If you had to pick the best method for solving Poisson’s Equation, which would it be and why?

A numerical method? An analytical method?
In the whole space? In a bounded domain? What type of boundary conditions? What type of forcing function?

The answer really depends!
If the RHS is Hölder, then continuity method + Schauder estimates. If the RHS is in L^2, then Lax-Milgram/Riesz representation.
the function u(x,y) is nonzero from 0<x<1 and 0<y<1 with Dirichlet BCs. I am looking for a best numerical/discrete method.

Rectangular domain as well

Edit: Dirichlet BCs such that u(0,y) = u(x,0) = 0

We will use the basic Poisson’s Equation form and the function f(x,y) being the RHS of the eqn will be:

f(x,y) = 2^(4a)*(1-x)^(a)*(1-y)^(a)*x^(a)*y^a

We will work on the domain of 0<=x<=1 and 0<=y<=1 so that the function forms a nice opening down paraboloid.

Edit: a can be whatever you’d like, but we will set it to a=1 for simplicity

Edit 2: I set the solution to be u(x,y) = 2^(4a)*(1-x)^(a)*(1-y)^(a)*x^(a)*y^(a).

f(x,y) (the RHS) is the laplacian of u(x,y). Then I tried different methods to numerically get back to u(x,y).

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