A parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations with a nonlinear damping term

Abstract In this work, we propose a parallel grad‐div stabilized finite element algorithm for the Navier–Stokes equations attached with a nonlinear damping term, using a fully overlapping domain decomposition approach. In the proposed algorithm, we calculate a local solution in a defined subdomain on a global composite mesh which is fine around the defined subdomain and coarse in other regions. The algorithm is simple to carry out on the basis of available sequential solvers. By a local a priori estimate of the finite element solution, we deduce error bounds of the approximations from our presented algorithm. We perform also some numerical experiments to verify the effectiveness of the proposed algorithm.