An inf-sup Stable and Residual-Free Bubble Element for the Oseen Equations

Abstract

We investigate the residual-free bubble method for the linearized incompressible Navier-Stokes equations. Starting with a nonconforming inf-sup stable element pair for approximating the velocity and pressure, we enrich the velocity space by discretely divergence-free bubble functions to handle the influence of strong convection. An important feature of the method is that the stabilization does not generate an additional coupling between the mass equation and the momentum equation as is the case for the streamline upwind Petrov-Galerkin method applied to equal-order interpolation. Furthermore, the discrete solution is piecewise divergence-free, a property which is useful for the mass balance in transport equations coupled with the incompressible Navier-Stokes equations.

A Review of Variational Multiscale Methods for the Simulation of Turbulent Incompressible Flows

“…In [55], an approximate RFB method for the Oseen equations is analyzed for the non-conforming Crouzeix-Raviart pair of finite element spaces. It was shown that the exact RFB method is identical to some SUPG-type stabilization in the case of constant coefficients.…”

Nonconforming local projection stabilization for generalized Oseen equations

“…Compared with the residual-free bubble method [2] , the stabilized term of the local projection method is simple (only one term). At the same time, it can be seen as the promotion of the residual-free bubble method.…”

“…We compare the local projection method with the residual-free bubble method [2] and the standard finite element method (see Tables 1-4). …”

“…Most importantly, the stabilized term is simple compared with the residualfree bubble method [2] , and it only contains the L 2 -projection term. …”

“…They can be sorted into two kinds. One kind of methods are based on the residual of the momentum equation, such as the streamline-upwind Petrov-Galerkin (SUPG) method, the least-square method, the residual-free bubble method [1][2][3] , and the least squares Petrov-Galerkin method [4] . However, the pressure and velocity derivatives in this residual vanish or are poorly approximated, causing the difficulties in the application of consistent stabilization.…”