Nonconforming and Other Methods - Finite Elements: Theory, Fast Solvers, and Applications in Elasticity Theory - page 157

§ 6. The Stokes Equation

The Stokes equation describes the motion of an incompressible viscous fluid in an
n -dimensional domain (with n =

2 or 3):

u + grad p =− f in ,

div u = 0in , ( 6 . 1 )

u = u 0 on ∂.

n

Here u : −→ R is the pressure. Since

we are assuming that the fluid is incompressible, div u = 0 when no sources or
sinks are present.

is the velocity field and p : −→ R

In order for a divergence-free flow to exist with given boundary values u 0 ,
by Gauss' integral theorem we must have

u 0 · νds = u · νds = div udx = 0 . ( 6 . 2 )

∂ ∂

This compatibility condition on u 0 is obviously satisfied for homogeneous bound-
ary values.

By an appropriate scaling we can assume that the viscosity is 1, which we
have already done in writing (6.1).

The given external force field f causes an acceleration of the flow. The pres-
sure gradient gives rise to an additional force which prevents a change in the
density. In particular, a large pressure builds up at points where otherwise a source
or sink would be created. From a mathematical point of view, the pressure can be
regarded as a Lagrange multiplier.

[ C 2 () ∩ C 0 ( ¯ ) ] n and p ∈ C 1 () ,
then we call u and p a classical solution of the Stokes problem. Note that (6.1)
only determines the pressure p up to an additive constant, which is usually fixed
by enforcing the normalization

If (6.1) is satisfied for some functions u ∈

pdx = 0 . ( 6 . 3 )