On 12/17/20 8:56 PM, Corbin Foucart wrote:
I have a problem with direction-dependent diffusion (that is, the diffusivity
\nu depends on the derivative direction but not spatial location) such that
the horizontal diffusivity and vertical diffusivity are different, leading to
a Laplacian as follows:
I am discretizing the problem with a mixed method, which defines the auxiliary
variable L = ∇u such that
So in my weak form, I end up with terms like the following:
(please excuse the abuse of index notation with repeated j three times).
phi_ij here refers to each scalar basis function in the tensor-valued finite
element space.
It seems using the deal.ii FEValuesExtractors, I can access the divergence of
each tensor-valued basis function, but I need to multiply each term in the
divergence with the appropriate value of \nu by component.
I suppose I could ask for the gradient of each phi and do the sum manually,but
I was wondering if there's a way to do it directly. Is there a standard way of
doing this in deal.ii?
You (hopefully) don't ever need to take the divergence of nu*(grad u) but
instead will want to integrate by parts. This is going to give you the
bilinear form
(nabla phi_i, nu nabla phi_j)
When you assemble this, you typically do
fe_values.shape_grad(i,q) * fe_values.shape_grad(j,q)
where both objects are of type Tensor<1,dim> (i.e., vectors). In your case,
you want to do
SymmetricTensor<2,dim> nu; // fill with whatever values you need
and then do
fe_values.shape_grad(i,q) * nu * fe_values.shape_grad(j,q)
step-20 does something similar (though for a different bilinear form).
Best
W.
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