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Purpose The purpose of this paper is to propose a new stabilized finite volume element method for the Navier-Stokes problem. Design/methodology/approach This new method is based on the multiscale enrichment and uses the lowest equal order finite element pairs P1/P1. Findings The stability and convergence of the optimal order in H1-norm for velocity and L2-norm for pressure are obtained. Originality/value Using a dual problem for the Navier-Stokes problem, the convergence of the optimal order in L2-norm for the velocity is obtained. Finally, numerical example confirms the theory analysis and validates the effectiveness of this new method.

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In this paper, we investigate fully discrete schemes for the Allen-Cahn and Cahn-Hilliard equations respectively, which consist of the stabilized finite element method based on multiscale enrichment for the spatial discretization and the semi-implicit scheme for the temporal discretization. With reasonable stability conditions, it is shown that the proposed schemes are energy stable. Furthermore, by defining a new projection operator, we deduce the optimal L 2 error estimates. Some numerical experiments are presented to confirm the theoretical predictions and the efficiency of the proposed schemes.

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