NavierStokes
============

This repository implements via finite element solvers for incompressible
Navier-Stokes (iNSE) equations in Arbitrary Lagrangian-Eulerian (ALE) formalism the schemes 
proposed on [RA20]_.

In particular, it aims to replicate the energy results shown in the article [RA20]_ 
for both Monolithic and Chorin-Temam solvers.

Solvers
-------

Both monolithic solver and a fractional step solver are implemented.

* Monolithic solver for the iNSE-ALE problem with linearized convective term and Taylor-Hood (P2/P1) stable finite element space.

* Fractional step solver for the iNSE-ALE problem with linealized convective term and P1/P1 finite element space. The Chorin-Temam schemes proposed is described in [RA20]_.

Flow model
----------

A rectangle domain is taken with fully Dirichlet homogeneous boundary conditions
and non-zero initial velocity profile. Further description of the problem also can be found 
in the reference.

Usage
----------

Since the repository aims to directly reproduce the results of the reference, 
no configuration files where implemented to further customize the problem. 
Nevertheless, the solvers are easily modified since its implementation is done via 
FEniCS [LO12]_.

An ipynb file is included to reproduce the results and recommended to use.

If desired to run the simulations only, execute::

    python main.py


Dependencies
------------

- Python >= 3.5 
- FEniCS >= 2019.1.0



Reference
^^^^^^^^^^

.. [RA20] Aróstica R., Bertoglio C. (2020) On monolithic and Chorin-Temam
   schemes for incompressible flows in moving domains. 
   Applied Mathematics Letters, doi: https://doi.org/10.1016/j.aml.2020.106830
   ISBN: 978-3-642-23099-8

.. [LO12] Logg A., Mardal K.-A., Wells G. N. (2012) Automated Solution of Differential
   Equations by the Finite Element Method.
   Springer, Berlin, Heidelberg, doi: https://doi.org/10.1007/978-3-642-23099-8