Talk:Discrete Poisson equation

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 $h^{4}$ 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

$\lambda _{a,b}=-4\left[\sin ^{2}({\frac {a\pi }{2(m+1)}})+\sin ^{2}({\frac {b\pi }{2(m+1)}})\right]$ for $a$ and $b$ ranging from $1,2,\cdots ,m$. 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 $\sin(\alpha x)\sin(\beta y)$. Lavaka 18:02, 20 November 2006 (UTC)

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)

In this page the A is written wrongly. The diagonal ( i,j+1 (when i=j) ) should be -1 for all. And same with (i,j-1). Dekay315 —Preceding undated comment added 12:01, 15 June 2018 (UTC)

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)

Correct Algebraic Signs?

Hi,

I fortunately found this page, when I was trying to implement a solver for a boundary value problem in image processing discipline. I noted, however, that the solutions produced by the solver showed an oposite sign as expected.

Recalculating the approach given on the wiki page, I realized that there is a sign flaw regarding the right-hand-side of the equation system. Since the block tridiagonal notion of A uses signs flipped with respect to the discretetized 2 dimensional Poisson equation (given in the first formula on the page) - the constant vector b has to use oposite signs on the derivatives ($g_{ij}$) as well.

So, the correct notion for the constant vector ${\begin{bmatrix}b\end{bmatrix}}$ would be:

$b={\begin{bmatrix}-dx^{2}g_{22}+u_{12}+u_{21}\\-dx^{2}g_{32}+u_{31}~~~~~~~~\\-dx^{2}g_{42}+u_{52}+u_{41}\\-dx^{2}g_{23}+u_{13}~~~~~~~~\\-dx^{2}g_{33}~~~~~~~~~~~~~~~~\\-dx^{2}g_{43}+u_{53}~~~~~~~~\\-dx^{2}g_{24}+u_{14}+u_{25}\\-dx^{2}g_{34}+u_{35}~~~~~~~~\\-dx^{2}g_{44}+u_{54}+u_{45}\\\end{bmatrix}}$

I changed the signs on the page accordingly.

BR

Smader (talk) 13:32, 13 February 2008 (UTC)

I checked the reference:

Cheny, Ward and David Kincaid, Numerical Mathematics and Computing 2nd Ed.,Brooks/Cole Publishing Company, Pacific Grove, 1985, page 448

and you are right. How did no one notice this until now?CFDFEM (talk) 17:37, 13 February 2008 (UTC)

Generally, the Discrete Poisson Equation takes the form

$-({\nabla }^{2}u)_{ij}=-{\frac {1}{dx^{2}}}(u_{i+1,j}+u_{i-1,j}+u_{i,j+1}+u_{i,j-1}-4u_{ij})=g_{ij}$

Without the negative in front, what is on the page after BR's correction is correct. I suspect that there was a typo that lead to the problem.

98.201.170.218 (talk) 19:21, 6 May 2009 (UTC)

Blanking

I have reverted the blanking of the article a number of times. First off, not every sentence needs a citation. In fact, for standard material like that in the section Discrete Poisson equation#On a two-dimensional rectangular grid, a general reference should be enough, as long as the material is verifiable. WP:SCICITE does not demand that each and every statement be cited individually, and I think that is clearly the case here. We don't even delete material that is in principle verifiable, that lacks in citations. If necessary, we tag it with a template like {{citation needed}}, in hopes that someone will provide a citation. We don't have a scorched-earth policy towards uncited material, and this is not original research. Anyone who argues otherwise either doesn't know enough to make that assessment, or is not acting in good faith: this is completely standard material.

Secondly, WP:NOT#HOWTO doesn't mean that we shouldn't include a discussion of methods for solution of a system in an encyclopedia article about a system of equations. It doesn't mean that any article with the phrase "how to solve" should be decimated. Rather it means that we don't provide instruction manuals. That is not what this is. Sławomir Biały (talk) 22:37, 20 November 2011 (UTC)

I really don't want to get into the middle of this, but I would like to re-emphasize one point Sławomir made: for scientific and mathematical articles in Wikipedia, the general rule is general reference in cases like this. One or a small number of references are given that cover a large portion of the article, with only a few inline citations on individual claims (where material is not from those general sources). This is unlike political or culture articles where there is not a general reference for the article and so many/most sentences are individually cited.

CRGreathouse (t | c) 00:19, 21 November 2011 (UTC)

I agree with the comment above and also wanted to emphasize another point. The section on Method of solution, which keeps getting removed per WP:NOT#HOWTO is not a how to. It neutrally lists multiple different algorithms that might be used to solve this system. Thenub314 (talk) 07:02, 22 November 2011 (UTC)

Correction

The matrix A for poisson equation is written wrongly. The diagonal (i,j+1(where i=j)) should be -1 for whole matrix and same with (i,j-1). Dekay315 Dekay315 (talk) 12:04, 15 June 2018 (UTC)