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By using local Fourier analysis, a simultaneous directions parallel method, which is a particular instance of the parallel fractional step algorithm, is shown to possess smoothing effects when applied to Poisson problems. The specific smoothing factor is determined and the expected factor values are found to be consistent with those obtained. The simultaneous directions approach is an advantageous alternative to other existing smoothers in the multigrid environment.

This paper is concerned with the analysis of the convergence and the derivation of error estimates for a parallel algorithm which is used to solve the incompressible Navier-Stokes equations. As usual, the main idea is to split the main differential operator; this allows to consider independently the two main difficulties, namely nonlinearity and incompressibility. The results justify the observed accuracy of related numerical results.

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