# finite volume for diffusion equation with anisotropic (tensor) coefficient

Consider the scalar PDE for $u$ with Dirichlet boundary conditions:

$\mathrm{div}(\mathcal{K}\nabla u) = f\; \forall x\; \in \Omega \subset R^2$,

$u = 0 \; \forall \; x\;\in \partial\Omega$

where $\mathcal{K} \in R^{2\times 2}$ is positive definite symmetric.

For a start I assume $\Omega$ is the unit square with a uniform rectangular mesh.

EDIT: I edited the question following Jed's comment to make it more specific.

If $\mathcal{K}$ is diagonal, it is simple enough. When it is not, I see that there it will require a computation of the "secondary gradient" (gradient along the face). How is that usually done ? And what I should do when I am near the boundary in this case ?

• This sounds like it could be a homework question. In any case, please read the previous questions on this topic and if something is still unclear, explain what you feel is missing from those questions. – Jed Brown Apr 14 '14 at 4:35
• @JedBrown: I did go through the notes at the link again. I have updated my question to be more specific. – me10240 Apr 14 '14 at 21:56
• In the MPFA finite-volume schemes, that gradient is implicit in integrating the tensor coefficients over elements. (I find these methods, as discussed in the Wheeler and Yotov paper, to be more principled.) For the reconstruction schemes, see the formulas on page 86 of Murthy's lecture notes (linked) and perhaps the least-squares gradient discussed in the preceding pages. A formula is sitting right there on the page next to the term "secondary gradient", so your question should really be more specific. – Jed Brown Apr 16 '14 at 4:45
• For strong anisotropy there are important subtleties there due to numerical pollution. I recommend looking at my paper "On numerical solution of strongly anisotropic diffusion equation on misaligned grids" NUMERICAL HEAT TRANSFER PART B-FUNDAMENTALS, 47(6):533–554, JUN 2005. – Maxim Umansky Apr 17 '14 at 1:42