A two-step stabilized finite element algorithm for the Smagorinsky model

•An efficient two-step stabilized finite element algorithm for the simulation of Smagorinsky model is presented.•The algorithm uses equal-order P1−P1 and P2−P2 finite elements pairs for the velocity and pressure in the first and second steps, respectively.•Optimal error estimate is derived for the presented algorithm.•Superiority of the proposed two-step stabilized finite element algorithm to the one-step stabilized algorithm with P2−P2 finite elements is demonstrated by numerical tests. This study considers an efficient two-step stabilized finite element algorithm for the simulation of Smagorinsky model, which involves solving a stabilized nonlinear Smagorinsky problem by the lowest equal-order P1−P1 finite elements and solving a stabilized linear Smagorinsky problem by the quadratic equal-order P2−P2 finite elements. We theoretically and numerically show that the present two-step algorithm can provide an approximate solution with basically the same accuracy as that of solving the stabilized P2−P2 finite element method, and represent a reduction in CPU time.