![]() Despite this, the method is easy to understand and can be a start for other methods of solving the equation. This allows us to use a relaxation method to solve it, although there are faster methods that one could find (for example, one could use Fourier transform). ![]() What is nice about this equation, apart from the superposition principle is that the solutions are harmonic functions which means that the value in a point is the average of the points around it (for a more rigorous explanation, please see the Wikipedia page). You can find the Laplace equation (or more general, the Poisson equation) in various topics, for example originating from Gauss law if you use the electric potential instead of the electric field, or in the heat equation in a steady state when the time derivative drops out (also see: diffusion equation). A web page is here 2 and another one here 3. The methods are easy and I suppose Wikipedia pages are a good start, but here there are several other links for help, I just googled them, you can find many more on the subject. So, this post is about the relaxation method combined with a multigrid method applied on the Laplace equation. It’s just too simple and besides, it’s nice to see the code action in the page. ![]() I didn’t feel the need and didn’t have the time and patience to write a C++ program for this topic. I might only post easy things like this one for a while… anyway, here it is, with only brief explanations and the JavaScript code. It looks like I will be quite busy for a while so I won’t have much time for the blog. ![]()
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June 2023
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