Incompressible finite element methods for Navier-Stokes equations with nonstandard boundary conditions in 𝑅³

Abstract

This paper is devoted to the steady state, incompressible Navier-Stokes equations with nonstandard boundary conditions of the form u ⋅ n = 0 {\mathbf {u}} \cdot {\mathbf {n}} = 0 , c u r l u × n = 0 \mathbf {curl}\;{\mathbf {u}} \times {\mathbf {n}} = {\mathbf {0}} , either on the entire boundary or mixed with the standard boundary condition u = 0 {\mathbf {u}} = {\mathbf {0}} on part of the boundary. The problem is expressed in terms of vector potential, vorticity and pressure. The vorticity and vector potential are approximated with curl-conforming finite elements and the pressure with standard continuous finite elements. The error estimates yield nearly optimal results for the purely nonstandard problem.