A parallel full domain partition method for Stokes and Navier–Stokes type variational inequalities with damping

Motivated by reducing the computational time and computer storage requirements in the numerical simulations, we present a parallel full domain partition method based on finite element approximations for Stokes and Navier–Stokes type variational inequalities with damping in this paper. Within this parallel method, each subproblem used to calculate an approximate solution is actually a global problem defined in the whole domain with the vast majority of the degrees of freedom associated with the particular subdomain that it is responsible for, making the present method easily implementable on the basis of existing black-box sequential solver without massive effort in recoding on the top of existing serial software. Errors of the approximate velocity in L2 norm and pressure in H−1 norm for the serial method are estimated. Based on these error estimate results and the theoretical tool of local a priori estimate for finite element solution, error estimates of the approximate solutions from the proposed method are derived. Correctness of the theoretical predictions and promise of the present method are illustrated by some results of numerics. It is numerically shown that by choosing suitable algorithmic parameters, our proposed parallel method can yield an approximate solution with an accuracy comparable to that of the one calculated by the serial method, and the computational time is reduced.