I want to solve the Forward Kolmogorov equation by Finite element method:
$Lf = (\nabla f)^T b + \frac{1}{2}tr((\nabla^2f)D)$
where $D = \begin{pmatrix} D_{11} & D_{12}\\ D_{21} & D_{22} \end{pmatrix}$ is the diffusion matrix.
For standard laplacian problem: $\Delta f = 0, x\in \Omega$, I know the weak formulation can be written as -$\int\nabla\phi\cdot\nabla fdx$ when assuming $\nabla f$ vanishes at the boundary and treats $\phi$ as test function. But how can we construct the weak formulation for this problem with weighted second order derivative. Specifically, how can we write $\frac{1}{2}(D_{11}\frac{\partial^2 f}{\partial x_1^2} + D_{22}\frac{\partial^2 f}{\partial x_2^2} + D_{12}\frac{\partial^2 f}{\partial x_1x_2}+D_{21}\frac{\partial^2 f}{\partial x_1x_2})$ into similar integral format?