How to write Neumann conditions with non-linear term of "u"?

Question

In the "Multiple Neumann, Robin, and Dirichlet Condition" of the tutorial.
It use a boundary condition that with "u":

$-\frac{\partial u}{\partial n}=p(u-q)$ on $\Gamma_R$

And the correspond code is :

a = inner(nabla_grad(u), nabla_grad(v))*dx + p*u*v*ds(0)
L = f*v*dx - g*v*ds(1) + p*q*v*ds(0)

Now i want a boundary condition that with nonlinear form of "u", for example:

$-\frac{\partial u}{\partial n}=p(u^2-q)$ on $\Gamma_R$

I changed the code to:

a = inner(nabla_grad(u), nabla_grad(v))*dx + p*u*u*v*ds(0)
L = f*v*dx - g*v*ds(1) + p*q*v*ds(0)

It cannot work. So what's the right way to do this? Thank you very much.

Answer 1

This is no more linear problem - observe that a depends also on u*u. You can try

• fixed-point iteration

  u0 = Function(V)
  a = inner(nabla_grad(u), nabla_grad(v))*dx + p*u0*u*v*ds(0)
  
  u = Function(V)
  until convergence:
       u0.assign(u)
       solve(a == L, u, bcs)
  

  where you need to supply convergence condition.

• Newton method

  u = Function(V)
  F = inner(nabla_grad(u), nabla_grad(v))*dx + p*u*u*v*ds(0) \
      - f*v*dx - g*v*ds(1) + p*q*v*ds(0)
  J = derivative(F, u)
  solve(F == 0, u, bcs, J)