Computational Electromagnetism

REFERENCES 189

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

on

scalar

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

one is

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

terminological difficulty.