A parallel grad-div stabilized finite element algorithm for the Stokes equations with damping

Abstract

This work studies a parallel grad-div stabilized finite element algorithm for the damped Stokes equations. In this algorithm, in the light of a fully overlapping domain decomposition technique, we solve a global grad-div stabilized problem to compute a local solution in an intersecting subdomain on a global composite mesh, which is fine in the subdomain and rough elsewhere, making the proposed algorithm easy to implement based on an available sequential solver. We derive error bounds of the approximate solutions from our presented algorithm by the theoretical tool of local a priori estimate for the grad-div stabilized finite element solution. Numerical results verify the validity of the theoretical analysis and demonstrate the benefits of the proposed algorithm. On the one hand, compared with the counterpart one excluding grad-div stabilization, this algorithm can reduce significantly the effect of pressure on the approximate velocities, and hence, yields much better approximate velocities in the case of small viscosities. On the other hand, it takes much less computational time in getting approximate solutions with a comparable accuracy than the standard grad-div stabilization method.