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.
Formally, consider a real Hilbert space X with the inner product ( ⋅ | ⋅ ) and the norm ∥ ⋅ ∥ . Let Y be a linear subspace of X and B : Y → X be a strongly monotone symmetric linear operator, that is, a linear operator satisfying
( B u | v ) = ( u | B v ) for all u , v in Y ( B u | u ) ≥ c ∥ u ∥ 2 for some constant c > 0 and all u in Y . The energetic inner product is defined as
( u | v ) E = ( B u | v ) for all
u , v in
Y and the energetic norm is
∥ u ∥ E = ( u | u ) E 1 2 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 → ∞ ( 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.
The operator B admits an energetic extension B E
B E : X E → X E ∗ defined on X E with values in the dual space X E ∗ that is given by the formula
⟨ B E u | v ⟩ E = ( u | v ) E for all
u , v in
X E . Here, ⟨ ⋅ | ⋅ ⟩ E denotes the duality bracket between X E ∗ and X E , so ⟨ B E u | v ⟩ E actually denotes ( B E u ) ( v ) .
If u and v are elements in the original subspace Y , then
⟨ B E u | v ⟩ E = ( u | v ) E = ( B u | v ) = ⟨ u | B | v ⟩ by the definition of the energetic inner product. If one views B u , which is an element in X , as an element in the dual X ∗ via the Riesz representation theorem, then B u 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 = B u . In different words, the original operator B : Y → X can be viewed as an operator B : Y → X E ∗ , and then B E : X E → X E ∗ is simply the function extension of B from Y to X E .
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 ≤ x ≤ b ) on the string be f ( x ) e , where e is a unit vector pointing vertically and f : [ a , b ] → 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
1 2 ∫ a b u ′ ( x ) 2 d x and the total potential energy of the string is
F ( u ) = 1 2 ∫ a b u ′ ( x ) 2 d x − ∫ a b u ( x ) f ( x ) d x . 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 ] → R in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product
( u | v ) = ∫ a b u ( x ) v ( x ) d x , with the norm being given by
∥ u ∥ = ( u | u ) . Let Y be the set of all twice continuously differentiable functions u : [ a , b ] → R with the boundary conditions u ( a ) = u ( b ) = 0. Then Y is a linear subspace of X .
Consider the operator B : Y → X given by the formula
B u = − u ″ , so the deflection satisfies the equation B u = f . Using integration by parts and the boundary conditions, one can see that
( B u | v ) = − ∫ a b u ″ ( x ) v ( x ) d x = ∫ a b u ′ ( x ) v ′ ( x ) = ( u | B v ) for any u and v in Y . Therefore, B is a symmetric linear operator.
B is also strongly monotone, since, by the Friedrichs' inequality
∥ u ∥ 2 = ∫ a b u 2 ( x ) d x ≤ C ∫ a b u ′ ( x ) 2 d x = C ( B u | u ) for some C > 0.
The energetic space in respect to the operator B is then the Sobolev space H 0 1 ( a , b ) . We see that the elastic energy of the string which motivated this study is
1 2 ∫ a b u ′ ( x ) 2 d x = 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 linear system of 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.
