Implementation of nonlinear term in FEM

Question

Although there are similar questions, I am also struggling with the implementation of the following term in "my own code" by Finite Element Method, namely, $\nabla \phi \cdot \nabla \phi$. $\phi$ is a state variable such as scalar potential. I think the descritization of this term is much more difficult than that of advection term in N-S equation. If possible, please tell me how to descritize and implement the term by FEM. The equation to solve is as follows, $\frac{\partial \phi}{\partial t} = (\nabla\phi)^2 + \nabla^2 \phi.$

Answer 1

You should rather think of what you will need in the following step, which is probably the numerical time integration of the semi-discrete equations.

1. If you are going to use a (semi) explicit time stepping scheme, all you need is a function that for a given $\phi_0$ assembles the vector $\langle (\nabla \phi_0)^2, v \rangle$, where $v$ are your test functions. In FEM packages like FEniCS this is straight forward.
2. If you going for an implicit scheme, you will probably apply a Newton or Picard iteration for the given iteration point $\phi_0$, you will need the forms or matrices associated with $\langle \nabla \phi_0 \nabla u, v \rangle$ where $u$ are the trial functions. This is a linear form that is easy to assemble with standard FEM methods.
3. For atmost generality you can tensorize your problem, express $(\nabla \phi)^2$ via $D_\nabla (\phi \otimes \phi)$ and assemble the associated linear form (linear in the Kronecker product). For an example considering the convection term $(u \cdot \nabla) u$ in the Navier-Stokes equations have a look at this python function.

Answer 2

Nonlinear differential equations are really no different than other nonlinear equations that you can solve with a Newton method. You may want to take a look at the discussion how this is done in a typical finite element code for a stationary problem here: http://dealii.org/developer/doxygen/deal.II/step_15.html

The code shows how to do it for a stationary problem, but the process is the same for every time step if you use a fully implicit method. Of course, the process becomes a lot simpler if you treat the nonlinear term as explicit in the time stepping scheme.