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

Abstract

SUMMARYA new regularization method is proposed for the Galerkin approximation of the incompressible NavierStokes 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 conÿned ows, and to make the discretized matrix positive deÿnite and then non-singular without the pressure pegging trick. E ects of the free parameters for the regularization on the solutions are numerically examined with a 2-D driven cavity ow 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 non-leaky 3-D driven cavity ows. Finally, the non-leaky 3-D cavity ows at di erent Re numbers are solved to compare with the literature data and to demonstrate the accuracy of the proposed method.

On the selective local stabilization of the mixed Q1–P0 element

“…[1,2]. In particular, such stabilizations on the Q 1 -P 0 element were studied in detail by Eguchi [16], Silvester and Kechkar [17], and Hughes and Franca [18]. But in this paper, we modify meshes to get stable Q 1 -P 0 elements when we do 'stabilization'.…”

A boundary‐domain integral equation method in viscous fluid flow

“…Among the three methods, FDM was ÿrst developed and is very simple to use although it requires very regular mesh [5]. In FEM, the penalty formulation is extensively used [6], but the determination of the penalty parameter is still controversial. The common feature of the three methods is that they are based on domain variable representations and local interpolation schemes, resulting in systems of equations that are highly spared matrices.…”