Editing Navier–Stokes equations (section)

===Discrete velocity===
With partitioning of the problem domain and defining [[basis function]]s on the partitioned domain, the discrete form of the governing equation is
<math display="block">\left(\mathbf{w}_i, \frac{\partial\mathbf{u}_j}{\partial t}\right) = -\bigl(\mathbf{w}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\bigr) - \nu\left(\nabla\mathbf{w}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{w}_i, \mathbf{f}^S\right).</math>

It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by [[Stokes' theorem]]. Discussion will be restricted to 2D in the following.

We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the [[Bending of plates|plate-bending]] literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,
<math display="block">\begin{align}
\nabla\varphi &= \left(\frac{\partial \varphi}{\partial x},\,\frac{\partial \varphi}{\partial y}\right)^\mathrm{T}, \\[5pt]
\nabla\times\varphi &= \left(\frac{\partial \varphi}{\partial y},\,-\frac{\partial \varphi}{\partial x}\right)^\mathrm{T}.
\end{align}</math>

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements.<ref>
{{Citation
 | last = Holdeman | first = J. T.
 | title = A Hermite finite element method for incompressible fluid flow
 | journal = Int. J. Numer. Methods Fluids
 | volume = 64 | pages = 376–408 | year = 2010
 | doi = 10.1002/fld.2154
 | issue = 4 |bibcode = 2010IJNMF..64..376H
| s2cid = 119882803
 }}</ref><ref>
{{Citation
 | last1 = Holdeman | first1 = J. T.
 | last2 = Kim | first2 = J. W.
 | title = Computation of incompressible thermal flows using Hermite finite elements
 | journal = Comput. Meth. Appl. Mech. Eng.
 | volume = 199 | pages = 3297–3304 | year = 2010
 | doi = 10.1016/j.cma.2010.06.036
 | issue = 49–52 |bibcode = 2010CMAME.199.3297H
}}</ref> The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces.
Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are [[#Nonlinearity|non-linear]], requiring iteration of the linearized equations.

Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.