Approximate the solution of $u(x,y) = xy^2$ of the Laplace equation in the rectangle $R = (0,\frac{3}{2}) \times (0,1)$, with steps $\Delta x = \Delta y = \frac{1}{2}$ and Dirichlet boundary conditions.
The numerical solution is exact in every point. WHY?
I have no idea what is the reason, but my guess is because it's kind of a trick question, and has something to do with the rectangle being open!
I started believing it was because if I'm using finite differences, then my grid would only contain two nodes of unknown values, but since the hypothesis tells us that the boundary conditions are of Dirichlet type, I would already know the exact value of those nodes! (but this just sounds too easy and kind of wrong)
Any help is appreciated!
EDIT:
Based on the comment by LutzL, I realized I missed some information. The Laplace equation I'm asking about is: $\Delta u := u_{xx} + u_{yy} = 0$. So, for this specific case it would be: $\Delta u = 2x = 0$.
The numerical method I'm using is finite central differences, so in terms of the approximate solution, it would be the following: \begin{align*} w_{i,j} = \frac{1}{4}\left(w_{i-1,j} + w_{i+1,j} + w_{i,j-1} + w_{i,j+1} \right) \end{align*}