Talk:Discrete Poisson equation

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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] 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(αx)sin(βy). 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)

[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)

[edit] 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 (gij) 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)