Energetic space

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

• ${\displaystyle (Bu|v)=(u|Bv)\,}$ for all ${\displaystyle u,v}$ in ${\displaystyle Y}$
• ${\displaystyle (Bu|u)\geq c\|u\|^{2}}$ for some constant ${\displaystyle c>0}$ and all ${\displaystyle u}$ in ${\displaystyle Y.}$

The energetic inner product is defined as

${\displaystyle (u|v)_{E}=(Bu|v)\,}$ for all ${\displaystyle u,v}$ in ${\displaystyle Y}$

and the energetic norm is

${\displaystyle \|u\|_{E}=(u|u)_{E}^{\frac {1}{2}}\,}$ for all ${\displaystyle u}$ in ${\displaystyle Y.}$

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

The energetic inner product is extended from ${\displaystyle Y}$ to ${\displaystyle X_{E}}$ by

${\displaystyle (u|v)_{E}=\lim _{n\to \infty }(u_{n}|v_{n})_{E}}$

where ${\displaystyle (u_{n})}$ and ${\displaystyle (v_{n})}$ are sequences in Y that converge to points in ${\displaystyle X_{E}}$ in the energetic norm.

The operator ${\displaystyle B}$ admits an energetic extension ${\displaystyle B_{E}}$

${\displaystyle B_{E}:X_{E}\to X_{E}^{*}}$

defined on ${\displaystyle X_{E}}$ with values in the dual space ${\displaystyle X_{E}^{*}}$ that is given by the formula

${\displaystyle \langle B_{E}u|v\rangle _{E}=(u|v)_{E}}$ for all ${\displaystyle u,v}$ in ${\displaystyle X_{E}.}$

Here, ${\displaystyle \langle \cdot |\cdot \rangle _{E}}$ denotes the duality bracket between ${\displaystyle X_{E}^{*}}$ and ${\displaystyle X_{E},}$ so ${\displaystyle \langle B_{E}u|v\rangle _{E}}$ actually denotes ${\displaystyle (B_{E}u)(v).}$

If ${\displaystyle u}$ and ${\displaystyle v}$ are elements in the original subspace ${\displaystyle Y,}$ then

${\displaystyle \langle B_{E}u|v\rangle _{E}=(u|v)_{E}=(Bu|v)=\langle u|B|v\rangle }$

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

A string with fixed endpoints under the influence of a force pointing down.

Consider a string whose endpoints are fixed at two points ${\displaystyle a<b}$ on the real line (here viewed as a horizontal line). Let the vertical outer force density at each point ${\displaystyle x}$ ${\displaystyle (a\leq x\leq b)}$ on the string be ${\displaystyle f(x)\mathbf {e} }$, where ${\displaystyle \mathbf {e} }$ is a unit vector pointing vertically and ${\displaystyle f:[a,b]\to \mathbb {R} .}$ Let ${\displaystyle u(x)}$ be the deflection of the string at the point ${\displaystyle x}$ under the influence of the force. Assuming that the deflection is small, the elastic energy of the string is

${\displaystyle {\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx}$

and the total potential energy of the string is

${\displaystyle F(u)={\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx-\int _{a}^{b}\!u(x)f(x)\,dx.}$

The deflection ${\displaystyle u(x)}$ minimizing the potential energy will satisfy the differential equation

${\displaystyle -u''=f\,}$

with boundary conditions

${\displaystyle u(a)=u(b)=0.\,}$

To study this equation, consider the space ${\displaystyle X=L^{2}(a,b),}$ that is, the Lp space of all square-integrable functions ${\displaystyle u:[a,b]\to \mathbb {R} }$ in respect to the Lebesgue measure. This space is Hilbert in respect to the inner product

${\displaystyle (u|v)=\int _{a}^{b}\!u(x)v(x)\,dx,}$

with the norm being given by

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

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

Consider the operator ${\displaystyle B:Y\to X}$ given by the formula

${\displaystyle Bu=-u'',\,}$

so the deflection satisfies the equation ${\displaystyle Bu=f.}$ Using integration by parts and the boundary conditions, one can see that

${\displaystyle (Bu|v)=-\int _{a}^{b}\!u''(x)v(x)\,dx=\int _{a}^{b}u'(x)v'(x)=(u|Bv)}$

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

${\displaystyle B}$ is also strongly monotone, since, by the Friedrichs's inequality

${\displaystyle \|u\|^{2}=\int _{a}^{b}u^{2}(x)\,dx\leq C\int _{a}^{b}u'(x)^{2}\,dx=C\,(Bu|u)}$

for some ${\displaystyle C>0.}$

The energetic space in respect to the operator ${\displaystyle B}$ is then the Sobolev space ${\displaystyle H_{0}^{1}(a,b).}$ We see that the elastic energy of the string which motivated this study is

${\displaystyle {\frac {1}{2}}\int _{a}^{b}\!u'(x)^{2}\,dx={\frac {1}{2}}(u|u)_{E},}$

so it is half of the energetic inner product of ${\displaystyle u}$ with itself.

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

${\displaystyle (u|v)_{E}=(f|v)\,}$ for all ${\displaystyle v}$ in ${\displaystyle X_{E}}$.

Next, one usually approximates ${\displaystyle u}$ by some ${\displaystyle u_{h}}$, a function in a finite-dimensional subspace of the true solution space. For example, one might let ${\displaystyle u_{h}}$ be a continuous piecewise linear function in the energetic space, which gives the finite element method. The approximation ${\displaystyle u_{h}}$ 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 ${\displaystyle u}$ and ${\displaystyle u_{h}}$, see Céa's lemma.

• Inner product space
• Positive-definite kernel

• Zeidler, Eberhard (1995). Applied functional analysis: applications to mathematical physics. New York: Springer-Verlag. ISBN 0-387-94442-7.

• Johnson, Claes (1987). Numerical solution of partial differential equations by the finite element method. Cambridge University Press. ISBN 0-521-34514-6.