Energetic space explained

In mathematics, more precisely in functional analysis, an energetic space is, intuitively, a subspace of a given real Hilbert space equipped with a new "energetic" inner product. The motivation for the name comes from physics, as in many physical problems the energy of a system can be expressed in terms of the energetic inner product. An example of this will be given later in the article.

Energetic space

Formally, consider a real Hilbert space

with the inner product

( ⋅ | ⋅ )

and the norm

\| ⋅ \|

. Let

be a linear subspace of

and

B:Y\toX

be a strongly monotone symmetric linear operator, that is, a linear operator satisfying

(Bu|v)=(u|Bv)

for all

u,v

(Bu|u)\gec\|u\|²

for some constant

c>0

and all

The energetic inner product is defined as

(u|v)_E=(Bu|v)

for all

u,v

and the energetic norm is

	1
	2

\|u\|

for all

The set

together with the energetic inner product is a pre-Hilbert space. The energetic space

X_E

is defined as the completion of

in the energetic norm.

X_E

can be considered a subset of the original Hilbert space

since any Cauchy sequence in the energetic norm is also Cauchy in the norm of

(this follows from the strong monotonicity property of

The energetic inner product is extended from

X_E

(u|v)_E=\lim_n\toinfty(u_n|v_n)_E

where

(u_n)

and

(v_n)

are sequences in Y that converge to points in

X_E

in the energetic norm.

Energetic extension

The operator

admits an energetic extension

B_E

B_E:X_E\to

	*
X
	E

defined on

X_E

with values in the dual space

	*
X
	E

that is given by the formula

\langleB_Eu|v\rangle_E=(u|v)_E

for all

u,v

X_E.

Here,

\langle ⋅ | ⋅ \rangle_E

denotes the duality bracket between

	*
X
	E

and

X_E,

\langleB_Eu|v\rangle_E

actually denotes

(B_Eu)(v).

and

are elements in the original subspace

then

by the definition of the energetic inner product. If one views

Bu,

which is an element in

as an element in the dual

X^*

via the Riesz representation theorem, then

will also be in the dual

	*
X
	E

(by the strong monotonicity property of

). Via these identifications, it follows from the above formula that

B_Eu=Bu.

In different words, the original operator

B:Y\toX

can be viewed as an operator

B:Y\to

	*,
X
	E

and then

B_E:X_E\to

	*
X
	E

is simply the function extension of

from

X_E.

An example from physics

Consider a string whose endpoints are fixed at two points

a<b

on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point

(a\lex\leb)

on the string be

f(x)e

, where

is a unit vector pointing vertically and

f:[a,b]\toR.

Let

u(x)

be the deflection of the string at the point

under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is

	1
	2

	b
\int
	a

u'(x)²dx

and the total potential energy of the string is

F(u)=

	1
	2

	b
\int
	a

u'(x)^2dx-

	b
\int
	a

u(x)f(x)dx.

The deflection

u(x)

minimizing the potential energy will satisfy the differential equation

-u''=f

with boundary conditions

u(a)=u(b)=0.

To study this equation, consider the space

X=L^2(a,b),

that is, the Lp space of all square-integrable functions

u:[a,b]\toR

in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

	b
(u\|v)=\int
	a

u(x)v(x)dx,

with the norm being given by

\|u\|=\sqrt{(u|u)}.

Let

be the set of all twice continuously differentiable functions

u:[a,b]\toR

with the boundary conditions

u(a)=u(b)=0.

Then

is a linear subspace of

Consider the operator

B:Y\toX

given by the formula

Bu=-u'',

so the deflection satisfies the equation

Bu=f.

Using integration by parts and the boundary conditions, one can see that

	b
(Bu\|v)=-\int
	a

u''(x)v(x)

	b
dx=\int
	a

u'(x)v'(x)=(u|Bv)

for any

and

Therefore,

is a symmetric linear operator.

\|u\|²=

	b
\int
	a

u^2(x)dx\leC

	b
\int
	a

u'(x)²dx=C(Bu|u)

for some

C>0.

The energetic space in respect to the operator

is then the Sobolev space

	1
H
	0(a,

b).

We see that the elastic energy of the string which motivated this study is

	1
	2

	b
\int
	a

u'(x)²dx=

	1
	2

(u|u)_E,

so it is half of the energetic inner product of

with itself.

To calculate the deflection

minimizing the total potential energy

F(u)

of the string, one writes this problem in the form

(u|v)_E=(f|v)

for all

X_E

Next, one usually approximates

by some

u_h

, a function in a finite-dimensional subspace of the true solution space. For example, one might let

u_h

be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation

u_h

can be computed by solving a system of linear equations.

The energetic norm turns out to be the natural norm in which to measure the error between

and

u_h

, see Céa's lemma.

References

Book: Zeidler , Eberhard . Applied functional analysis: applications to mathematical physics . New York: Springer-Verlag . 1995 . 0-387-94442-7 .
Book: Johnson , Claes . Numerical solution of partial differential equations by the finite element method . Cambridge University Press . 1987 . 0-521-34514-6.

Energetic space explained

Energetic space

Energetic extension

An example from physics

See also

References