7.5 BACK TO MAGNETOSTATICS 215
Let q~ be the vector of degrees of freedom (one for each node, including
this time those contained in S), and • = IR N (there are N nodes). We
denote
T~Jn -'- ~D ~
hi. grad
W n n u ~[0 ~S n.
h i
Wn,
TIJ= {TlJn " n e N}, and let G, R,
Ml(~t )
be the same matrices as in
Chapter 5. Still denoting by P the extension to • (obtained by filling-in
with zeroes) of the matrix P of (48), we finally get the following ap-
proximation for (50):
(51) (GtMI(~t)G + P)q0 + 11 j= 0.
Although the matrix P of (38) is full, the linear system (51) is reasonably
sparse, because P only concerns the "S part" of vector q~.
Remark 7.4. The linear system is indeed an
approximation
of (50), and not
its interpretation in terms of degrees of freedom, for (Pq~, qY) is just an
approximation of fs Pq°s q°'s on the subspace @re(D), not its restriction, as
in the Galerkin method. (This is another example of "variational
crime".) ~)
Remark 7.5. There are other routes to the discretization of P. Still using
magnetic charges (which is a classic approach, cf. [Tz]), one could place
them differently, not on S but inside D [MW]. One might, for example
[Ma], locate a point charge just beneath each node of S. (The link
between q and ¢p would then be established by collocation, that is to
say, by enforcing the equality between q0 and the potential of q at
18
nodes. ) Another approach [B2] stems from the remark that interior
and exterior Dirichlet-to-Neumann maps (call them
Pint
and Pxt) add to
something which is easily obtained in discrete form, because of the relation
(Pint + Pext)q ) =
q = K-1% Since, in the present context, we must mesh D
anyway, a natural discretization
Pint
of
Pint
is available, thanks to the
"static condensation" trick of Exer. 4.8: One minimizes the quantity
fD I grad(Y~n ~ N(D)
q)n
Wn )2 with respect to the
inner
node values q~n' hence
a quadratic form with respect to the vector ~ (of surface node DoFs), the
matrix of which is
Pint"
A reasoning similar to the one we did around
(46-47) then suggests B K -1B
t as
the correct discretization of K <, hence
finally P
=
Pext = B K -1B
t --
Pint
(which ensures the symmetry of
Pext,
but
does not eliminate the difficulty evoked in Exer. 7.3). And (lest we
forget...) for some simple shapes of S (the sphere, for example), P is
known in closed form, as the sum of a series. 0
18See, e.g., [KP, ZK]. These authors' method does provide a symmetric P, but has
other drawbacks. Cf. [B2] for a discussion of this point.