Abstract
In this paper, we develop and analyze the L ∞ stability and convergence analysis for non-singular finite element and finite volume solutions for the stationary 3D Navier-Stokes equations. We obtain optimal esti- mates for the gradient of velocity and the pressure in the L ∞-norm by applying the stabilization of a macro-element and technical lemmas including weighted L 2-norm estimates for the regularized Greens functions associated with the Stokes problem. Moreover, using the finite element solutions as interpolations, the relationship between the finite element method and the finite volume method is used to obtain the interesting super-close convergence rate with O ( h 3/2) in the L 2-norm and the optimal rate with O ( h ) in the L ∞-norm between the finite element method and the finite volume method for the velocity gradient and the pressure. Furthermore, optimal error estimates in the L ∞-norm are derived for the first time for the velocity gradient and pressure without a logarithmic factor O (|log h |) for the stationary 3D Naiver-Stokes equations.
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