Computational Electromagnetism

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).