Robin bcs with local gradient dependence

[dokken]

@bragostin

Form has no parts with arity 2.

This comes from

@bragostin

a, L = lhs(F), rhs(F)

which you do not need, as F is properly defined at

@bragostin

F = (
    - inner( k * grad(T), grad(s)) * dx 
    - HTC * (T - Tref) * s * ds(1)
    + f * s * ds(2)
    )

You can remove

@bragostin

a, L = lhs(F), rhs(F)

@bragostin

F = a - L

In general, solving a non-linear problem with a non-integer power on the unknown is challenging, see for instance:

Default absolute tolerance and relative tolerance

I write the following in the context of FE modelling rather than numerical analysis. I.e. this should be interpreted as tips and tricks. Perhaps someone else can contribute something more rigorous, or their own experience. Do you know what to expect the residual of your system to be to machine precision? Adjust your absolute tolerance accordingly. E.g: A Poisson problem with a diffusion coefficient of 1 may yield a residual to machine precision around 10^{-12}. Linear elasticity with a Young’…

You need a good initial guess, which i guess (completely random) at:

T.interpolate(Expression("3*x[1]+x[0]", degree=1))

which makes the problem solvable.