Normal unit vector in variational formulation

variational formulation
1

Hi everybody,
I need to define two subdomains, each of which has its own governing PDE, and an interface coupling condition between the two. The variational formulation resembles

... + \int_{\Gamma_s} (\mathbf{n}_1 \cdot \mathbf{u}) \overline{q} ds + \int_{\Gamma_s} \mathbf{n}_1 \cdot \mathbf{\overline{v}} ds = \int_{\Gamma}(\partial_{\mathbf{n}_2} p) \overline{q} ds + ...

Here, \overline{q} and \mathbf{\overline{v}} are test functions, p and \mathbf{u} are trial functions, \mathbf{n}_1 is the outward surface-normal unit vector on \Gamma_s (interface), and \mathbf{n}_2 is the outward surface-normal unit vector on \Gamma (external boundaries).

My question is, what is the proper way of defining \mathbf{n}_1 and \partial_{\mathbf{n}_2}?

I thought of using

n_1 = FacetNormal(mesh)
n_2 = FacetNormal(mesh)

a = ... + inner(n_1, u)*q*ds(1) + inner(n_1, v)*ds(1)
L = inner(grad(p),n_2)*q*ds(0) + ...

but Iā€™m not sure if the ā€œFacetNormal()ā€ function was designed for this. Can someone explain it?
Thanks!

2

FacetNormal is exactly what you are looking for. And the terms you implemented are correct for the variational form above.

But as you said that Gamma_s is an interface, I would expect it to be a interior surface measure dS which you integrate over, and in this case you have to restrict the quantities using, e.g.,

inner(n_1('+'), u('+'))*q('+')*dS(1)

as described here Integrating over an interior surface

Also, no need to define n_1 and n_2 here, you can just use one object for these (it is the same anyway, and is restricted by the appropriate (surface) measures)

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