[dune-functions] Function space of constant functions

[Simon Praetorius]

Hi Christian,

> At least for the first two points I'd rather think your discretization
> needs something to include constraints based on functionals?!
> Regarding the Cahn-Hilliard example I don't this model, so I can't
> judge. I found an arxiv articel on this scheme, but it was unclear for
> me, where your global value comes into play. Can you perhaps elaborate
> on this?
>
> Best
> Christian
>
I'm just experimenting a little bit with some schemes, see, e.g.

 > A New Class of Efficient and Robust Energy Stable Schemes for 
Gradient Flows, Jie Shen, Jie Xu, and Jiang Yang, 
https://arxiv.org/abs/1710.01331

Essentially you have a Cahn-Hilliard equation and a scalar equation for 
the evolution of a free-energy integral

dt phi = laplace(mu)
mu = -eps*laplace(phi) + r/sqrt(E(phi)) * U(phi)
dt r = 1/(2sqrt(E(phi))) * int dx{ U(phi) * dt phi }

with

E(phi) = int dx{ f(phi) } where f is some double-well or logarithmic 
potential or something like that
r = sqrt(E(phi)) (a scalar quantity)
U(phi) = dE/dphi

So, for exact r this is the classical CahnHilliard equation, but due to 
the evolution of r it is slightly different. I don't want to judge on 
this scheme whether it is good or not, I just wanted to implemented this 
in a simple framework to try out. This is typically solved using some 
FFT based approaches but I've tried it in a FEM context.

If you don't use a discretization module that does provide already the 
*constraints based on functionals* than this scalar basis could be a 
simple workaround. I was not specifically "designed" for the purpose on 
constraints, but it turned out to be applicable for this as well. You 
always have multiple ways to implement the same problem. This is just one.

Best wishes,
Simon