Schwarz’s decomposition method for incompressible flow problems.

*(English)*Zbl 0652.76022
Domain decomposition methods for partial differential equations, 1st Int. Symp., Paris/France 1987, 333-349 (1988).

[For the entire collection see Zbl 0649.00019.]

Three-dimensional fluid dynamics computation imply the solution of very large linear systems and there seems to be an agreement that direct methods are too expensive to handle them efficiently. Among the possible alternative paths that can be followed, domain decomposition comes quite naturally to the mind: if many small systems could be properly connected, computing effort could be split into pieces that could even be dealt with in parallel. There is by no means a unique way to do so. We shall describe in this paper a Schwarz’s type (overlapping domains) decomposition algorithm for the numerical solution of the Navier-Stokes equations of incompressible flows. We shall consider its performance in linear Stokes problem, then in the nonlinear case. Finally we shall discuss possible variants, some of which are presently under test.

Three-dimensional fluid dynamics computation imply the solution of very large linear systems and there seems to be an agreement that direct methods are too expensive to handle them efficiently. Among the possible alternative paths that can be followed, domain decomposition comes quite naturally to the mind: if many small systems could be properly connected, computing effort could be split into pieces that could even be dealt with in parallel. There is by no means a unique way to do so. We shall describe in this paper a Schwarz’s type (overlapping domains) decomposition algorithm for the numerical solution of the Navier-Stokes equations of incompressible flows. We shall consider its performance in linear Stokes problem, then in the nonlinear case. Finally we shall discuss possible variants, some of which are presently under test.

##### MSC:

76D05 | Navier-Stokes equations for incompressible viscous fluids |

76D07 | Stokes and related (Oseen, etc.) flows |

76M99 | Basic methods in fluid mechanics |