Convergence and optimization of the parallel method of simultaneous directions for the solution of elliptic problems

For the solution of elliptic problems, fractional step methods and in particular alternating directions (ADI) methods are iterative methods where fractional steps are sequential. Therefore, they only accept parallelization at low level. In [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673-708], Lu et al. proposed a method where the fractional steps can be performed in parallel. We can thus speak of parallel fractional step (PFS) methods and, in particular, simultaneous directions (SDI) methods. In this paper, we perform a detailed analysis of the convergence and optimization of PFS and SDI methods, complementing what was done in [T. Lu, P. Neittaanmäki, X.C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, RAIRO Modél. Math. Anal. Numér. 26 (6) (1992) 673-708]. We describe the behavior of the method and we specify the good choice of the parameters. We also study the efficiency of the parallelization. Some 2D, 3D and high-dimensional tests confirm our results.