Differentiation summations by indexed function

[ognik]

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 a link 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 $ \pd{E}{{\phi}_{ij}} $

I think that I can just differentiate inside the summations?
What do I do with the i-1 and j-1 terms?
------------------
Thanks for reading.

 

[jvicens]

The variational principle is a powerful tool in solving PDEs numerically. It allows us to find the function that minimizes a given functional (in this case, the energy functional E) and satisfies certain boundary conditions. In this case, we are looking at a discretized version of an elliptical PDE on a 2D grid.

To answer your first question, the starting equation is the energy functional E, which is a measure of the total energy of the system. It is obtained by integrating the energy density (the integrand inside the integral) over the entire region of interest. The integrand itself is a combination of the kinetic energy (the first term) and the potential energy (the second term). The goal is to find the function phi that minimizes this energy functional.

The next step involves using the variational principle, which states that the variation of the energy functional is zero for the true solution. This means that the true solution will make the energy functional stationary. By "stationary", we mean that the variation of the energy functional with respect to the function phi is zero. This is where the boundary conditions come into play - the variations must respect the Dirichlet boundary conditions.

The variation of the energy functional is then calculated, and using integration by parts, we reach the final equation that you have mentioned. The line integral is over the boundary of the region of interest, and the unit vector n is used to ensure that the variation respects the boundary conditions.

To answer your second question, the equation that follows is just a discretized version of the energy functional, where the continuous variables have been replaced by discrete variables. The summations represent the contributions from each grid point, and the h term is the grid spacing. This equation is used for numerical calculations.

To take the differential with respect to phi, you can just differentiate inside the summations. The i-1 and j-1 terms will be taken care of by the chain rule. I hope this helps to clarify the steps in the discretization process.