A new positive-definite regularization of incompressible Navier-Stokes equations discretized with Q1/P0 finite element

A new regularization method is proposed for the Galerkin approximation of the incompressible Navier-Stokes equations with Q1/P0 element, by newly introducing a square-type linear form into the variational divergence-free constraint regularized with the global pressure jump (GPJ) method. The addition of the square-type linear form is intended to eliminate the hydrostatic pressure mode appearing in confined flows, and to make the discretized matrix positive definite and then non-singular without the pressure pegging trick. Effects of the free parameters for the regularization on the solutions are numerically examined with a 2-D driven cavity flow problem. Furthermore, the convergences in the conjugate gradient iteration for the solution of the pressure Poisson equation are compared among the mixed method, the GPJ method and the present method for both leaky and nonleaky 3-D driven cavity flows. Finally, the non-leaky 3-D cavity flows at different Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method.