A parallel subgrid stabilized method for the steady natural convection problem

Based on local finite element approximations, this paper proposes a parallel subgrid stabilized method for the steady natural convection problem with moderate to high Rayleigh numbers, where full domain partition is utilized for parallelization, and the stabilization terms for penalizing the dominant convection-included instability in the problem are based on two elliptic projections. In this method, each processor calculates independently a local stabilized solution in its own subdomain adopting a global grid with a small mesh size h around the interested subdomain and a large mesh size H elsewhere. It is easy-to-implement based upon existing sequential codes and has a better parallel performance due to no communication between processors during the computing process. By virtue of the theoretical tool of the local a priori estimates for the subgrid stabilized solutions, we analyze optimal error bounds of the approximate solutions for the present method, and derive scalings of the algorithmic parameters with respective to the mesh sizes and stabilization parameters. We finally validate the theoretical convergence rates and demonstrate its efficiency by a series of numerical results. It is shown that our present method can provide an optimal convergence rate with the same order as the counterpart method excluding subgrid stabilizations and the standard subgrid stabilized method; however, it has a significant improvement in computational time in comparison with the standard method.