Optimal error estimate of the penalty finite element method for the time-dependent Navier-Stokes equations

Source abstract

A fully discrete penalty finite element method is presented for the two-dimensional time-dependent Navier-Stokes equations. The time discretization of the penalty Navier-Stokes equations is based on the backward Euler scheme; the spatial discretization of the time discretized penalty Navier-Stokes equations is based on a finite element space pair ( X h , M h ) (X_h,M_h) which satisfies some approximate assumption. An optimal error estimate of the numerical velocity and pressure is provided for the fully discrete penalty finite element method when the parameters ϵ , Δ t \epsilon ,~\Delta t and h h are sufficiently small.