I am not very familiar with analytic solution of PDEs. Here is the problem I don't know how to solve:
Let $\Omega$ be the unit square $(0,1)^2$, we consider the elliptic equation
$-div(k(x,y) \nabla u(x,y))=1$ in $\Omega$,
$u(x,y)=0$ on $\partial \Omega$,
where $k(x,y) = .2(2.5 + 1.5\sin (2\pi x))(2.5 + 1.5\cos (2\pi y)).$
Does we have an explicit analytic solution for this equation?