180
CHAPTER 6 The "Curl Side": Complementarity
There is more to say about complementarity. In particular, there is
an obvious problem of duplication of efforts: Computations of q0 and a
are independent, though their results are closely related. No use is made
of the knowledge of q0 when solving for a, which seems like a waste,
since as one rightly suspects, and as the following discussion will make
clear, solving for a is much more expensive than solving for q0. There is
a way to save on this effort, which is explained in detail in Appendix C.
We now address the more urgent question of whether edge elements are
really mandatory in the vector potential approach.
6.3 WHY NOT STANDARD ELEMENTS ?
As we saw, edge elements are able to solve a problem that was quite
difficult with nodal vectorial elements, namely, to obtain a vector potential
formulation in terms of explicit
independent
degrees of freedom. This is a
good point, but not the whole issue, for difficult does not mean impossible,
and if bilateral estimates, or any other consequence of the hypercircle
trick, are the objective, one must concede that it
can
be achieved with
standard nodal elements, scalar-valued for q0, vector-valued for a. All
that is required is approximation "from inside" (within the functional
space), Galerkin style. It's thus a gain in simplicity or in accuracy, or
both, that we may expect of edge elements.
To make a fair comparison, let us suppose that, after proper selection
of independent variables, the vector potential approach with nodal
elements consists in looking for a in some subspace of IPlm, that we
shall denote AF(Ip1). Let us similarly rename AF(w 1) the above ~F,
to stress its relationship with edge elements. In both methods, the quantity
fD
~-1 I rot a 12
is minimized, but on different subspaces
of
[L2rot(D),
which results in linear systems of the same form Ma - b, but with
different M and b, and a different interpretation for the components of
a. Let us call
Mp
and M w, respectively, the matrix M in the case of the
IP 1
and the W 1 approximation. Both are symmetric and (a priori)
nonnegative definite. We shall find the nodal vectorial approach inferior
on several counts. But first...
6.3.1 An apparent advantage: M p is regular
Indeed, let a
~ ~F
(ip1) be such that rot a = 0, which is equivalent to
m
Ma = 0. Then a = grad ~. Since a is piecewise linear with respect to the
6.3 WHY NOT STANDARD ELEMENTS?
181
mesh
m and
continuous (in all its three scalar components), q0 is
piecewise quadratic
and
differentiable, two hardly compatible conditions.
The space p2 of piecewise quadratic functions on m is generated by the
products
WnWm,,
where n and m span the nodal set N. Therefore,
grad ~ - ~ m,n ~ N CG
n
grad(w Wn), and since the products WnW m are not
differentiable, the normal component of this field has a nonzero jump
across all faces. (This jump is affine with respect to coordinates over a
face.) Demanding that all these jumps be 0 is a condition that considerably
constrains the c~'s (in practice, only globally quadratic ~, as opposed to
mesh-wise quadratic, will comply). Consequently, the kernel of rot in
AF(Ip 1) will be of very low dimension, and as a rule reduced to 0,
because of additional constraints imposed by the boundary conditions.
So unless the mesh is very special, one may expect a regular
M p,
whereas
the matrix M is singular, since W 1 contains gradients (cf. Prop. 5.4,
asserting that WID grad W°).
Good news? We'll see. Let us now look at the weak points of the
nodal vectorial method.
6.3.2 Accuracy is downgraded
When working with the same mesh, accuracy is downgraded with nodal
1 F 1
elements, because rot(Afm(IP )) C rot(A m(W )), with as a rule a
strict
inclusion. Minimizing over a smaller space will thus yield a less accurate
upper bound in the case of nodal elements, for a given mesh m. (Let's
omit the subscript m for what follows, as far as possible. Recall that N,
E, F, T refer to the number of nodes, etc., in the mesh.)
The inclusion results from this:
Proposition
6.4.
For a given mesh m, any field
u ~ IP 1
is sum of some field
in W ~ and of the gradient of some piecewise quadratic function, i.e.,
IP 1 c W 1 q-
grad
p2.
Proof.
Given u e IP 1, set u e
= ~e X" U,
for all e e E, and let v =
~e ~ ~ Ue We be the field in W 1 which has these circulations as edge DoFs.
Then, both u and v being linear with respect to coordinates, rot(u- v)
is piecewise constant. But its fluxes through faces are 0, by construction
(again, see Fig. 5.5), so it vanishes. Hence u = v + V% where q0 is such
that Vq0 be piecewise linear, that is, q0 e p2.
As a corollary, rot IP 1 c rot W 1, hence the inclusion, as far as (but this
is what we assumed for fairness) AFm(IP~)
c a w.
As a rule, this is strict
inclusion, because the dimension of rot IP ~ cannot exceed that of IP 1,
182 CHAPTER 6 The "Curl Side": Complementarity
which is 3N (three scalar DoFs per node), whereas the dimension of
rot W 1 is (approximately the same as) that of the quotient W1/grad(W°),
which is E - N + 1, that is, 5 to 6 N, depending on the mesh, as we know
(cf. 4.1.1).
Exercise
6.4. Check that IP 1 is
not
contained in W 1.
Exercise
6.5. Show that W 1 c IP ~ + grad p2 does
not
hold.
6.3.3 The "effective" conditioning of the final matrix is worsened
The importance of the
condition number
of a matrix, that is, the ratio of its
extreme eigenvalues, is well known. This number determines to a large
extent the speed of convergence of iterative methods of solution, and the
numerical accuracy of direct methods.
This is true, that is, in the case of
regular
matrices. But if a symmetric
nonnegative definite matrix M is singular, this does not preclude the
use of iterative methods to solve Ma = f. All that is required is that f be
in the range of M, so that (Ma, a)- 2(f, a) have a finite lower bound.
Then any "descent" method (i.e., one that tries to minimize this function
by decreasing its value at each iteration) will yield a minimizing sequence
u k that may not converge, but
does converge modulo
ker(M), and this may
be just enough. To be definite, suppose the matrix M is a principal
submatrix of RtM(~-I)R, as was shown to be the case with edge elements.
The quadratic form to be minimized is indeed bounded from below, and
the desired convergence is that of Rak, not a k. By working out the
simple example of the iterative method
U n + 1 -- Un-
P (Man - f), which is
easy in the basis of eigenvectors of M, one will see that what counts, as
far as convergence modulo ker(M) is concerned, is the ratio of extreme
strictly positive eigenvalues. Let us call this
effective
conditioning, denoted
by K(M).
We now show that
K(Mp) >> K(Mw) ,
thus scoring an important point
for edge elements. This will be quite technical, unfortunately.
Both matrices can be construed as approximations of the "curl-curl"
operator, rot(~t-~rot ), or rather, of the associated boundary-value problem.
Their higher eigenvalues have similar asymptotic behavior, when the
mesh is refined. (I shall not attempt to prove this, which is difficult, 6
but it can easily be checked for meshes with a regular, repetitive layout.)
So we should compare the first positive eigenvalue ~,l(Mw) with its
homologue Kl(Mp). As we noticed, Mp is regular, in general. But zero
6The difficulty lies in
stating
the claim with both precision and generality.
6.3 WHY NOT STANDARD ELEMENTS?
183
is an eigenvalue of the curl-curl operator, and as a rule, spectral elements
of an operator are approximated (when the mesh is repeatedly refined
while keeping flatness under control, which we informally denote as
"m --~ 0") by those of its discrete matrix counterpart, which
does not
contain O. Therefore,
when m ~ 0,
~I(Mp)
tends to 0. But for M w, the
situation is quite different: This matrix is singular, since it contains the
vectors a = G~ (with
~n
¢ 0 for all n not in sB). No need in
consequence for the eigenvalue 0 to be approximated "from the right",
as was the case for
M p.
And indeed, limm+ 0 Kl(Mw) > 0. This can be seen by applying the
Rayleigh quotients theory, according to which
KI(M w) = inf{(Mwa, a) : l al - 1, (a, a') = 0 V a' ~ ker(Mw)}.
In terms of the associated vector fields, this orthogonality condition means
(34)
fD a.
grad ~'- 0 V ~' ~ ~t '° ¢~ W°,
where ~t'°= {~
L2grad(D) • ~/ =0 on
sb}. Let
al(m )
be the field whose
DoFs form the eigenvector
a I
corresponding to ;~1, and al its limit
when m ~ 0. Equation (34) holds for a 1. Therefore (take the projections
on W°m of ~' in (34), and pass to the limit),
D
al" grad ~'= 0 V ~'~ ~t '°,
and a 1
is thus divergence-free. Hence
__ ~-1 2 .~0
lim=__,0 X~(M w) inf{ID I rot a I • a e , div a = 0, ID I a12 = 1},
and this Rayleigh quotient is strictly positive.
So we may conclude that )ffMw)/~:(Mp) tends to 0: Effective condi-
tioning is asymptotically better with edge elements.
6.3.4 Yes, but...
Is the case over? Not yet, because the defendants have still some arguments
to voice. Ease in setting up boundary conditions? Yes, but think of all
these standard finite element packages around. Reusing them will save
much effort. Bad conditioning? Yes, but asymptotically so, and we don't
go to the limit in practice; we make the best mesh we can, within limits
imposed by computing resources; this mesh may not be the same for
both methods, since the number of degrees of freedom will be different,
so the comparison may well be of merely academic interest. Same thing
about the relative loss of accuracy: Given the same resources, we may
184 CHAPTER 6 The "Curl Side": Complementarity
use more refined meshes in the case of nodal elements, since the number
of degrees of freedom is lower, apparently. After all, the number of
edges E is much higher than 3N, isn't it?
Let us count again. Assume the mesh is first done with bricks, each
of these being further divided into five tetrahedra (cf. Exer. 3.7). Thus, T
5N. Then F ~ 10 N (four faces for each tetrahedron, shared by two),
and the Euler-Poincar6 formula, that is, as we know,
N-E + F- T = x(D),
shows that E ~ 6N. So indeed, the number of DoFs in the edge element
method (about E ~ 6N) will be twice as high as in the nodal vectorial one
(~ 3N).
These figures, which ignore boundary conditions, are quite ap-
proximate (cf. [Ko] for precise counting). The ratios are valid for big
meshes only. Still, the conclusion is neat: Tetrahedral edge elements
generate more degrees of freedom than classical elements.
But is that really topical? The most meaningful number, from the
point of view of data storage and CPU time, is not the size of the matrix,
but the number of its nonzero entries. It happens this number is
smaller,
for a given mesh, for M w than for Mp, against pessimistic expectations.
For let us count the average number of entries on a given row of
Mp:
This equals the number of DoFs that may interact with a given nodal
one, that is, if we denote by v i the basis vectors in a Cartesian frame, the
number of couples {m, j} for which ~D -1
rot (v i
Wn). rot (vj Win) ~ 0, for
a given {n, i}. But rot (v~ Wn) = -- V i X grad w n, so this term vanishes if
the supports of w n and w m don't intersect, so two DoFs may interact
only when they belong to the same node, or to nodes linked by an edge.
As each node is linked with about 12 neighbors, there are 38 extra-diagonal
non-vanishing terms on a row of
Mp, on
average.
As for M w, the number of extra-
diagonal non-vanishing terms on the row
of edge e is the numbertof edges e' for
which the integral ~
D
~t rot w e . rot w e,
differs from 0, that is, edges belonging to
a tetrahedron that contains e. On the
average, e belongs to 5 tetrahedra (because
it belongs to 5 faces, since F ~ 5E/3, each
face having 3 edges). This edge thus has
15 "neighbors" (inset): 10 edges which
share a node with e, and 5 opposite in their common tetrahedron. So
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