Solve the 2d Laplace equation in a square whose sides have length $L$, under the Dirichlet boundary conditions $u(0, y) = \sin(\frac{\pi y}{L})$, $u(L, y) = \sin(\frac{2\pi y}{L})$, $u(x, 0) = \sin(\frac{\pi x}{L})$, $u(x, L) = \sin(\frac{2\pi x}{L})$
(2d Laplace equation:$\partial_x^2 u + \partial_y^2 u = 0$)