Finite element methods for elliptic systems with constraints (Q2760345)

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.