Analysis of mixed and stabilized space-time finite element methods for the Navier-Stokes equations

Abstract

The performance and the accuracy of the stable mixed space-time finite element methods for the incompressible Navier-Stokes equations in a bounded domain in 1R are investigated in this work. The mixed method is based on the recently developed space-time mini element, which consists of piecewise linear functions for the pressure and of piecewise linear functions enriched with a bubble function for the velocity. This element is stable in the sense that, the underlying pair of discrete spaces for velocity and pressure satisfies the so-called u condition. We assess the accuracy of the underlying mixed approximation and compare its performance with that of the stabilized Galerkin/least-squares space-time (GLS/ST) Both methods are based on the Galerkin method with the use of simplex-type meshes. Numerical results for some 2D problems are presented using both Cartesian and cylindrical frames of reference. Introduction \HE solution of time-dependent problems, such as the incompressible Navier-Stokes equations *Phd Student t Professor * Professor Copyright © 2001 by Donatien N'dri. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. by finite element methods, leads to the use of finite elements in space and finite differences in time. In contrast, the space-time formulation performs the discretization in space and time concurrently by blending the space and time variables into a spacetime finite element. This space-time element has one extra dimension, and the finite element interpolation functions depend on both the space and time variables. The introduction by Jamet (for parabolic equations) of interpolation functions continuous in space but discontinuous in time led to the so-called time-discontinuous Galerkin method. The choice of the approximation spaces for the velocity and the pressure is a fundamental issue for the discretization of the incompressible NavierStokes equations. The approximation spaces must satisfy a priori a compatibility condition known as the condition. If the mixed discretization is not stable, stabilizing techniques are required. This led to the development of stabilized finite element methods based on equal order interpolations, like the Galerkin least-squares space-time (GLS/ST) methods. On the other hand, to the best of our knowledge, there is no published work on the construction of approximation spaces for velocity and pressure which satisfy the inf-sup condition in the context of the space-time finite element method, and our previous paper was the first attempt in that direction. The goal of this paper is to assess the perforAmerican Institute of Aeronautics and Astronautics (c)2001 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization. mance of the stable mixed space-time finite element when compared with the stabilized Galerkin leastsquares space-time (GLS/ST) The paper is organized as follows:We first define the problem, followed by a review of the Galerkin We recall the space-time mini element and the stabilized Galerkin least-squares space-time (GLS/ST) method and conclude with some 2D numerical examples which are the basis of the comparison. Governing Equations We consider a viscous, incompressible Newtonian fluid which occupies at time t G (0,T), a bounded region Q C K, with boundary T = dfi. The primary degrees of freedom are u(x,£), the velocity of the fluid and the pressure p(x,t). The conservation of mass and momentum of the fluid are expressed by the equations V-u = 0 on Q, Vt€(0,T) ,