I have a system of the form
$$A(u)f(u)=b$$
where $A$ is basically a matrix originating from the Finite Element Method.
I try to solve it using the Newton method:
$$R = A(u_{i}) f(u_{i}) - b $$ $$u_{i+1}=u_{i} - J^{-1}R$$
where $J$ is the Jacobian of the residuum $R$ with respect to $u$.
At a first glance, it works fine. The residuum drops over several orders of magnitudes within just a few iterations. It seems that it converges to an error small enough for my application.
But still - after a couple of iterations, it stagnates/oscillates at a small error. I wonder whether this is normal or whether I should suspect a bug in my code.
What I found out so far:
- The remaining error is around $10^{-7}$ with entries in $J$ around $10^{-8}$ and in $u$ around $10^{3}$. I suspect that this is not yet extreme enough to cause problems with limited floating point precision.
- The remaining error becomes smaller when the derivatives $dA/du$ and $df/du$ become larger.
I'm of course happy with an answer to my problem, but I'm also happy with some further reading on the topic.
