# Finite element method for the Navier-Stokes equation with local conservation properties

I was wondering if anyone here is familiar with a finite element formulation for the Navier-Stokes equation that also satisfies the mass conservation locally, much like what the finite volume method does. I am more interested in having a robust method even if it involves extra dissipation. My attempts so far to solve the Navier-Stokes equation with SUPG and continuous finite elements have been a bit frustrating. I found this approach to be too finicky.

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If youâ€™re solving the incompressible Navier-Stokes equations, Iâ€™d recommend looking into divergence conforming Discontinuous Galerkin (DG) methods. These methods are non-conforming and are similar in some ways to finite volume methods. Upwinding can be incorporated naturally through the specification of the numerical flux. Provided the finite element spaces are chosen such that the divergence operator maps the velocity space into the pressure space, mass is conserved exactly. Exact mass conservation is advantageous for a number of reasons, one of which being that the error estimates for the velocity field are independent of the pressure field, and thus the pressure approximation cannot pollute the velocity approximation.

I am planning on adding a FEniCSx demo of this type of method soon. In the meantime, you may find these references helpful:

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These references are very useful. Thanks!

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Hello, where can I see the following FEniCSx demo about Discontinuous Galerkin (DG) methods in the future?

look at this paper by Marien Hanot to appear in SIAM J Numerical Analysis
https://arxiv.org/pdf/2106.05146.pdf
and the corresponding github repo
https://github.com/mlhanot/Navier-Stokes-feec

Check out this PR. It implements a divergence conforming discontinuous Galerkin method for the Naiver-Stokes equations. Itâ€™s not polished yet but works fine! Soon it will be possible to implement hybrid discontinuous Galerkin schemes too