Finite element methods for elliptic systems with constraints
type: Item
instance of: scholarly article
zbMATH Open document ID: 1003.76049
publication date: 2001-12-19T00:00:00Z
Mathematics Subject Classification ID: 76M10
Mathematics Subject Classification ID: 76N15
Mathematics Subject Classification ID: 65N30
zbMATH DE Number: 1684504
zbMATH Keywords: elliptic systems
zbMATH Keywords: genealized Stokes problem
zbMATH Keywords: compressible nonisothermal flow
zbMATH Keywords: homogeneous Dirichlet boundary conditions
zbMATH Keywords: conforming finite elements
zbMATH Keywords: convergence
zbMATH Keywords: Babuska-Brezzi condition
zbMATH Keywords: chemical vapor deposition reactors
zbMATH Keywords: Uzawa's method
author: Manfred Dobrowolski
MaRDI profile type: Publication
DOI: 10.1002/(SICI)1099-1506(199903)6:2%3C115::AID-NLA151%3E3.0.CO;2-8
title: Finite element methods for elliptic systems with constraints
published in: Numerical Linear Algebra with Applications
review text: The article deals with generalized Stokes problem in a bounded domain \(\Omega \subset\mathbb{R}^n\) with homogeneous Dirichlet boundary condition on \(\partial \Omega \). The generalization consists in the facts that instead of the pressure gradient, i.e. \(\partial _i p\), the author uses \(b_{ij}\partial _j p\), and instead of the continuity equation \(\partial _iu^i = 0\) the author considers the equation \(c_{ij}\partial _j u^i = 0\). It is proved that the problem is weakly solvable if and only if it is elliptic. The problem is further approximated by the standard conforming finite elements, and the convergence of this method is proved under the assumption that certain bilinear forms defined by means of the matrices \(b_{ij}\) and \(c_{ij}\) satisfy the Babuska-Brezzi condition. NEWLINENEWLINENEWLINEThe method is applied to a compressible nonisothermal flow. The density is assumed to be a function of temperature, and the pressure is one of the unknowns. This model is near to incompressible Navier-Stokes equations, and it can describe e.g. metalorganic chemical vapor deposition reactors. The author uses Uzawa's method to solve the resulting system of nonlinear discrete equations.
description: scientific article; zbMATH DE number 1684504