For the solution of the incompressible Navier-Stokes equations in domains of arbitrary shape, we use a finite volume method on structured boundary fitted grids. References [38,17,51,43,54,55] describe the discretization in detail and [39,54] discuss the capability of the method to accurately solve a number of laminar and turbulent flows. A Schwarz type domain decomposition iteration [42] in combination with GMRES [41] acceleration is used. In [10], significant reductions in computing time can be obtained using the GMRES acceleration procedure, see [11] and [10].
However, since the method described in [10] requires accurate solution of subdomain problems, it appears that the computing time can be much larger than with single-block solution for the same number of unknowns. Also, it is not known beforehand how accurate the subdomain problems must be solved. The required subdomain solution accuracy may be quite high, especially when grid cells are very much stretched near block interfaces, and a too low accuracy generally gives wrong results. A possible solution to both problems is to abandon the assumption of exact subdomain solution and to allow (very) inaccurate subdomain solution. Since the preconditioner may now vary in each iteration, GMRES acceleration may no longer be applied. Instead, a method based on GCR [23] is used.
Considerable reductions in computing time can be obtained in this way for a 2-dimensional advection-diffusion equation, see [9]. Approximate subdomain solution using a single iteration with ILUD factorization reduced multi-block computing time to almost that of single-block computing time. This encouraged us to extend this approach to the Navier-Stokes equations. Theoretical results and numerical experiments are presented to illustrate the effect of inaccurate solution of subdomain problems for the incompressible Navier-Stokes equations.
Parallel computing is of increasing importance. Therefore it is important to compare the parallel (additive) domain decomposition algorithms with the best multiplicative algorithms, which are known to be faster than additive algorithms. Therefore, we pay much attention to multiplicative algorithms.