Talk:Discrete Poisson equation
From Wikipedia, the free encyclopedia
It would be nice to expand this page a bit so that it has real information, not just as a page that gets people "started in the right direction". Things that would be nice to add:
- for the case of a square grid, derive the condition number of the matrix
- for the case of a square grid and disc, mention or derive the fast poisson methods (those involving the FFT; see, eg. Arieh Iserles' book).
- for the case of a square grid, the eignevalues and eigenfunctions
- various stencils and their accuracy, e.g. standard 5 pt, and the 9 pt and modified 9 pt
- mention the discrete laplace equation and how it is simpler (and how the 5 pt stencil gets h4 accuracy, similar to the modified 9 pt stencil)
- make the matrices a bit more general, using kronecker product notation
- perhaps a mention to the form of the discrete Laplacian, e.g. block TST (Toeplitz Symmetric Tridiagonal)
- Lavaka 02:30, 18 September 2006 (UTC)
- I might not be the right person to address some of the things mentioned above which is beyond what I have seen with this subject. For instance, in terms of the eignevalues of this system, I am not aware if there is an expression that easily gives them. I don't see anything mentioned in my numerical methods books. I have seen discussion of FFT as a solution method, but I want to apply it before I am comfortable elaborating more on it here.Slffea 20:46, 19 November 2006 (UTC)
-
- There is no expression for the eigenvalues of the discrete poisson equation for arbitrary domains, but over a rectangular or square grid, with uniform spacing, it is pretty simple. Let m be the number of interior grid points, and let the domain be the unit grid, all the eigenvalues are in the form
![\lambda_{a,b} = -4\left[\sin^2(\frac{a \pi}{2(m+1)}) + \sin^2(\frac{b \pi}{2(m+1)})\right]](../../../math/1/3/d/13dbdb3df36c0d7463a5d3ace7ad4cac.png)
for a and b ranging from 
. For large m and when a and b are small, this is close to the spectrum of the continuous laplacian. You can derive this if you assume the eigenfunctions are in the form sin(αx)sin(βy). Lavaka 18:02, 20 November 2006 (UTC)
- There is no expression for the eigenvalues of the discrete poisson equation for arbitrary domains, but over a rectangular or square grid, with uniform spacing, it is pretty simple. Let m be the number of interior grid points, and let the domain be the unit grid, all the eigenvalues are in the form
-
-
-
- This is a good example of what I mean. I have gone though my numerical methods books as well as papers I Xeroxed out of some journals, and I don't see this expression. It seems to me having all the eigenvalues should be very helpful in solving the Poisson equation, so I am wondering if there is a method out there that takes advantage of the above. Obviously, I have to research this further.Slffea 23:18, 20 November 2006 (UTC)
-
-
[edit] less algebra, and more explanation please
This article is pretty poor IMHO. How about explaining what it is without resorting to algebra, and also explaining its applications... --Rebroad 10:21, 19 November 2006 (UTC)
- I added an "Applications" section for where it is encountered in Computational fluid dynamics, which is the only place I have used this discretization. I don't think I can do anything about the algebra though.Slffea 20:46, 19 November 2006 (UTC)
