In the present work, we describe a new algorithm, capable of considerably improving the convergence of waveform relaxation, that is based on a special multicycle or nested organization of the iteration process. Mathematically, it can be classified as a polynomially accelerated method applicable to systems containing "black boxes." The convergence area is shown to be wider for a larger number of iterative cycles, the only parameter of the algorithm. The method is able to suppress or reduce the effects of both the local and global instantly operating feedback. The proposed algorithm can either make the convergence uniform or at least increase the time windows. It can be used selectively in time and in "space," whenever the conventional relaxation methods cannot cope with high-gain global feedback.