# Newton's method stagnates at small error

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.

• What is the PDE and the geometry you are looking at? Commented Jun 8, 2017 at 13:21
• How do you compute $J^{-1}R$? And how do you define "error"? Commented Jun 8, 2017 at 19:25
• What is the (estimated) condition number of your Jacobian? What is your machine precision? Commented Jun 9, 2017 at 16:34
• Thanks for your answers. I will try to find out more as soon as possible. Commented Jun 10, 2017 at 8:48

I'd suspect a bug in the code unless you understand exactly why the seen behavior is expected.

As for a particular references on how to write nonlinear solvers (globally convergent newton solvers), I was very happy working with:

1. Newton Methods for Nonlinear Problems,
Deufelhard, Peter; Springer, 2006 (Second Printing)
ISBN 978-3-642-23898-7 (Softcover)

2. Reference implementations for (1),
http://elib.zib.de/pub/elib/codelib/NewtonLib/index.html

3. A Family of Newton Codes for Systems of Highly Nonlinear Equations (1991); Nowak, U., Weimann, L.; ?,
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.43.3751

The NLEQ_ERR solver outlined in (1),(2) works very well for my applications.

Another nice book with simple non-linear examples (including the (tangent) stiffness matrices of each newton step is):

• Advanced Topics in Finite Element Analysis of Structures: With Mathematica and MATLAB Computations; Bhatti M. Asghar; Wiley; ISBN: 978-0-471-64807-9