How to solve the possion equation with nolinear term by finite element method?

Question

$$-\Delta u+u^3=f\ \text{in}\ \Omega$$ $$u=0\ \text{on}\ \partial{\Omega}$$ the difficult here is how to handle the nonliear term $u^3$?

Answer 1

The two most common options are

1. Picard iteration (or the fixed point iteration)
2. Newton's method

Let me briefly explain the Picard iteration since it is simpler to implement. The weak formulation for the problem is of the form $$ (\nabla u, \nabla v) + (u^3,v) = (f,v) $$ for every suitable test function $v$. Then by making an initial guess, say $u_0=0$, we may repeatedly solve $u_{k} \in H^1_0(\Omega)$ from $$ (\nabla u_k, \nabla v) + (u_{k-1}^2 u_k, v) = (f,v) \quad \forall v \in H^1_0(\Omega). $$ This gives us a sequence of solutions hopefully converging to the solution $u$ of the nonlinear problem.

In case of any convergence problems you might have to set $\widetilde{u}_k = (1-\alpha)u_k+\alpha u_{k-1}$ with $0<\alpha<1$, say $\alpha = 0.5$, and then use $\widetilde{u}_k$ during the next iteration.