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Curl-conforming finite element methods for Navier-Stokes equations with nonstandard boundary conditions in \({\mathcal R}^ 3\). (English) Zbl 0702.76037
The Navier-Stokes equations theory and numerical methods, Proc. Conf., Oberwolfach/FRG 1988, Lect. Notes Math. 1431, 201-218 (1990).
Summary: [For the entire collection see Zbl 0695.00018.]
This paper is devoted to the steady-state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form: \[ u\times n=0,\quad p+(1/2)u\cdot u=0, \] or \[ u\cdot n=0,\quad curl u\cdot n=0,\quad curl curl u\cdot n=0, \] or \[ u\cdot n=0,\quad curl u\times n=0. \] The problem is formulated in the primitive variables: velocity and pressure, and the divergence-free condition is imposed weakly by the equation \((\nabla q,v)=0\). Thus, while more regularity is required for the pressure, owing to the boundary conditions, the velocity needs only have a smooth curl. Hence, the velocity is approximated with curl conforming finite elements and the pressure with standard continuous finite elements. The error analysis gives optimal results.

76D05 Navier-Stokes equations for incompressible viscous fluids