Chapter 10.03: Lesson: Elliptic PDEs: Liebermann Method: Part 2 of 2

Learn the Lieberman method of solving elliptic PDEs

[AJ-etvf]

I successfully implemented this in Python and finally discovered the "right" error tolerance for this to work, that is, the max-infinity-norm of the percent error values should be used as demonstrated in this video. I was using the root-mean-squared error norm previously and it all resulted in the iterations exiting very early even though the values at the interior nodes haven't converged yet towards the approximate solution. Because of this, I had to resort to iterating solutions towards the maximum iterations I set it, e.g. 10 thousand, 20 thousand, etc, which makes it very inefficient.

Now I see that using the norm in a RMS-sense exits early because many values converge quickly and meet the desired criterion and thus dominating the RMS-norm, leaving the solutions unconverged. Hence, using the max-infinity-norm is the appropriate error tolerance criterion so that all values will converge and the iterations will appropriately stop once convergence is "detected".

Again, thank you very much for your videos sir. Very helpful and informative for me.

[davidc.x.pardede]

Thank you so much, keep up the great work!

[isupportargentina]

So after you've found a value at a node, you can immediately make use of that value going forwards?

[bliengme]

There is a minor type in the final table showing T values after 1, 2 and 9 iterations. For 9 iterations, T(3, 3) should read 131.2828 not 131.2525..