6.3 WHY NOT STANDARD ELEMENTS? 185
there are about 15 extra-diagonal nonzero entries per row, that is, about
90 N terms of this kind in M w, against 3 x 38 = 114N in
Mp, a
sizable
advantage in favor of edge elements.
This rather satisfying conclusion should not mask the obvious problem
with
all
vector potential methods, whatever the finite elements: a large
number of DoFs, relatively. In the same conditions as previously, the q0
method only generates 12N off-diagonal nonzero terms. In Appendix C,
we shall see how some savings are possible, but only to some extent.
Complementarity has its price.
6.3.5 Conclusion
Focusing on magnetostatics as I do in this book has obvious shortcomings,
but also the advantage of delimiting a narrow field in which theory can
deploy itself, and any fundamental observation in this limited area has
all chances to be valid for magnetics in general. This seems to be the case
of the nodal elements vs edge elements debate. But of course, theory will
not carry the day alone, and numerical experience is essential. We have
a lot of that already. It began with M. Barton's thesis [Br] (see an account
in [BC]), whose conclusions deserve a quote:
When the novel use of tangentially continuous edge-elements for the
representation of magnetic vector potential was first undertaken, there was
reason to believe it would result in an interesting new way of computing
magnetostatic field distributions. There was only hope that it would result
in a significant improvement in the state-of-the-art for such computations.
As it has turned out, however, the new algorithm has significantly out-
performed the classical technique in every test posed. The use of elements
possessing only tangential continuity of the magnetic vector potential allows
a great many more degrees of freedom to be employed for a given mesh as
compared to the classical formulation; and these degrees of freedom result
in a global coefficient matrix no larger than that obtained from the smaller
number of degrees of freedom of the other method. (...) It has been
demonstrated that the conjugate gradient method for solving sets of linear
equations is well-defined and convergent for symmetric but underdetermined
sets of equations such as those generated by the new algorithm. As predicted
by this conclusion, the linear equations have been successfully solved for all
test problems, and the new method has required significantly fewer iterations
to converge in almost all cases than the classical algorithm.
One may also quote this [P&], about scattering computations:
Experience with the 3D node-based code has been much more discouraging:
many problems of moderate rank failed to converge within N iterations
186 CHAPTER 6 The "Curl Side": Complementarity
(...). The Whitney elements were the only formulation displaying robust
convergence with diagonal PBCG 7 iterative solution.
Countless objections have been raised against edge elements. The
most potent one is that W 1, contrary to IP 1, does not contain globally
linear vector fields, like for instance x --~ x, and thus lack "first-order
completeness". This is both true and irrelevant. In magnetostatics, where
the object of attention is not the unknown a but its curl, we already
disposed of the objection with Prop. 6.4. But even in eddy-current
computations (where, as we'll see in Chapter 8, edge elements are natural
approximants for the field h), it does no good to enlarge W 1 to the
space spanned by the wnVw m (which does include linear fields). See
[B4, DB, Mk].
The debate on this and other issues relative to the edge elements vs
nodal elements contest winds its way and will probably go on for long,
but it would be tedious to dwell on it. As one says, those who ignore
history are bound to live it anew. A lot remains to be done, however:
research on higher-order edge elements (to the extent that [Ne] does not
close the subject), error analysis [Mk, MS, Ts], edge elements on other
element forms than tetrahedra, such as prisms, pyramids, etc. [D&].
The problem of singularity addressed in Remark 6.4 is crucial, in
practice, when there is a distributed source-term, as is the case in
magnetostatics. If the discretization of the right-hand side is properly
done, one will obtain a system of the
form
RtM2(~-I)R a
=
Rtk, where a
is the DoF-vector of the vector potential, and k a given vector, and in
this case the right-hand side Rtk is in the range of RtM2(~-I)R. But
otherwise, the system has no solution, which the behavior of iterative
algorithms tells vehemently (drift, as evoked in 6.3.3, slowed convergence,
if not divergence). This seems to be the reason for the difficulties
encountered by some, which can thus easily be avoided by making sure
that the right-hand side behaves [Re]. There, again, tree-cotree techniques
may come to the rescue.
Finally, let us brush very briefly on the issue of singularity. Should
the edge-element formulation in a be "gauged", that is, should the
process of selecting
independent
variables be pushed further, to the point
of having only
non-redundant
DoFs? This can be done by extracting
spanning trees. Such gauging is necessary with nodal elements, but not
with edge elements. Experiments by Ren [Re] confirm this, which was
already suggested by Barton's work.
7This refers to the "preconditioned biconjugate gradient" algorithm [Ja].
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