Proving convergence of 5 point scheme for the Poisson equation

Question

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?

Answer 1

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.

Answer 2

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