We propose and analyze a finite element method for approximating solutions to the Navier-Stokes-alpha model (NS-α) that utilizes approximate deconvolution and a modified grad-div stabilization and greatlyimproves accuracy in simulations. Standard finite element schemesfor NS-α suffer from two major sources of error if their solutions are considered approximationsto true fluid flow: (1) the consistency error arising from filtering; and (2) the dramatic effect of the large pressure erroron the velocity error that arises from the (necessary) use of the rotational form nonlinearity.The proposed scheme “fixes” these two numericalissues through the combined use of a modified grad-div stabilization that acts in both the momentum and filter equations, and an adapted approximate deconvolution technique designed to work with the altered filter. We provethe scheme is stable, optimally convergent, and the effect of the pressureerror on the velocity error is significantly reduced. Several numerical experiments are given that demonstrate theeffectiveness of the method.
