Finite element least-squares methods for a compressible stokes system

Kee-Hwan Kim ,Sang Dong Kim ,Seoungwoo Shin

doi:10.1155/s0161171204204124

Abstract

The least-squares functional related to avorticityvariable or avelocity fluxvariable is considered for two-dimensional compressible Stokes equations. We show ellipticity and continuity in an appropriate product norm for each functional.

Highlights

Let Ω be a convex polygonal domain in R2
Since the continuity equation is of hyperbolic type containing a convective derivative of p, we further assume that the boundary condition for the pressure is given on the inlet of the boundary where the characteristic function β points into Ω, that is, p = 0 on Γin, (1.2)
There was a study on a mixed finite element theory for a compressible Stokes system, but there are a few trials dealing with a compressible Stokes system like (1.1) using leastsquares method

Summary

Introduction

Let Ω be a convex polygonal domain in R2. Consider the stationary compressible Stokes equations with zero boundary conditions for the velocity u = (u1, u2)t and pressure p as follows:−μ∆u + ∇p = f in Ω, ∇ · u + β · ∇p = g in Ω, u = 0 on ∂Ω, (1.1)where the symbols ∆, ∇, and ∇· stand for the Laplacian, gradient, and divergence operators, respectively (∆u is the vector of components ∆ui); the number μ is a viscous constant; f is a given vector function; β = (U , V )t is a given C1 function. Consider the stationary compressible Stokes equations with zero boundary conditions for the velocity u = (u1, u2)t and pressure p as follows: In order to provide ellipticity for each functional, we assume the H1 and H2 regularity assumptions for the compressible Stokes equations.

Objectives

Results

Conclusion