The Use Of Non-Conforming Finite Elements In Three-Dimensional Viscous Incompressible Flow.

Abstract

In several papers published since the early eighties, the author demonstrated
that some mixed finite element methods to solve two-dimensional viscous incompressible
flow equations in primitive variables with a conforming velocity, have non-conforming
three-dimensional analogues. Parallelly he established that some classical non-conforming
two-dimensional methods in other formulations admit non trivial equivalent extensions to
the three-dimensional case. In this work, while recalling some of the above mentionned
examples, the author exhibits a case where a fundamental property of a three-dimensional
non-conforming method does not hold for its analogous two-dimensional version. This
property is shown to play a crucial role in connection with the Navier-Stokes equations in
terms of a vector potential with a vanishing gradient on the boundary of the flow domain.