Two-grid parallel stabilized finite element method based on overlapping domain decomposition for the Stokes problem

Abstract

In this paper, one non-iterative two-grid parallel stabilized finite element method based on overlapping domain decomposition is developed and investigated for the Stokes problem. The lowest-order finite element pairs P1−P1 are utilized to approximate the velocity and pressure, respectively. The difference between a consistent and an under-integrated mass matrices is chosen as the stabilized term to circumvent the so-called inf-sup condition. Algorithm distributes legitimately residuals into local domains with a partition of unity. Error estimates are rigorously established. Theoretical results indicate that patches of diameter 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 our theoretical results.