70 CHAPTER 3 Solving for the Scalar Magnetic Potential
A little more abstractly, suppose a problem has been cast in the form
Ax = b, where b symbolizes the data, x the solution, and A some
mapping of type
SOLUTION --~ DATA.
Solving the problem means, at a
high enough level of abstraction, finding 7 the inverse A -1, which may
not be defined for some values of b (those corresponding to sharp
corners, let's say, for illustration). But if there is a solution x n for each
element
b n
of some sequence that converges toward b, it's legitimate to
define the limit x = lim n _, ~ x n as the solution, if there is such a limit, and
if there isn't, to invent one.
That's the essence of completion. Moreover,
attributing to Ax the value b, whereas A did not make sense, a priori,
for the
generalized solution
x, constitutes a prolongation of A beyond its
initial domain, a thing which goes along with completion (cf. A.4.1).
Physicists made much mileage out of this idea of a generalized solution,
as the eventual limit of a parameterized family, before the concepts of
modern functional analysis (complete spaces, distributions, etc.) were
elaborated in order to give it status.
Summing up: We now attribute the symbol O* to the completion of
the space of piecewise smooth functions in D, null on sh0 and equal to
some constant
on
Shl , with respect to the norm IIq011,
=
[~D ~ I grad
q) i2] 1/2.
Same renaming for
(I)I
(which is now the closure of the previous one in
O*). Equation (1), or Problem (3), has now a (unique) solution. The next
item in order s is to
solve
for it.
3.3
DISCRETIZATION
But what do we mean by that? Solving an equation means being able to
answer specific questions about its solution with controllable accuracy,
whichever way. A century ago, or even more recently in the pre-computer
era, the only way was to represent the solution "in closed form", or as
the sum of a series, thus allowing a numerical evaluation with help of
formulas and tables. Computers changed this: They forced us to work
from the outset with
finite
representations. Eligible fields and solutions
7An unpleasantly imprecise word. What is required, actually, is some
representation
of
the inverse, by a formula, a series, an algorithm.., anything that can give
effective
access to
the solution.
SWhether Problem (3) is well posed (cf. Note 1.16) raises other issues, which we
temporarily bypass, as to the continuous dependence of q~ on data: on I (Prop. 3.2 gave
the answer), on ~ (cf. Exers. 3.17 and 3.19), on the dimensions and shape of the domain
(Exers. 3.18 and 3.20).

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