Velocity Projection-Based Stabilized Finite Element Method for Steady Navier-Stokes Equations using equal order interpolation

Abstract:

A new type of velocity $L^2$ projection-based stabilized finite element method for steady Navier-Stokes equations is proposed and analyzed. Velocity and pressure are approximated equal order element $P_1/P_1$. To overcome the violation of discrete inf-sup condition when equal order elements are used, pressure projection stabilized term is added. Velocity projection-based stabilized method directly increases the $L^2$-stability instead of $H^2$-stability. The main advantages of the proposed methods lies in that, all the computations are performed at the same element level, without the need of nested meshes and the projection of the gradient of velocity. It is showed that this discrete model is stable and has a unique branch of nonsingular solutions , given the continuous Navier-Stokes equations has an unique branch of nonsingular solutions . Moreover, error estimates are derived, and numerical experiments show that the method is valid.