214 CHAPTER 7 Infinite Domains
code (cf. p. 225), and although not totally satisfactory (cf. Exer. 7.3), it
does make the computation of double integrals simple.
Simple does not mean trivial however, and care is required for terms
of K, which are of the form
KrT. =1~ fTdyfT.dX Ix-y1-1,
where T and T' are two non-intersecting triangles in generic position in
3-space. The internal integral is computed analytically, and the outer
one is approximated by a quadrature formula, whose sophistication must
increase when triangles T and T' are close to each other. Any programmer
with experience on integral or semi-integral methods of some kind has
had, at least once in her life, to implement this computation, and knows
it's a tough task. Unfortunately, the details of such implementations are
rarely published (more out of modesty than a desire to protect shop
secrets). Some indications can be gleaned from [AR, C1, R&].
Remark 7.3. As anticipated earlier, the "naive" discretization of (38),
yielding P = (1/2 + H)K -1, would be inconsistent (the dimension of K is
not what is expected for P, that is, N(S)). But the more sophisticated
expression (B/2 + H) K -1
g t
would not do, either, since this matrix is not
symmetric, and the symmetrization in (48), to which we were led in a
natural way, is mandatory. 0
Exercise 7.3. Show, by a counter-example, that matrix P may happen to
be singular with the above choice for the ~i"
7.5 BACK TO MAGNETOSTATICS
We may now finalize the description of the "finite elements and integral
method in association" method of 7.3.3, in the case when the unknown is
the scalar potential. Let cI~(D) be the space of restrictions to D of the
scalar potentials in space ~. We denote again by q0 s the trace of q0 on
S. Thanks to the operator P, the Euler equation (15) is equivalent to the
following problem:
find ~ ~
~(D)
such that
(49) ~D ~t (h j + grad q0). grad q0' + ~t 0 ~s (n. h j + Pq0 s) q0' s = 0 V q0' ~ ~(D).
Let m be a mesh of D. Then ~m(D) = {q~" q0 = ~ n~ N ~n Wn} is the natural
approximation space for ~(D). Hence the following approximation of
(49),
find q~ ~
~(D)
such that, for all q~' ~
cI~ (D),
(50)
fD ~t (h j + grad q0). grad q0' + ~t 0 ~s (n. h j + Pq0 s) q0' s = 0.
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.
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