Energetic space
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 [math]\displaystyle{ X }[/math] with the inner product [math]\displaystyle{ (\cdot|\cdot) }[/math] and the norm [math]\displaystyle{ \|\cdot\| }[/math]. Let [math]\displaystyle{ Y }[/math] be a linear subspace of [math]\displaystyle{ X }[/math] and [math]\displaystyle{ B:Y\to X }[/math] be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
- [math]\displaystyle{ (Bu|v)=(u|Bv)\, }[/math] for all [math]\displaystyle{ u, v }[/math] in [math]\displaystyle{ Y }[/math]
- [math]\displaystyle{ (Bu|u) \ge c\|u\|^2 }[/math] for some constant [math]\displaystyle{ c\gt 0 }[/math] and all [math]\displaystyle{ u }[/math] in [math]\displaystyle{ Y. }[/math]
The energetic inner product is defined as
- [math]\displaystyle{ (u|v)_E =(Bu|v)\, }[/math] for all [math]\displaystyle{ u,v }[/math] in [math]\displaystyle{ Y }[/math]
and the energetic norm is
- [math]\displaystyle{ \|u\|_E=(u|u)^\frac{1}{2}_E \, }[/math] for all [math]\displaystyle{ u }[/math] in [math]\displaystyle{ Y. }[/math]
The set [math]\displaystyle{ Y }[/math] together with the energetic inner product is a pre-Hilbert space. The energetic space [math]\displaystyle{ X_E }[/math] is defined as the completion of [math]\displaystyle{ Y }[/math] in the energetic norm. [math]\displaystyle{ X_E }[/math] can be considered a subset of the original Hilbert space [math]\displaystyle{ X, }[/math] since any Cauchy sequence in the energetic norm is also Cauchy in the norm of [math]\displaystyle{ X }[/math] (this follows from the strong monotonicity property of [math]\displaystyle{ B }[/math]).
The energetic inner product is extended from [math]\displaystyle{ Y }[/math] to [math]\displaystyle{ X_E }[/math] by
- [math]\displaystyle{ (u|v)_E = \lim_{n\to\infty} (u_n|v_n)_E }[/math]
where [math]\displaystyle{ (u_n) }[/math] and [math]\displaystyle{ (v_n) }[/math] are sequences in Y that converge to points in [math]\displaystyle{ X_E }[/math] in the energetic norm.
Energetic extension
The operator [math]\displaystyle{ B }[/math] admits an energetic extension [math]\displaystyle{ B_E }[/math]
- [math]\displaystyle{ B_E:X_E\to X^*_E }[/math]
defined on [math]\displaystyle{ X_E }[/math] with values in the dual space [math]\displaystyle{ X^*_E }[/math] that is given by the formula
- [math]\displaystyle{ \langle B_E u | v \rangle_E = (u|v)_E }[/math] for all [math]\displaystyle{ u,v }[/math] in [math]\displaystyle{ X_E. }[/math]
Here, [math]\displaystyle{ \langle \cdot |\cdot \rangle_E }[/math] denotes the duality bracket between [math]\displaystyle{ X^*_E }[/math] and [math]\displaystyle{ X_E, }[/math] so [math]\displaystyle{ \langle B_E u | v \rangle_E }[/math] actually denotes [math]\displaystyle{ (B_E u)(v). }[/math]
If [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] are elements in the original subspace [math]\displaystyle{ Y, }[/math] then
- [math]\displaystyle{ \langle B_E u | v \rangle_E = (u|v)_E = (Bu|v) = \langle u|B|v\rangle }[/math]
by the definition of the energetic inner product. If one views [math]\displaystyle{ Bu, }[/math] which is an element in [math]\displaystyle{ X, }[/math] as an element in the dual [math]\displaystyle{ X^* }[/math] via the Riesz representation theorem, then [math]\displaystyle{ Bu }[/math] will also be in the dual [math]\displaystyle{ X_E^* }[/math] (by the strong monotonicity property of [math]\displaystyle{ B }[/math]). Via these identifications, it follows from the above formula that [math]\displaystyle{ B_E u= Bu. }[/math] In different words, the original operator [math]\displaystyle{ B:Y\to X }[/math] can be viewed as an operator [math]\displaystyle{ B:Y\to X_E^*, }[/math] and then [math]\displaystyle{ B_E:X_E\to X^*_E }[/math] is simply the function extension of [math]\displaystyle{ B }[/math] from [math]\displaystyle{ Y }[/math] to [math]\displaystyle{ X_E. }[/math]
An example from physics
Consider a string whose endpoints are fixed at two points [math]\displaystyle{ a\lt b }[/math] on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point [math]\displaystyle{ x }[/math] [math]\displaystyle{ (a\le x \le b) }[/math] on the string be [math]\displaystyle{ f(x)\mathbf{e} }[/math], where [math]\displaystyle{ \mathbf{e} }[/math] is a unit vector pointing vertically and [math]\displaystyle{ f:[a, b]\to \mathbb R. }[/math] Let [math]\displaystyle{ u(x) }[/math] be the deflection of the string at the point [math]\displaystyle{ x }[/math] under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is
- [math]\displaystyle{ \frac{1}{2} \int_a^b\! u'(x)^2\, dx }[/math]
and the total potential energy of the string is
- [math]\displaystyle{ F(u) = \frac{1}{2} \int_a^b\! u'(x)^2\,dx - \int_a^b\! u(x)f(x)\,dx. }[/math]
The deflection [math]\displaystyle{ u(x) }[/math] minimizing the potential energy will satisfy the differential equation
- [math]\displaystyle{ -u''=f\, }[/math]
with boundary conditions
- [math]\displaystyle{ u(a)=u(b)=0.\, }[/math]
To study this equation, consider the space [math]\displaystyle{ X=L^2(a, b), }[/math] that is, the Lp space of all square-integrable functions [math]\displaystyle{ u:[a, b]\to \mathbb R }[/math] in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
- [math]\displaystyle{ (u|v)=\int_a^b\! u(x)v(x)\,dx, }[/math]
with the norm being given by
- [math]\displaystyle{ \|u\|=\sqrt{(u|u)}. }[/math]
Let [math]\displaystyle{ Y }[/math] be the set of all twice continuously differentiable functions [math]\displaystyle{ u:[a, b]\to \mathbb R }[/math] with the boundary conditions [math]\displaystyle{ u(a)=u(b)=0. }[/math] Then [math]\displaystyle{ Y }[/math] is a linear subspace of [math]\displaystyle{ X. }[/math]
Consider the operator [math]\displaystyle{ B:Y\to X }[/math] given by the formula
- [math]\displaystyle{ Bu = -u'',\, }[/math]
so the deflection satisfies the equation [math]\displaystyle{ Bu=f. }[/math] Using integration by parts and the boundary conditions, one can see that
- [math]\displaystyle{ (Bu|v)=-\int_a^b\! u''(x)v(x)\, dx=\int_a^b u'(x)v'(x) = (u|Bv) }[/math]
for any [math]\displaystyle{ u }[/math] and [math]\displaystyle{ v }[/math] in [math]\displaystyle{ Y. }[/math] Therefore, [math]\displaystyle{ B }[/math] is a symmetric linear operator.
[math]\displaystyle{ B }[/math] is also strongly monotone, since, by the Friedrichs's inequality
- [math]\displaystyle{ \|u\|^2 = \int_a^b u^2(x)\, dx \le C \int_a^b u'(x)^2\, dx = C\,(Bu|u) }[/math]
for some [math]\displaystyle{ C\gt 0. }[/math]
The energetic space in respect to the operator [math]\displaystyle{ B }[/math] is then the Sobolev space [math]\displaystyle{ H^1_0(a, b). }[/math] We see that the elastic energy of the string which motivated this study is
- [math]\displaystyle{ \frac{1}{2} \int_a^b\! u'(x)^2\, dx = \frac{1}{2} (u|u)_E, }[/math]
so it is half of the energetic inner product of [math]\displaystyle{ u }[/math] with itself.
To calculate the deflection [math]\displaystyle{ u }[/math] minimizing the total potential energy [math]\displaystyle{ F(u) }[/math] of the string, one writes this problem in the form
- [math]\displaystyle{ (u|v)_E=(f|v)\, }[/math] for all [math]\displaystyle{ v }[/math] in [math]\displaystyle{ X_E }[/math].
Next, one usually approximates [math]\displaystyle{ u }[/math] by some [math]\displaystyle{ u_h }[/math], a function in a finite-dimensional subspace of the true solution space. For example, one might let [math]\displaystyle{ u_h }[/math] be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation [math]\displaystyle{ u_h }[/math] 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 [math]\displaystyle{ u }[/math] and [math]\displaystyle{ u_h }[/math], see Céa's lemma.
See also
References
- Zeidler, Eberhard (1995). Applied functional analysis: applications to mathematical physics. New York: Springer-Verlag. ISBN 0-387-94442-7. https://books.google.com/books?id=bFqRaovfY4MC&q=%22energetic+space%22.
- Johnson, Claes (1987). Numerical solution of partial differential equations by the finite element method. Cambridge University Press. ISBN 0-521-34514-6.
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