# Solution predictors for accelerating convergence in nonlinear FEM

I am looking for the details of commonly-used predictors for accelerating the convergence of iterations using Newton-Raphson scheme for nonlinear problems in FEM. I am looking specifically for static problems.

Note: Such predictors based on velocity and/or acceleration are used for transient problems but they are not applicable for pure static problems.

Appreciate any inputs.

• Is it even possible to accelerate convergence of Newton's method? I wasn't aware that that's possible. – Wolfgang Bangerth Oct 18 '20 at 3:22
• @WolfgangBangerth, Yes. If we can compute the initial guess (the predictor) such that it is closer to the actual solution, then the number of iterations certainly decrease. Such an approach also allows us to use bigger load steps. – Chenna K Oct 18 '20 at 14:09
• Sure, but that happens before you even start the Newton iteration. So if I understand you right, you're looking for something that produces a good initial guess from which to start a Newton iteration? I think if so, you will have to explain in more detail what the set up is that you're trying to solve, and what information is available to compute a predictors. – Wolfgang Bangerth Oct 19 '20 at 4:26
• @WolfgangBangerth I consider estimation of the initial guess as also part of the NR scheme. I do not want to impose any constraints to start with. I think it's better to work the other way. If such techniques for accelerating the convergence do exist, then what information and/or additional operations do they require? We can consider the example of a beam/plate with a compressible Neo-Hookean model with either a body force or a specified non-zero displacement on its faces. – Chenna K Oct 20 '20 at 15:51
• @Chenna_K: You're not going to find anything if you look for "convergence acceleration". "Convergence" is an asymptotic property that considers how iterations progressively get closer to the solution. An example of convergence acceleration is Anderson mixing and its variations. But what you're looking for is really "how do I choose a good starting point for Newton's method". That's of course a good question, but you should phrase the question differently. – Wolfgang Bangerth Oct 20 '20 at 20:31