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

variational formulation
1

Hi,

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:
https://bitbucket.org/nate-sime/leopart/src/master/source/FormsStokes.py#lines-138
https://bitbucket.org/nate-sime/leopart/src/253d42a713d6c35c479c99ea1c64b6121d67b577/source/cpp/stokesstaticcondensation.cpp#lines-204

Thanks for your help!
Bin

2

Please supply more information about your question. Is there any particular issues with extending Leopart/Geopart?
If so, supply a minimal code example illustrating your issue. Follow the guidelines in: Read before posting: How do I get my question answered?

3

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

4

\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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