Two-Scale Picard Stabilized Finite Volume Method for the Incompressible Flow (Advances in Applied Mathematics and Mechanics, 2014)
In this paper, we consider a two-scale stabilized finite volume method for the two-dimensionalstationary incompressible flow approximated by the lowest equal-order element pair $P_1-P_1$which does not satisfy the inf-sup condition. The two-scale method consists of solving a small non-linear systemon the coarse mesh and then solving a linear Stokes equations on the fine mesh. Convergence of the optimalorder in the $H^1$-norm for velocity and the $L^2$-norm for pressure is obtained. The error analysis showsthere is the same convergence rate between the two-scale stabilized finite volume solution and the usual stabilized finite volume solution on a fine mesh with relation  $h =\mathcal{O}(H^2)$.  Numerical experiments completelyconfirm theoretic results. Therefore, this method presented in this paper is of practical importance inscientific computation.