I am currently trying to solve the following PDE using various numerical methods - direct and iterative.
$$ \nabla ^2 u = \rho $$
where $u = u(x,y)$ and $\rho(0.5,0.5) = 2$ (zero elsewhere). The domain is $0<x,y<1$ and $u = 0$ on the boundaries.
I am fairly sure that I have written my numerical algorithms correctly, however I do not have the knowledge to interpret the output in relation to what the "true" solution is.
I have found that for all grids the discretized laplacian at (0.5,0.5) is equal to two within floating point precision - this is what I expect should be the case. My main issue is that I do not witness convergence in any sense that I understand, as the grid spacing decreases the values of the solution just get smaller and smaller. It seems to me that if I could solve on a grid with infinitely small spacing (this is my understanding of what the "true" solution is?) then I would just have a constant solution that takes the value zero at all points in the domain??
I would appreciate any answer that could explain what the analytical solution to this PDE is and any insight into what behaviour I should look for in terms of convergence for my numerical methods - I am clear how to describe the convergence of the iterative methods but not the convergence with respect to the "true" solution of the PDE. My gut instinct is that this particular problem is not well behaved in some sense!!
Thanks for any help!
Your numerical methods will generally converge to a weak solution, which will also be a strong solution if one exists. Here the weak solution is the zero function, and there is no strong solution (which should be unsurprising, since the forcing on the right side is not continuous). You can check that the weak solution is the zero function quite easily, because when you multiply by any test function, the right side will integrate to zero (since it is only nonzero at one point), so the left side should also integrate to zero as well.