The FEM is usually used with a weak form of PDE. But for the non-divergence form elliptic operator $$ -a_1(x,y) \frac{\partial^2}{\partial x^2} - a_2(x,y) \frac{\partial^2}{\partial y^2} $$ or another non-divergence form $$ -\frac{\partial^2}{\partial x^2}a_1(x,y)\cdot - \frac{\partial^2}{\partial y^2} a_2(x,y)\cdot $$
is it inevitable to involve the derivatives $\partial_x a_1(x,y), \partial_y a_2(x,y)$? Is it better to use FD (finite difference) or collocation methods instead of Galerkin FE
Updates: Since I just came back from Swiss Numerical Colloquium 2013, there was a fantastic talke given by Endre Suli on the DG approximation of the Hamilton-Jacobi-Bellman equation which is fully non-linear and involves a non-divergence form elliptic equation which has non-smooth coefficients. He gave very good reviews on the FD methods and proposed the high order DG method. His paper can be found here. I hope interested persons would also enjoy his paper. Just for fun!