### Solving a set of 5 non linear PDEs simultaneously

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6 months ago by
Hi,
I am new to Fenics and I would like to know the best approach to solve a set of 5 non linear PDEs together. These equations are not reducible and has to be solved together. Generating the weak form manually is very difficult.
Community: FEniCS Project
What do you mean by "generating the weak form manually"? You'll have to at least write down the weak form before FEniCS can help. Do you only have a strong form? Or do you only mean that you don't know how to implement the weak form in FEniCS?
written 6 months ago by Alexander G. Zimmerman
I would say he meant "compiling" the weak form...
written 6 months ago by Michal Habera
I am not sure how the weak form would be implemented in Fenics when there are a set of governing equations and they are coupled. I am looking at five PDEs and appropriate boundary conditions.
written 6 months ago by Aswin Rajeevan
Step 1 is to derive (outside of FEniCS) the weak/variational form. If you have that, then Michal's answer about mixed finite elements should have the info you need. It's indeed where I learned to do this.

If you don't have the weak form yet, then let us know and I (unless someone beats me to it) can link you to a tutorial about deriving a weak form for a system of PDE's.
written 6 months ago by Alexander G. Zimmerman
Yes please, I would like to look at it. I am only familiar with the weak formulation for single differential equation.
written 6 months ago by Aswin Rajeevan

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6 months ago by
The essence is the use of MixedElement. See nice tutorial http://www.logg.org/anders/2016/05/11/systems-of-chemical-reactions-in-fenics/
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6 months ago by
Based on our conversation attached to your question, I understand that the fundamental question here is how to derive the weak form given a system of PDE's in strong form.

Perhaps someone can link to a nice tutorial on this which involves FEniCS; but I learned this back when using deal.II, and this was my favorite explanation: https://www.dealii.org/8.5.0/doxygen/deal.II/group__vector__valued.html

Also to supplement the good example that Michal has already shared, here's a simple steady incompressible Navier-Stokes example which I threw together for a brief presentation to my research group: https://github.com/geo-fluid-dynamics/introduction-to-fenics/tree/master/navier-stokes