The steps above can be abbreviated as a fixed-point Gauss-Seidel problem  with respect to the interface displacement, which is to be solved until convergence of the corresponding residuals  . Typical convergence norms are   which makes the norm independent on the interface mesh size or relative convergence . Note that the cost of evaluation of   is high and it may contain noise/errors (for example due to not fully converged solvers).

Various methods can be used to ensure and accelerated convergence of the root-finding problem. A (dynamic) under-relaxation is . For example Aitken under-relaxation is given by . 

--uriencoded--\mathbf r_%7B\Gamma, i+1%7D = \mathbf R( \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D) =
\mathbf H(\mathbf d%5e%7Bn+1%7D_%7B\Gamma, i+1%7D) - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D= \widetilde%7B\mathbf d%7D%5e%7Bn+1%7D_%7B\Gamma, i+1%7D - \mathbf%7B d%7D%5e%7Bn+1%7D_%7B\Gamma, i%7D