Local and parallel stabilized finite element algorithms based on the lowest equal-order elements for the steady Navier–Stokes equations

Abstract

Based on a fully overlapping domain decomposition approach, local and parallel stabilized finite element algorithms are proposed and investigated for the steady incompressible Navier–Stokes equations, where the inf−sup unstable lowest equal-order P1−P1 finite element pairs are used and the stabilized term is based on two local Gauss integrations defined by the difference between a consistent and under-integrated matrix of pressure interpolants. In these algorithms, each processor computes a local stabilized solution in its own subdomain using a global grid that is locally refined around its own subdomain, making the algorithms have low communication cost and easy to implement based on a sequential solver. Using the technical tool of the local a priori estimate for the stabilized solution, error bounds of the proposed algorithms are derived. Theoretical and numerical results show that, the algorithms can yield an approximate solution with an accuracy comparable to that of the standard stabilized finite element solution with a substantial decrease in CPU time.