5
$\begingroup$

So, we are solving the Biharmonic equation ($\Delta^2 u = f$) on a rectangle by solving the Poisson equation ($\nabla^2 u = f$) two times. We have nice boundary conditions, $u = 0$ and $\Delta u = 0$ on the boundary.

We use a 5 point scheme where

$\frac{1}{h^2} \delta_x^2u_p + \frac{1}{k^2} \delta_y^2u_p = f_p + \tau_p$

and we get the following expression for the truncation error:

$\tau_p = \frac{1}{12}h^2\partial_x^4u_p + \frac{1}{12}k^2\partial_y^4u_p$

Assuming "nice" initial value function $f$ we want to prove that the method is convergent, ie. that the global error somehow goes to zero when the step sizes $h$ and $k$ goes to zero.

Any clues on how to proceed to show this?

$\endgroup$

migrated from math.stackexchange.com Apr 16 '12 at 16:22

This question came from our site for people studying math at any level and professionals in related fields.

  • $\begingroup$ I'm going to migrate this question to the Computational Science.SE site. There will be a link that appears below the question here that you can follow to the new location of your question. If you need help associating an account on Computational Science.SE, you can flag your question for moderator attention, and someone over there will help out. $\endgroup$ – Zev Chonoles Apr 16 '12 at 16:22
2
$\begingroup$

From what I can remember, proving stability for the five point stencil has a classic proof involving the spectral radius because the eigenvalues are a simple trigonometric function. While I haven't got my numerical analysis books handy, I was able to find the proof in p. 43 of this note (this is probably the notes which ended up as the previously mentioned LeVeque book)

Note that you probably want to look at the 1D case first since the note is very brief on the 2D case. I suppose it depends if you want to take the eigenvalue identity for granted, the rest is quite straightforward.

$\endgroup$
1
$\begingroup$

Have you tried some good books about finite differences? Pretty sure that they show how to prove that a given method is convergent or not.

J. Strikwerda, Finite Difference Schemes and Partial Differential Equations, SIAM.

http://pages.cs.wisc.edu/~strik/strik.html

R. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady State and Time Dependent Problems, SIAM.

http://faculty.washington.edu/rjl/fdmbook/index.html

$\endgroup$
  • $\begingroup$ I have a copy of J. Strikwerda's book, and I am currently reading up on the different ways to show stability (which I need to show). Until now I have found it to focus on time dependant one-dimensional problems. I also have borrowed a copy of A. Iserles, A First Course in the Numerical Analysis of Differential Equations. Will come back with updates if I figure it out, thanks. $\endgroup$ – burk Apr 16 '12 at 18:58

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.