A two-step Jacobi-type iterative method

In multigrid methods, it is preferred to employ smoothing techniques which are convergent. In practice, the standard Jacobi and Gauss-Seidel methods are the choices for “smoothers” However, it is known that if the spectral radius condition is violated, then convergence is not guaranteed for these methods. In this paper we develop a simple two-step Jacobi-type method which has better convergence properties and which can be employed as a convergent “smoother” wherever the standard iterative methods fail. We provide the convergence proofs and demonstrate the applicability of the method on a variety of problems.