How to avoid calling solve(F==0,...) in a time loop?

General
1

I have so far solved the problem by calling solve(F == 0, T, BCS) successfully solving the nonlinear heat transfer equation, but that was only in smaller time, in a larger time step to solve the docker always stuck (did not report any errors).
according to https://fenicsproject.discourse.group/t/the-solution-speed-is-very-slow/11576, I try to Avoid calling solve F==0 inside the loop, but I still have many questions:

I try to define the nonlinear variational problem before the time loop by,This comes from http://home.simula.no/~hpl/homepage/fenics-tutorial/release-1.0-nonabla/webm/nonlinear.html#tut-nonlinear-newton-algebraic :

.....
T = TrialFunction(V)
v = TestFunction(V)

F = mu*c*T*v*dx + dt*k(T_n)*dot(grad(T), grad(v))*dx - mu*c*T_n*v*dx - dt*f*v*dx - dt*sum(integrals_N)

T_= Function(V) 
F = action(F,T_)

J = derivative(F, T_, T)
problem = NonlinearVariationalProblem(F, T_, bcs, J)
solver = NonlinearVariationalSolver(problem)
prm = solver.parameters
prm['newton_solver']['absolute_tolerance'] = 1E-8
prm['newton_solver']['relative_tolerance'] = 1E-7
prm['newton_solver']['maximum_iterations'] = 100
prm['newton_solver']['relaxation_parameter'] = 0.8

t = 0
for n in range(num_steps):
    
    solver.solve()
.....

1.Can the above method avoid calling F==0 in the time loop?
2.What happens to the NonlinearVariationalProblem?Is it partially assembled and calculated?
3.Am I still solving the linearization problem with a direct solver in this form?If so, how do I through iteration method to solve, whether to need to manually go to https://jsdokken.com/dolfinx-tutorial/chapter4/newton-solver.html
Please forgive my stupidity!

2

It does not call solve in the loop, so yes.

Nonlinear variational problem compiles the forms, generates integration kernel, creates the required structures for assembly etc.

You are still solving the same problem, just caching matrices, solvers etc. if you want more control, you would Need to write a custom newton solver. There are many examples on the forum on how to do that with legacy dolfin

3

Thanks for your constant help, I will try to implement the Newton iteration process manually to speed up the solution.