A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations

Abstract

The tradìtíonal splitting-up method or fractíonal step method is stuítable for sequentìal compulìng. Thís means that the computing of the present fractional step needs the value of the previous fractional steps. In thìs paper we propose a new splitting-up scheme for which the computing of the fractional steps is índependent of each other and therefore can be computed by parallel processors. We have proved the convergence estimates of this scheme both for steady state and nonsteady state linear and nonlinear problems. To use .finite element method to solve Navier-Stokes problems it is always dfficult to handle the zero-divergent finíte element spaces. Here, by using the splitting-up method we can use the usual finite element spaces to solve it. Moreover, the proposed method can solve the steady and nonsteady state Navíer-Stokes problem by only solving some one dimensional línear systems. All these one dimensional systems are índependent of each other, so they can be computed by parallel processors.