zbMATH — the first resource for mathematics

Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in \(R^ 3\). (English) Zbl 0666.76053
The steady, incompressible Navier-Stokes equations are studied, first with boundary conditions \(\bar u\cdot \bar n=0\), \(rot\quad \bar u\times \bar n=0\) on the entire boundary. The weak formulation is obtained in terms of vector-potential, vorticity and pressure. Assuming that the solution has a particular regularity which in general seems to be not obtained with usual theorems, the weak formulation is approximated with curl-conforming finite elements developed by J. C. Nedelec [Numer. Math. 35, 315-341 (1980; Zbl 0419.65069)]. The divergence free-condition is approximated by Greens formula. The assumptions of regularity are used to obtain the error estimates and convergence of the discrete problem. Finally the case of the above boundary conditions mixed with the standard condition \(\bar u=0\) on a part of boundary is studied.
Reviewer: G.Pasa

76D05 Navier-Stokes equations for incompressible viscous fluids
35J65 Nonlinear boundary value problems for linear elliptic equations
35Q30 Navier-Stokes equations
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
Full Text: DOI