On a finite element method for unsteady compressible viscous flows

Abstract

In this article we present an analysis of a finite element method for solving two-dimensional unsteady compressible Navier-Stokes equations. Under the time-stepping size restriction ⌬t Յ Ch, we prove the existence and uniqueness of the numerical solution and obtain an a prior error estimate uniform in time.

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

“…Orders of accuracy in the error estimates (3.13) are the same as those for the two-dimensional case that were proved in [22]. The corresponding proof of part 1 in [22] does not involve specific cases in spatial dimensions. Therefore the proof of these estimates under the induction hypotheses in Section 3 in [22] is also valid for three-dimensional case here.…”

“…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 .…”

“…Part 2 is to prove (3.8) and (3.16) for n. The finite element approximation (3.7) is a direct extension of the two-dimensional finite element approximation analyzed in [22] to the threedimensional case. Orders of accuracy in the error estimates (3.13) are the same as those for the two-dimensional case that were proved in [22]. The corresponding proof of part 1 in [22] does not involve specific cases in spatial dimensions.…”

“…The corresponding proof of part 1 in [22] does not involve specific cases in spatial dimensions. Therefore the proof of these estimates under the induction hypotheses in Section 3 in [22] is also valid for three-dimensional case here. We will not repeat the same tedious derivation, but only outline the procedure here.…”

The analysis of a finite element method for the three-species Lotka–Volterra competition-diffusion with Dirichlet boundary conditions

“…We start by working with a similar procedure to Liu's [13,14] for the case of compressible Navier-Stokes flow problems. We assume that the initial conditions by choosing A 0 i,h as the Ritz projection of A 0 i onto the FE subspace V h to the numerical algorithm (8) is considered.…”