Computational Electromagnetism

2.3 WEAK FORMULATIONS 41

the description of the (physical) field as a connector between geometrical

entities.

Which somewhat devalues differential operators, too: grad, rot and

div, in this light, appear as auxiliaries in the expression of conservation

relations, as expressed by the Ostrogradskii and Stokes theorems. Their

failure to make sense locally is thus not to be taken too seriously.

Proper form is given to the foregoing ideas in

differential geometry.

There, one forgets about the scalar or vector fields and one focuses on the

mappings they represent (and thus, to some extent, hide). Fields of

linear mappings of type

GEOMETRIC_ELEMENT --~ REAL

are called

differential forms,

of

degree 0

to 3 according to the dimension of the

geometric objects they act upon, and under regularity assumptions which

are milder than for the scalar or vector proxies, one defines a unique

operator d, the

exterior differential,

which is realized as grad, rot, or div,

depending on the dimension.

All

laws of electromagnetism can be cast

in this language (including constitutive laws, which are mappings from

p-forms to (3 - p)-forms, with p = 0 to 3).

The moderate approach we now follow does not go so far, and keeps

the fields as basic objects, but stretches the meaning of the differential

operators, so that they continue to make sense for some discontinuous

fields. The main idea is borrowed from the theory of distributions: Instead

of seeing fields as collections of pointwise values, we consider how they

act on other fields, by integration. But the full power of the theory of

distributions is not required, and we may eschew most of its difficulties.

2.3 WEAK FORMULATIONS

First, some notation. Symbols

C k

and C a for smoothness have already

been introduced, compact support 9 is usually denoted by a subscripted

0, and blackboard capitals are used in this book to stress the vector vs

scalar opposition when referring to spaces of fields. Putting all these

conventions together, we shall thus have the following list of infinite-

dimensional linear spaces:

• ck(E3 ) • The vector space of all k-smooth functions in E3,

• ok(E3 )" All k-smooth vector fields in E3,

9The

support

of a function, real- or vector-valued, is the closure of the set of points

where it doesn't vanish. Cf. A.2.3.

42 CHAPTER 2 Magnetostatics: "Scalar Potential" Approach

• c0k(D),

~0 k(D)"

Same, with compact support contained in D,

being understood that domain D can be all E3, and finally,

• ck(w ), ck (D),

for the vector spaces of restrictions to D (the closure

of D), of k-smooth functions or vector fields. In all

of these, k can be replaced by ~o. (In the_ inset, the

supports of a q)l E C0~(D) and a q)2 E C ~(D), which is

thus the restriction of some function defined beyond

D, whose support is sketched.)

" W

2.3.1 The "divergence" side

Now, let's establish a technical result, which generalizes integration by

parts. Let D be a regular domain (not necessarily bounded), S its

boundary, b a smooth vector field, and q0 a smooth function, both with

compact support in

E 3

(but their supports may extend beyond D).

Form u = q0 b. Ostrogradskii's theorem asserts that SD div u = Js n. u,

with n pointing outwards, as usual. On the other hand, we have this

vector analysis formula,

div(q0 b) = q~ div b + b. grad q0.

Both things together give

(9)

SD q0

div b =

-

~D b. grad q0 + Ss n. b q),

a fundamental formula.

By (9), we see that a 1-smooth divergence-free field b in D is

characterized by

(10)

S D b.

grad q) - 0 V q0 ~ C01(D),

since with q0 = 0 on the boundary, there is no boundary term in (9). But

(10) makes sense for fields b which are only

piecewise

smooth. 1° We

now take a bold step:

Definition

2.1.

A piecewise smooth field b which satisfies

(10)

will be said to

be

divergence-free,

or solenoidal,

in the weak sense.

The q)'s in (10) are called

test functions.

1°All that is required is the integrability of b. grad q0 in (10), so 0-smoothness, that is,

continuity, of each "piece" of b is enough.