also a surface operator by the same argument. The formula
~s Us" grads q°s = -~s q°s divs Us + ~s q°s v. u s
(where v now refers to the rim of S), is proved by the same technique as
in dimension 3, but (35) suggests to also introduce a rot operator acting
surface fields, as follows:
rot s q0 s = - n x grad s q0s,
hence the formula
~s q°s rots Us = ~s Us" r°tsq°s- ~s q°s I:. u S,
the nice symmetry of which compensates for the slight inconvenience s
of overloading the symbol rot s .
REFERENCES and Bibliographical Comments
The search for complementarity, a standing concern in computational electro-
magnetism [HP, HT], is related to the old "hypercircle" idea of Prager and Synge
[Sy]. In very general terms, this method consisted in partitioning the set of
equations and boundary conditions into two parts, thus defining two orthogonal
subsets in the solution space, the solution being at their intersection. Finding
two approximations within each of these subsets allowed one (by the equivalent
of the Pythagoras theorem in the infinite dimensional solution space, as was
done above in 6.3.1) to find the center and radius of a "hypercircle" containing
the unknown solution, and hence the bounds (not only on quadratic quantities
such as the reluctance, but on linear functionals and, even better, on pointwise
quantities [Gr]). After a period of keen interest, the idea was partly forgotten,
then revisited or rediscovered by several authors [DM, HP, LL, Ny, R&, Va .... ],
Noble in particular [No], who is credited for it by some (cf. Rall [Ra], or Arthurs
[An], who also devoted a book to the subject JAr]). Thanks to the Whitney
elements technology, we may nowadays adopt a different (more symmetrical)
partitioning of equations than the one performed by Synge on the Laplace problem.
(The one exposed here was first proposed in [B1].) On pointwise estimates,
which seem to stem from Friedrichs [Fr], see [Ba], [Co], [Ma], [St], [Sn].
8The risk of confusion, not to be lightly dismissed, will be alleviated by careful definition
of the types of the fields involved: The first rot is
VECTOR ~ SCALAR
(fields), the other
SCALAR --> VECTOR.
Various devices have been proposed to make the distinction,
including the opposition rot vs Rot, but they don't seem to make things more mnemonic.
A. Di Carlo has proposed to denote the second operator by "grot", which would solve this