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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!

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Does your application already lead to this formulation? –  shuhalo Feb 18 '13 at 12:35
    
I had no application in mind. But it appears somewhere, e.g. the Kolmogorov equation. I also heard before that some process in biology is modelled also in non-divergence form. –  Hui Zhang Feb 18 '13 at 19:45
    
Even if the diffusion tensor is discontinuous, you can still multiply it by a test function and integration by parts element by element assuming the triangulation resolves the interface of $a$. –  Shuhao Cao Jul 15 '13 at 6:12
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You can of course re-write the equation in a form that is more amenable to the FEM: $$ -\nabla \cdot \left(\begin{pmatrix}a_1 & 0 \\ 0 & a_2\end{pmatrix} \nabla u\right) + \begin{pmatrix} \partial_x a_1 \\ \partial_y a_2\end{pmatrix} \cdot \nabla u = f. $$ This is an advection-diffusion problem. This form also gives you a better idea of whether the problem is advection dominated or diffusion dominated and consequently whether it will be necessary to stabilize the discretization or not.

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Suppose $a_1, a_2$ are discontinuous, what can you do? Is the Galerkin weak form always superior to collocation strong form? –  Hui Zhang Feb 18 '13 at 21:22
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You are taking too many steps at once. You need to first consider what that means for the continuous equation to begin with, for example if the solution exists at all. I'm no expert in this, but imagine that $f$ is a continuous function. Then if you take the original, non-standard form, for the product of $a$ and the second derivatives of $u$ to be equal to the continuous $f$, you need that the second derivatives of $u$ are discontinuous as well. That means that $u\not\in C^2$, so not a classical solution. Would the collocation method converge in that case? –  Wolfgang Bangerth Feb 19 '13 at 3:45
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