Nonlinearvariationalsolver not converge

Question

I tried to sover the next nonlinear partial diferential equation with fenics

$\psi + c_0 \nu \|\nabla \psi\| = \psi_0$

bc = DirichletBC(V, psi0, psi0_boundary)      

psi, v = (Function(V), TestFunction(V))
a = psi*v*dx + dt*nu*dot(grad(psi),grad(psi))**(1./2)*v*dx 
L = psik*v*dx
F = a - L   
J = derivative(F, psi) 
pde = NonlinearVariationalProblem(F, psi, bc, J)
solver = NonlinearVariationalSolver(pde)
solver.solve()

but I get the next message:

*** Error: Unable to solve nonlinear system with NewtonSolver.
*** Reason: Newton solver did not converge. Bummer.
*** Where: This error was encountered inside NewtonSolver.cpp.
*** Process: 0

What is the problem?

Comments

You need to post complete code that runs.

Answer 1

Consider that Jacobian of the term $\int |\nabla\psi| \, v \,\mathrm{d}x$ is

$$
\int \frac{\nabla\psi}{|\nabla\psi|}\cdot\nabla\xi \, v \,\mathrm{d}x .
$$

Setting $\psi$ being non-constant everywhere prior to Newton iteration may help.

Alternatively you may try to supply your own Jacobian which will somewhat regularize the problem and keep converging it.

Comments

How I can to set my own Jacobian?

---

This is obvious, isn't it? You did

J = derivative(F, psi) 
pde = NonlinearVariationalProblem(F, psi, bc, J)

Just throw away derivative(F, psi) and supply some bilinear form J on $V \times V$.