[Leopart,Geopart] Extend Stokes to Brinkman model using HDG


I’m trying to extend the Stokes model to Brinkman model:
\mu\nabla ^2\mathbf{u}-\nabla p=\mathbf{f}

\mu\nabla ^2\mathbf{u}-\nabla p-\kappa\mathbf{u}=\mathbf{f}

The variational formulation of Brinkman model should be:
\int_{\Omega}{\mu\nabla \mathbf{u}\cdot \nabla \mathbf{v}}-\int_{\Omega}{p\left( \nabla \cdot \mathbf{v} \right)}-\int_{\Omega}{\kappa\mathbf{u}\cdot \mathbf{v}}=\int_{\Omega}{\mathbf{f}\cdot \mathbf{v}}+\int_{\partial \Omega _N}{\mathbf{g}\cdot \mathbf{v}}
where \mathbf{g}=\nabla \mathbf{u}\cdot \mathbf{n}+p\mathbf{n}

The new term should be added is -\int_{\Omega}{\kappa \mathbf{u}\cdot \mathbf{v}}.
The fenics expression should be -k * inner(u, v)

I’m not familiar with HDG, so what should I modify the Leopart/Geopart library for this?

such as:

Thanks for your help!

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Thanks for your rapid response. I was editing the question.

\int_\Omega \mathbf{u} \cdot \mathbf{v}\;\mathrm{d}\mathbf{x} = \sum_{\kappa \in \mathcal{T}_h} \int_\kappa \mathbf{u} \cdot \mathbf{v}\;\mathrm{d}\mathbf{x} is just a reaction term with no facet integrals. So, in the HDG formulation it just appears in the local-to-local components of the matrix multiplied by \mathbf{v} and not \overline{\mathbf{v}}. It should be straightforward to extend the functionality for your needs.

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