New stabilized finite element method for time‐dependent incompressible flow problems

Abstract A new stabilized finite element method is considered for the time‐dependent Stokes problem, based on the lowest‐order P 1 − P 0 and Q 1 − P 0 elements that do not satisfy the discrete inf–sup condition. The new stabilized method is characterized by the features that it does not require approximation of the pressure derivatives, specification of mesh‐dependent parameters and edge‐based data structures, always leads to symmetric linear systems and hence can be applied to existing codes with a little additional effort. The stability of the method is derived under some regularity assumptions. Error estimates for the approximate velocity and pressure are obtained by applying the technique of the Galerkin finite element method. Some numerical results are also given, which show that the new stabilized method is highly efficient for the time‐dependent Stokes problem. Copyright © 2009 John Wiley & Sons, Ltd.