A new stabilized finite element method for the transient Navier–Stokes equations

This paper is concerned with the development and analysis of a new stabilized finite element method based on two local Gauss integrations for the two-dimensional transient Navier–Stokes equations by using the lowest equal-order pair of finite elements. This new stabilized finite element method has some prominent features: parameter-free, avoiding higher-order derivatives or edge-based data structures, and stabilization being completely local at the element level. An optimal error estimate for approximate velocity and pressure is obtained by applying the technique of the Galerkin finite element method under certain regularity assumptions on the solution. Compared with other stabilized methods (using the same pair of mixed finite elements) for the two-dimensional transient Navier–Stokes equations through a series of numerical experiments, it is shown that this new stabilized method has better stability and accuracy results.