The analysis of a finite element method for the Navier-Stokes equations with compressibility

Abstract

A finite element formulation is developed for the two dimensional nonlinear time dependent compressible Navier-Stokes equations on a bounded domain. The existence and uniqueness of the solution to the numerical formulation is proved. An error estimate for the numerical solution is obtained.

On a finite element method for unsteady compressible viscous flows

“…The assumption (2.4) here is the same as the assumption (3.5) in [27]. Let T h be a partition consisting of a regular isoparametric family of n-simplices with diameter not greater than h; let W h ʚ H 0 1 (⍀) and Q h ʚ Q be finite dimensional subspaces of piecewise polynomials of degree k and m, respectively, associated with T h and V h ϭ W h 2 .…”

“…Liu [26] discussed a finite element method with streamline diffusion for compressible Navier Stokes equations and obtained an a priori error estimate. Kellogg and Liu [27] analyzed a standard finite element method for two-dimensional compressible Navier-Stokes equations. The existence of the solution was proved, and an a priori error estimate was obtained under the time-stepping restriction ⌬t Յ Ch 2 , where h is the element diameter of a quasi-uniform triangulation on ⍀.…”

On a finite element method for three‐dimensional unsteady compressible viscous flows

“…As noted in Section 3, the finite element approximation (3.7) and the error estimates (3.13) are valid for both two-and three-dimensional cases. Considering there was no numerical computation given in our previous two-dimensional work [21,22], we will be therefore testing both two-and three-dimensional examples. To represent curved boundary segments accurately, we use the mapping by the linear blending function method over curved quadrilateral elements discussed by Szabo and Babuska [31].…”

“…Here P٪ is the state function of the flow; 1 , *, H, and M are constants. The assumption (3.4) here is the same as the assumption (3.5) in [21] and the assumption (2.4) in [22]. Let T h be a partition consisting of a regular isoparametric family of n-simplices with diameter not greater than h; Let W h ʚ H 0 1 (⍀) and Q h ʚ Q be finite dimensional subspaces of piecewise polynomials of degree k and m, respectively, associated with T h , and V h ϭ W h 3 .…”

“…The existence of the solution was proved and a priori error estimates were obtained. In this article, we consider the finite element method developed in [21] for the three-dimensional unsteady compressible Navier-Stokes equations. We prove there exists a unique solution and the numerical solution is convergent when ⌬t Յ Ch 3/ 2 .…”