- #1
ognik
- 643
- 2
Homework Statement
Hi - looking at 'discretizing elliptical PDEs'.
I understand the normal lattice approach, but this approach uses the variational principle. I have a couple of questions please. The text says:
$$ \: Given\: E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\frac{1}{2}\left(\nabla \phi\right)^{2} - S\phi \right] $$
"It is easy to show that E is stationary under all variations $\delta \phi$ that respect the Dirichlet boundary conditions imposed. Indeed the variation is $$ \delta E=\int_{0}^{1} \,dx\int_{0}^{1} \,dy\left[\nabla \phi .\nabla \delta \phi - S \delta \phi \right] $$
..which upon integrating the the second derivative by parts becomes...
$$ \delta E=\int_{C} \,dl\: \delta \phi \vec{n} . \nabla \phi + \int_{0}^{1} \,dx \: \int_{0}^{1} \,dy\: \delta \phi \: \left[-\nabla^2 \phi - S \right] $$
... where the line integral is over the boundary of the region of interest (C) and n is the unit vector to the boundary."
Sadly, while I recognize all the words and symbols, I cannot follow it at all; I also have no idea where they get the starting equation or what it means. (The sometimes problem with computational physics is that it assumes certain background knowledge). I am hoping someone can expand enough (or provide links) so that I can follow all the above steps in detail?
The second question follows a short bit later in the text, apparently the above 'easily' leads to the following equation:
$$ E= \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n}\left[\left({\phi}_{ij} - {\phi}_{i-1,j}\right)^2 + \left({\phi}_{ij} - {\phi}_{i,j-1}\right)^2 -{h}^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}{S}_{ij}{\phi}_{ij}\right] $$
I probably don't need to understand how they get to the above equation - I'd just prefer to understand as much as I can, so again please help me to follow it.
What I really MUST do is now take a differential of the above w.r.t. $$ {\phi}_{ij} ,\: IE\: \frac{\partial{E}}{{\partial{\phi}_{ij}}} $$
Homework Equations
$$ E= \frac{1}{2} \sum_{i=1}^{n}\sum_{j=1}^{n}\left[\left({\phi}_{ij} - {\phi}_{i-1,j}\right)^2 + \left({\phi}_{ij} - {\phi}_{i,j-1}\right)^2 -{h}^{2}\sum_{i=1}^{n}\sum_{j=1}^{n}{S}_{ij}{\phi}_{ij}\right] $$
The Attempt at a Solution
I don't even know where to start in terms of the text above, but I should be able to do the partial derivative of the relevant eqn - with a little help help on the following 2 queries first please:
I think that I can just differentiate inside the summations, is that right?
What do I do with the i-1 and j-1 terms when differentiating w.r.t. ∅ij?
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Thanks for reading.