Energetic space

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 c0 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 normenergetic 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.

I commented out the below text, since it is not clear what norm one uses to talk about convergence and boundedness. I will think more about it.

That is, B_E is that linear functional which acts like B but has a domain of X_E—that is, its domain includes all limit points, u, of the domain of B for which Bun is bounded as u_n\to u.

An example from physics

Consider a string whose endpoints are fixed at two points a 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 Locally integrable function (\text{L}^1_\text{loc}) on [a, b] that are twice continuously differentiable from 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 C0.

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.

References

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