarchical splitting

ND

w

g 2 ( ; Th ) ; ND2( ; Th ) = ND1( ; Th )  ND 6

(15)

ND

where the hierarchical surplus space g 2( ; Th ) is given by

g 2( ; Th ) := (Id ? 1)ND 2 ( ; Th ) : ND

(16)

Lemma 3 Define the symmetric positive semidefinite bilinear form b : ND 2 ( ; Th )  ND2( ; Th ) 7! R by b(qh; qh) := a(q1h; q1h) + a(q2h; q2h), q1h 2 ND1( ; Th), q2h 2 g 2 ( ; Th ), qh := q1h + q2h . Then b(qh; qh )  a(qh ; qh ) for all qh 2 ND 2 ( ; Th ). ND bh := F?T 1 qh 2 Proof. Pick some qh 2 ND 2 ( ; Th ) and an arbitrary T 2 Th and set q ND2(Tb). We decompose g 2 (Tb) : qbh = qb1h + qb2h ; qb1h 2 ND1(Tb); qb2h 2 ND As (15) is a direct sum, we can resort to the equivalence of all norms on finite dimensional spaces to see





kqbhk2L (Tb)  qb1h 2L (Tb) + qb2h 2L (Tb) : 2

2

2

When desired, the constants of this equivalence can be swiftly computed as the eigenvalues of a small matrix [4]. It is important to realize that thanks to (7), we know that bh = 0, then b1h = 0 and b2h = 0. As a consequence, the same if argument as above shows

curl q

curl q

curl q





kcurl bqhk2L (Tb)  curl qb1h 2L (Tb) + curl qb2h 2L (Tb) : 2

2

2

Be aware that the splitting (15) is based on degrees of freedom. Hence, it can be done element by element and is respected by the transformation FT . In sum, using (9), we locally (and globally ) get the asserted equivalence





kqhk2L (T )  q1h 2L (T ) + q2h 2L (T )



kcurlqhk2L (T )  curlq1h 2L (T ) + curlq2h 2L (T ) : 2

2

2

2

2

2

The crucial observation is that the equivalence holds separately for k
kL2 ( ) and k
kL2( ), so that it does not matter, whether  = 0 on T . 2 It is essential that b(
;
) from the previous lemma fully decouples 1 ( ; Th ) and g 2 ( ; Th ): As r( h ) = 0 for h 2 1 ( ; Th ), the defect equation (14) for b(
;
) can be restricted to the hierarchical surplus space: Seek eh 2 g 2( ; Th ) such that

curl

ND

q

q ND

ND

e ND

g 2 ( ; Th ) : b(eeh; q2h) = r(q2h ) 8q2h 2 ND

(17)

However, solving (17) still encounters a large linear problem in the space of hierarchical surpluses. Therefore, we exploit lemma 2 once more to perform localization. For

7

ND

that sake we pick a suitable basis of g 2 ( ; Th ). On each element T with barycentric coordinate functions i , i = 1; 2; 3; 4, the basis functions are given by e 0ij :=i j + j i ; 1  i < j  4 ; a) :=  e(ijk j + k j i ; 1  i < j < k  4 ; k i b) e(ijk :=k i j + i j k ; 1  i < j < k  4 :

b grad grad b grad grad b grad grad e0ij is “associated” with the edge connecting vertices i and j , whereas The function b be(ijka) and be(ijkb) belong to the face spanned by vertices i, j , and k. By straightforward a) = 0, and  b (b) e0ij = 0, 1 b e (ijk computations we see that 1b 1 eijk = 0. A global basis of g 2 ( ; Th ) then reads ND g 2 ( ; Th ) ; fbe0e ; be(Fa); be(Fb); e edge of Th; F face of Th g  ND and it defines a direct decomposition of the hierarchical surplus space

g 2( ; Th ) = ND with

X

edge e

n o

g 2(e) := Span be0e ND

ND2(T )

=

ND1(T )

g 2(e) + ND

X

face F

g 2 (F ) ; ND

(18)

o

n

g 2 (F ) := Span b e (Fa); be(Fb) : ; ND

X

+

|e

g 2(e) ND

+ {z

g 2 (T ) ND

X F

ND2(F ) }

Figure 2: Location of degrees of freedom related to the basis functions involved in the localization procedure

curl ND

g 2 (e) = 0 for all edges e, and, Two facts ought to be mentioned: First, e(Fa) and e(Fb) are linearly independent. This means, if some eh 2 second, g 2 ( ; Th ) is -free, it will be split into -free localized contributions. As in the proof of lemma 2, this is the key for separately getting the equivalence for the (semi)-norms k
kL2 (T ) and k
kL2(T ). Thus, using affine equivalence techniques as before, we can prove the following lemma:

ND

curl b

curl curl

curl b

curl

8

u

P e P F e g 2 ( ; Th ), q g 2 (e), qeFh 2 eh = e q eh + F q eh , eqh 2 ND q 2 ND

Lemma 4 For eh g 2 (F ), set

ND

c(qeh; qeh) :=

X edges e

b(qeeh; eqeh) +

X faces F

b(qeFh ; qeFh ) :

q q

Then c(
;
) is a symmetric positive definite bilinear form, which fulfills c(eh ; eh )  b(eh; eh) on g 2( ; Th ). Hence, in (17) we can replace b(
;
) by c(
;
) and still get an eh , whose energy norm is equivalent to that of the true error. The gain is that solving c(eh ; eh ) = r( h ), eh 2 g 2 ( ; Th ), involves only small local problems: According to the dimensions of the subspaces occurring in the decomposition (18), we face a scalar equation for each edge e of the mesh and a 2  2 linear system for each face F . In sum,

ND

q q

q

e

e q

ND

eeh =

q

r(be0e ) be0 + X F be(a) + F be(b) ; b F e0 e0 e faces F a F edges e a(be ; be ) X

where aF ; bF are determined by

!  F

a(be(Fa); be(Fa)) a(be(Fa); be(Fb)) a(be(Fb); be(Fa)) a(be(Fb); be(Fb))

(19)

!

a = r(be(Fa)) : bF r(be(Fb))

b b curl e e

If the edge e is located in the interior of the non-conducting region, we get a( e0e ; e 0e ) = eh will 0 and r( e0e ) = 0. The first term in (19) can simply be dropped then and of the discretization error. still provide information about the norm of the Obviously, it takes just a few elementary computations per element to get eh and, further, the energy norm of eh on each element. The previous considerations have led us to the final result: Theorem 1 Provided that the saturation assumption (13) holds with 0 < h < 0 < 1 uniformly in h, the quantity keh kA from (19) provides an efficient and reliable estimator for the energy norm of the discretization error ? h . The bid for localization could have been characterized as search for a spectrally equivalent preconditioner for b(
;
) in the surplus space. Using the splitting (18) to simplify the defect equation in the surplus space amounts to applying a (block)-Jacobi preconditioner to (17). It goes without saying that any other symmetric (local) preconditioner can also be employed, in particular, a symmetric Gauß-Seidel sweep. Lemma 4 makes sure that we can count on the desired spectral equivalence also for symmetric Gauß-Seidel preconditioning. Though the related bilinear form eludes a concise statement, the computation of eh is no more expensive than in for the Jacobi preconditioner. We can even expect improved constants in the norm equivalence. Remark. If the finite element solution h is computed by an iterative scheme, we have to take into account an additional iteration error. Then Galerkin orthogonality is not exactly satisfied and the above equivalences have to be augmented by terms incorporating the iteration error (see [6]).

b

curl

e

e

u u

e

u

9

4 Numerical experiments Throughout the numerical experiments we use lowest order edge elements on an unstructured tetrahedral grid. The stiffness matrix and load vector corresponding to (3) are computed using Gaussian quadrature of order 5. Interpolation of boundary values is of the same order. Iteration errors can be neglected as numerous multigrid steps are carried out to compute h . To gauge the quality of the error estimator we rely on the effectivity index " := ^=, which gives the ratio between the estimated and the true energy norm of the discretization error. Here,  2 := k kA and ^2 := keh kA . This quantity reflects the quality of the global estimate. For a good error estimator, the effectivity index is to approach a constant rapidly as refinement proceeds. We point out that, since we can only expect equivalence of the estimated energy of the error and its true energy, the effectivity index may be far off the ideal value 1. In our first experiment the coefficients  and are kept constant all over the domain;  is always set to 1. The smooth analytic solution = (0; 0; sin(x1)) is enforced by the choice of the right hand side and Dirichlet boundary values. The computations are conducted on uniformly refined meshes. The resulting effectivity indices are reported in table 1.

u

e

e

u

Level



0

1

2

3

4

5

Jacobi-based hierarchical error estimator = 10?4 0.45 0.63 0.67 0.69 0.69 0.70 = 10?2 0.45 0.63 0.67 0.69 0.69 0.70 =1 0.46 0.67 0.72 0.75 0.76 0.77 = 102 0.65 1.01 1.01 1.04 1.06 1.07 4 = 10 0.74 1.27 1.40 1.43 1.33 1.19 Gauß-Seidel-based hierarchical error estimator = 10?4 0.55 0.82 0.92 0.96 0.98 0.99 = 10?2 0.55 0.82 0.92 0.96 0.98 0.99 =1 0.56 0.83 0.92 0.95 0.97 0.98 2 = 10 0.71 0.87 0.92 0.91 0.90 0.90 = 104 0.72 0.88 0.93 0.94 0.94 0.93

Table 1: Effectivity indices " for the two variants of the hierarchical error estimator on the unit cube (Exp. 1).

For both versions of the hierarchical error estimator the effectivity indices quickly approaches a stable asymptotic value as h ! 0, so that they qualify as efficient and reliable a-posteriori error estimators. This is all but true independently of the value of , which is good news, as may reflect the size of the current timestep and can vary heavily, hence. Moreover, it is conspicuous that the Gauß-Seidel approach by far outstrips Jacobi, when a good guess for the true energy of the error is desired.

10

Our second 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. As far as the coefficients are concerned, this experiment comes fairly close to the arrangements in realistic eddy current computations. In particular, we choose as follows:



(x) = 10 ::

max fjx1 ? 0:5j; elsewhere

jx2 ? 0:5j; jx3 ? 0:5jg  0:25

u n

We impose homogeneous Dirichlet boundary conditions  = 0 on the boundary and a smooth right hand side = (1; 1; 1). To estimate the true errors, we carried out two more refinement steps than reported in tables 2 and 3, respectively, and compare the discrete solutions to those obtained on the finest levels. The results are collected in table 2 for uniform refinement. We did the same experiment on meshes generated by adaptive red-green refinement. The latter relies on an averaging strategy, which marks elements in the set

f

(

1 X 2 ^T

A^ := T 2 Th : ^T2 >  n

e

)

T T 2Th

;

for refinement. Here, T2 is the energy of eh on a single element adaptive refinement are supplied in table 3. Level Jacobi-based Gauß-Seidel-based

0

1

2

3

0.56 0.83

0.72 0.93

0.77 0.96

0.79 0.97

T . The figures for

Table 2: Effectivity indices for the hierarchical error estimators in Exp. 2.

Level Jacobi-based Gauß-Seidel-based

0

1

2

3

4

0.56 0.84

0.71 0.93

0.73 0.94

0.72 0.94

0.72 0.97

Table 3: Effectivity indices in the adaptive case (Exp. 2). The message sent by the second experiment is the same as for the first. On top of that it highlights the efficacy of the error estimators also in the case of discontinuous coefficients.

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, Analysis of three dimensional electromagnetic fileds using edge elements, J. Comp. Phys., 108 (1993), pp. 236–245.

11

[3] H. A MMARI , A. B UFFA , AND J.-C. N E´ D E´ LEC, A justification of eddy currents model for the Maxwell equations, tech. rep., IAN, University of Pavia, Pavia, Italy, 1998. [4] R. BANK, Hierarchical bases and the finite element method, Acta Numerica, 5 (1996), pp. 1–43. [5] R. BANK AND A. W EISER, Some a posteriori error estimators for elliptic partial differential equations, Math. Comp., 44 (1985), pp. 283–301. [6] R. B ECK , R. H IPTMAIR, R. H OPPE , AND B. W OHLMUTH, Residual based a-posteriori error estimators for eddy current computation, Tech. Rep. 112, SFB 382, Universit¨at T¨ubingen, T¨ubingen, Germany, March 1999. To appear in M2 AN. [7] J. B EY, Tetrahedral grid refinement, Computing, 55 (1995), pp. 355–378. [8] O. B IRO AND K. R ICHTER, CAD in electromagnetism, in Advances in Electronics and Electron Physics, P. Hawkes, ed., vol. 82, Academic Press, 1991, pp. 1–96. [9] F. B ORNEMANN, An adaptive multilevel approach to parabolic equations I. General theory and 1D-implementation, IMPACT Comput. Sci. Engrg., 2 (1990), pp. 279–317. [10]

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[11] F. B ORNEMANN , B. E RDMANN , 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. [12] A. B OSSAVIT, A rationale for edge elements in 3D field computations, IEEE Trans. Mag., 24 (1988), pp. 74–79. [13]

, Solving Maxwell’s equations in a closed cavity and the question of spurious modes, IEEE Trans. Mag., 26 (1990), pp. 702–705.

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