# Dynamic Successive Over/under Relaxation (SOR) with several variables

I am solving a partial differential algebraic equation (PDAE) system which has the following dependent variables:

$f=f(X,T)$ and $g=g(T)$, along with a few others

My current method for coupling is to use fixed-point iteration with constant under-relaxation factors for the two coupled variables $f$ and $g$. The issue is that convergence (determined from the norm of the residuals) is too slow. To resolve this I have tried to implement dynamic under-relaxation using Aitken's method. However, this doesn't seem to offer any speed-up - it actually often requires more iterations to converge than using constant relaxation factors. This is the case even if I apply upper and lower bounds on the relaxation factors calculated by Aitken's method. Reducing the starting relaxation factors to small values (~0.01) also doesn't seem to help much.

My initial thoughts are that the Aitken's method is not accounting for the coupling between the two variables - the relaxations factors $\omega_f$ and $\omega_g$ are calculated with no consideration for the variation of each other. I have tried having a look at the literature, but I can only find the application of Aitken's method for dynamic relaxation for a single variable.

Any suggestions for improving the performance of Aitken's method in this case would be much appreciated