Non-iterative two-grid parallel scheme based upon overlapping domain decomposition for the Stokes problem

In this paper, one non-iterative two-grid parallel scheme (NITGPS) based upon an overlapping domain decomposition is proposed and investigated for the Stokes problem. The crucial idea is to spread legitimately residuals into local domains with a partition of unity. Once the coarse grid approximation is derived, local problems defined on the fine grid are totally independent and there need little communication. On the other hand, some local a priori estimates are obtained to rigorously establish error estimates. Theoretical results indicate that patches of diameter \(H|ln H|(|ln H|-ln|ln H|)\) $H|lnH|(|lnH|-ln|lnH|)$ or H|lnH| are sufficient to preserve optimal convergence rates with a proper configuration between the coarse mesh size H and the fine mesh size h in 2-D or 3-D case, respectively. Finally, some numerical experiments are reported to support the theoretical results.