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 $$X$$ with the inner product $$(\cdot|\cdot)$$ and the norm $$\|\cdot\|$$. Let $$Y$$ be a linear subspace of $$X$$ and $$B:Y\to X$$ be a strongly monotone symmetric linear operator, that is, a linear operator satisfying


 * $$(Bu|v)=(u|Bv)\, $$ for all $$u, v$$ in $$Y$$
 * $$(Bu|u) \ge c\|u\|^2$$ for some constant $$c>0$$ and all $$u$$ in $$Y.$$

The energetic inner product is defined as
 * $$(u|v)_E =(Bu|v)\,$$ for all $$u,v$$ in $$Y$$

and the energetic norm is
 * $$\|u\|_E=(u|u)^\frac{1}{2}_E \, $$ for all $$u$$ in $$Y.$$

The set $$Y$$ together with the energetic inner product is a pre-Hilbert space. The energetic space $$X_E$$ is defined as the completion of $$Y$$ in the energetic norm. $$X_E$$ can be considered a subset of the original Hilbert space $$X,$$ since any Cauchy sequence in the energetic norm is also Cauchy in the norm of $$X$$ (this follows from the strong monotonicity property of $$B$$).

The energetic inner product is extended from $$Y$$ to $$X_E$$ by
 * $$ (u|v)_E = \lim_{n\to\infty} (u_n|v_n)_E$$

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 $$B$$ 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


 * $$\langle B_E u | v \rangle_E = (u|v)_E$$ for all $$u,v$$ in $$X_E.$$

Here, $$\langle \cdot |\cdot \rangle_E$$ denotes the duality bracket between $$X^*_E$$ and $$X_E,$$ so $$\langle B_E u | v \rangle_E$$ actually denotes $$(B_E u)(v).$$

If $$u$$ and $$v$$ are elements in the original subspace $$Y,$$ then


 * $$\langle B_E u | v \rangle_E = (u|v)_E = (Bu|v) = \langle u|B|v\rangle$$

by the definition of the energetic inner product. If one views $$Bu,$$ which is an element in $$X,$$ as an element in the dual $$X^*$$ via the Riesz representation theorem, then $$Bu$$ will also be in the dual $$X_E^*$$ (by the strong monotonicity property of $$B$$). Via these identifications, it follows from the above formula that $$B_E u= Bu.$$ In different words, the original operator $$B:Y\to X$$ 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 $$B$$ from $$Y$$ to $$X_E.$$

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 $$x$$ $$(a\le x \le b)$$ on the string be $$f(x)\mathbf{e}$$, where $$\mathbf{e}$$ is a unit vector pointing vertically and $$f:[a, b]\to \mathbb R.$$ Let $$u(x)$$ be the deflection of the string at the point $$x$$ under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is


 * $$\frac{1}{2} \int_a^b\! u'(x)^2\, dx$$

and the total potential energy of the string is


 * $$F(u) = \frac{1}{2} \int_a^b\! u'(x)^2\,dx - \int_a^b\! u(x)f(x)\,dx.$$

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 $$u:[a, b]\to \mathbb R$$ in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product


 * $$(u|v)=\int_a^b\! u(x)v(x)\,dx,$$

with the norm being given by


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

Let $$Y$$ be the set of all twice continuously differentiable functions $$u:[a, b]\to \mathbb R$$ with the boundary conditions $$u(a)=u(b)=0.$$ Then $$Y$$ is a linear subspace of $$X.$$

Consider the operator $$B:Y\to X$$ 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


 * $$(Bu|v)=-\int_a^b\! u''(x)v(x)\, dx=\int_a^b u'(x)v'(x) = (u|Bv) $$

for any $$u$$ and $$v$$ in $$Y.$$ Therefore, $$B$$ is a symmetric linear operator.

$$B$$ is also strongly monotone, since, by the Friedrichs's inequality


 * $$\|u\|^2 = \int_a^b u^2(x)\, dx \le C \int_a^b u'(x)^2\, dx = C\,(Bu|u)$$

for some $$C>0.$$

The energetic space in respect to the operator $$B$$ is then the Sobolev space $$H^1_0(a, b).$$ We see that the elastic energy of the string which motivated this study is


 * $$\frac{1}{2} \int_a^b\! u'(x)^2\, dx = \frac{1}{2} (u|u)_E,$$

so it is half of the energetic inner product of $$u$$ with itself.

To calculate the deflection $$u$$ minimizing the total potential energy $$F(u)$$ of the string, one writes this problem in the form


 * $$(u|v)_E=(f|v)\,$$ for all $$v$$ in $$X_E$$.

Next, one usually approximates $$u$$ 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 $$u$$ and $$u_h$$, see Céa's lemma.