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1332:{\displaystyle 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}}}
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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
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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
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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
Knowledge, 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
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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
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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.
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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.
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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
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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:
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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
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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.
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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
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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
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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
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I agree with the comment above and also wanted to emphasize another point. The section on Method of solution, which keeps getting removed per
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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.
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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.
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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
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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).
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This article is pretty poor IMHO. How about explaining what it is without resorting to algebra, and also explaining its applications... --
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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).
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doesn't mean that we shouldn't include a discussion of methods for solution of a system in an encyclopedia article
1606:{\displaystyle -({\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}}
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be the number of interior grid points, and let the domain be the unit grid, all the eigenvalues are in the form
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is not a how to. It neutrally lists multiple different algorithms that might be used to solve this system.
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perhaps a mention to the form of the discrete
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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method out there that takes advantage of the above. Obviously, I have to research this further.
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mention the discrete laplace equation and how it is simpler (and how the 5 pt stencil gets
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page) - the constant vector b has to use oposite signs on the derivatives (
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for the case of a square grid, derive the condition number of the matrix
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make the matrices a bit more general, using kronecker product notation
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I added an "Applications" section for where it is encountered in
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for the case of a square grid, the eignevalues and eigenfunctions
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Discrete
Poisson equation#On a two-dimensional rectangular grid
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and you are right. How did no one notice this until now?
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Generally, the
Discrete Poisson Equation takes the form
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1366:Numerical Mathematics and Computing 2nd Ed.
598:{\displaystyle \sin(\alpha x)\sin(\beta y)}
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485:{\displaystyle 1,2,\cdots ,m}
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40:WikiProjects
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653:—Preceding
112:Mathematics
103:mathematics
59:Mathematics
30:Start-class
1774:Categories
1752:Correction
1680:Secondly,
1657:WP:SCICITE
1653:verifiable
780:would be:
1738:Thenub314
512:and when
1758:Dekay315
1643:Blanking
650:Dekay315
676:Rebroad
655:undated
139:on the
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1348:Smader
690:Slffea
629:Slffea
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1776::
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