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 .