Stable space-time formulation for the Navier-Stokes equations

Abstract

A space-time finite element method for the timedependent incompressible Navier-Stokes equations in a bounded domain in IB’ is presented. The method is based on the time-discontinuous Galerkin method with the use of simplex-type meshes together with the requirement that the space-time finite element discretization for the velocity and the pressure satisfy the inf-sup stability condition of Brezzi and Babuska. The finite element discretization for the pressure consists of piecewise linear functions, while piecewise linear functions enriched with a bubble function are used for the velocity. Numerical results for some 20 problems are presented. Introduction T HE solution of time-dependent problems such as the incompressible Navier-Stokes equations by finite element methods leads to the mixing of finite elements in space and finite differences in time. In contrast, the space-time formulation performs the discretizations in space and time concurrently by blending the space and time variables into a space-time finite element. This space-time element has one extra dimension, and the finite element interpolation functions are dependent on both the space and time variables. In other words, this procedure can be viewed as an extension of the ‘Phd Student tprofessor SProfessor Copyright @ 1999 by Donatien N’dri. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. finite element method over the time domain. Although the concept was originally introduced by O&n1 an,d FJ-iec1_,2 the fimf ~,--,G~*’ “A.., SLI 0” *Lulll~llLal resuits can be traced back to Bonnerot and Jamet314 for the Stefan problem. Improving on the time-continuous interpolation used in their previous formulation, Jarnet later introduced another approach. His idea was to permit the unknown to be discontinuous with respect to time. This procedure, known as the time-discontinuous Galerkin method, allows the space-time domain to be organized into a series of “slabs” S, = R x (tn, &+I), where R is the underlying spatial domain and t, is a discrete time level. The fully discretized equations are then solved on one space-time slab at a time, leading to a time-marching procedure in which the solution for the current slab provides the initial condition for the next one. The time-discontinuous Galerkin method has been successfully applied to numerous problems such as heat conduction, elastodynamics, structural acoustics, etc. Unfortunately, like the standard Galerkin method, the time-discontinuous Galerkin method has been shown to produce spurious numerical oscillations in convection-dominated problems. To eliminate these oscillations, Varoglu and Finn63 7 first employed the method of characteristics to modify the spatial discretization at each time step. Johnson and Saranens and Hansbo and Szepessyg introduced the streamline diffusion method (SD). These formulations enhance stability through the addition of a small least-squares term to the