[Getfem-users] Interface Conditions in GetFem++

[Domenico Notaro]

Dear GetFem++ Users,

I am wondering how to implement compatibility conditions for the unknown velocity at the interface between elements.

I have the following coupled 1D-3D perfusion problem (i.e. a 1D network embedded in a 3D tissue):

+ Momentum conservation: Darcy equation for both u1D-p1D, u3D-p3D;

+ Mass conservation: divergence equation for u1D, u3D with a forcing term f=f(p1D, p3D) accounting for the coupling;

+ Interface condition for u1D at network's bifurcations.

that I want to discretize with (P1,dP0) and (RT1,dP0) elements for the 1D and 3D problem, respectively.

I started from a primal pressure formulation (Darcy equations post-processed) and I am trying to extend that to a mixed pressure-velocity formulation. If I understood well the given starting code, the interface conditions for the primal pressure formulation (continuity of p1D over the bifurcation) are not explicitly implemented because we have p1D in H1 and hence C0.

Now, in a mixed pressure-velocity form I lost continuity of the pressure and I want to impose conditions at each network interface for the 1D velocity, like "sum u1D,i = 0  with i spanning over the inward and outward branches".

How can I address this issue? Is there any way to add a matrix for compatibility constraints or to use specific finite element spaces?

Best regards,

Domenico

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