My question is about understanding the derivation of the weak form of a variational problem (to be used for the solution via the finite element method).
The problem is as follows (it is an image processing problem, $\mathbf{u}$ is a 2D vector field of displacements I is an image function (see Horn-Schunk optical flow algorithm)):
Minimize the following functional w.r.t. the vector field $\mathbf{u}$
$$E(\mathbf{u}) = \int_\Omega \left\|\mathbf{u}\cdot\nabla I + \frac{\partial I}{\partial t}\right\|^2dx + \lambda\int_\Omega \nabla \mathbf{u} : \nabla \mathbf{u} dx$$
(: denotes the Frobenius inner product http://en.wikipedia.org/wiki/Frobenius_inner_product#Frobenius_product)
The apparent weak form of the problem is given in the following form (http://code.google.com/p/debiosee/wiki/DemosOptiocFlowHornSchunck):
$$E(\mathbf{u}) = \int_\Omega \left(\mathbf{u}\cdot\nabla I + \frac{\partial I}{\partial t}\right) \left(\mathbf{v}\cdot\nabla I\right) dx + \lambda\int_\Omega\nabla\mathbf{u}:\nabla\mathbf{v}dx$$
Where $\mathbf{v}$ is the test function.
What I would like to know is if there is a direct way to come to this conclusion from the original problem or the way to go is to formulate the weak form of the Euler-Lagrange equation of the original functional?
Additionally I would like to know if any of you knew about a vector calculus textbook you could recommend for mastering these kinds of operations in their vector form without working out the derivation componentwise?
Thanks a lot in advance!
Peter
Best Answer
To answer your first equation, there is no direct way by looking at the functional to derive the variational problem. The only way I know is to use the method same as deriving Euler-Lagrange equation like you said. $\newcommand{\b}{\mathbf}$
Let $\b{v}\in V$. $V$ is the test space. Normally the difference between $V$ and the space $\b{u}$ lies in is the boundary condition. Since $\b{u} = \mathrm{argmin}_{\b{v}} E(\b{v})$, adding any perturbation would result the functional being bigger, we have for any $\b{v}\in V$: $$ \lim_{\epsilon\to 0}\frac{d}{d\epsilon}E(\b{u}+\epsilon\b{v}) = 0 $$ First compute: $$ \begin{aligned} E(\b{u}+\epsilon\b{v}) &= \int_\Omega \left|(\b{u}+\epsilon\b{v})\cdot\nabla I + \frac{\partial I}{\partial t}\right|^2 + \lambda\int_\Omega \nabla (\b{u}+\epsilon\b{v}): \nabla (\b{u}+\epsilon\b{v}) \\ &= \int_\Omega \left\{(\b{u}+\epsilon\b{v})\cdot\nabla I + \frac{\partial I}{\partial t}\right\}^2 + \lambda\int_\Omega \nabla \b{u}: \nabla \b{u} + \lambda \epsilon^2\int_\Omega \nabla \b{v}: \nabla \b{v} + 2\lambda \epsilon\int_\Omega \nabla \b{u}: \nabla \b{v} \end{aligned} $$ I am replace your notation of $\|\cdot\|$ by absolute value since it is a scalar inside. Taking derivative with respect to $\epsilon$: $$ \begin{aligned} \frac{d}{d\epsilon}E(\b{u}+\epsilon\b{v}) &= 2\int_\Omega \left((\b{u}+\epsilon\b{v})\cdot\nabla I + \frac{\partial I}{\partial t}\right) \frac{d}{d\epsilon}\left((\b{u}+\epsilon\b{v})\cdot\nabla I + \frac{\partial I}{\partial t}\right) \\ &\quad + 2\lambda \epsilon\int_\Omega \nabla \b{v}: \nabla \b{v}+2\lambda\int_\Omega \nabla \b{u}: \nabla \b{v} \\ &= 2\int_\Omega \left((\b{u}+\epsilon\b{v})\cdot\nabla I + \frac{\partial I}{\partial t}\right) \left(\mathbf{v}\cdot\nabla I\right) + 2\lambda \epsilon\int_\Omega \nabla \b{v}: \nabla \b{v}+2\lambda\int_\Omega \nabla \b{u}: \nabla \b{v} \end{aligned} $$ Letting $\epsilon \to 0$: $$ 0=\lim_{\epsilon \to 0}\frac{d}{d\epsilon}E(\b{u}+\epsilon\b{v}) = 2\int_\Omega \left(\b{u}\cdot\nabla I + \frac{\partial I}{\partial t}\right) \left(\mathbf{v}\cdot\nabla I\right) +2\lambda\int_\Omega \nabla \b{u}: \nabla \b{v} $$ And this gives us the variational problem: For any $\b{v}\in V$: $$ \int_\Omega \left(\b{u}\cdot\nabla I \right) \left(\mathbf{v}\cdot\nabla I\right) +\lambda\int_\Omega \nabla \b{u}: \nabla \b{v} = \int_\Omega \frac{\partial I}{\partial t} \left(\mathbf{v}\cdot\nabla I\right) $$ This is in the form of $B(\b{u},\b{v}) = F(\b{v})$, which can be the bilinear form for finite element method.
About the second question, there are many resources on the internet covering vector calculus identities. Instead of component-wise derivation, working on the differential operator as a whole not only eases the computation a lot, but gives you an insight of the physical meaning. For example, see the Vector Calculus Identities page on Wikipedia.