# Finite element method for the Navier-Stokes equation with local conservation properties **URL:** \ **Category:** Uncategorized\ **Created:** [August 6, 2022, 6:20am UTC](https://fenicsproject.discourse.group/t/finite-element-method-for-the-navier-stokes-equation-with-local-conservation-properties/8929 "2022-08-06T06:20:25Z")\ **Posts on this page:** 1\ **Showing post:** 2
**Author:** ![jpdean](https://yyz2.discourse-cdn.com/free1/user_avatar/fenicsproject.discourse.group/jpdean/32/3389_2.png) [@jpdean](https://fenicsproject.discourse.group/u/jpdean)\ **Post date:** [August 8, 2022, 9:47am UTC](https://fenicsproject.discourse.group/t/finite-element-method-for-the-navier-stokes-equation-with-local-conservation-properties/8929/2 "2022-08-08T09:47:53Z")
If you’re solving the incompressible Navier-Stokes equations, I’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: > **[A Note on Discontinuous Galerkin Divergence-free Solutions of the...](https://link.springer.com/article/10.1007/s10915-006-9107-7)** > > We present a class of discontinuous Galerkin methods for the incompressible Navier–Stokes equations yielding exactly divergence-free solutions. Exact incompressibility is achieved by using divergence-conforming velocity spaces for the approximation... > **[On the Divergence Constraint in Mixed Finite Element Methods for...](https://epubs.siam.org/doi/10.1137/15M1047696)** > > The divergence constraint of the incompressible Navier--Stokes equations is revisited in the mixed finite element framework. While many stable and convergent mixed elements have been developed throughout the past four decades, most classical methods... > **[High order exactly divergence-free Hybrid Discontinuous Galerkin Methods for...](https://www.sciencedirect.com/science/article/abs/pii/S004578251630264X)** > > In this paper we present an efficient discretization method for the solution of the unsteady incompressible Navier–Stokes equations based on a high or… > **[A Hybridizable Discontinuous Galerkin Method for the Navier–Stokes Equations...](https://link.springer.com/article/10.1007/s10915-018-0671-4)** > > We introduce a hybridizable discontinuous Galerkin method for the incompressible Navier–Stokes equations for which the approximate velocity field is pointwise divergence-free. The method builds on the method presented by Labeur and Wells (SIAM J Sci... --- _[View the full topic](https://fenicsproject.discourse.group/t/finite-element-method-for-the-navier-stokes-equation-with-local-conservation-properties/8929)._