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Chapter 2
Preliminaries
The classical process for the numerical solution of partial differential equations (describing
a physical phenomenon, in our case the Navier-Stokes equations describing the flow of
an incompressible fluid, or the related Oseen or Stokes equations) is to derive a weak
formulation, provide analysis, discretize the system (in our case with finite elements), and
finally to solve the resulting linear algebra problems.
This first chapter contains the parts of this process, from the problem formulation to
(non-multigrid) solution methods for the arising linear systems.
2.1 Navier-Stokes Equations
Our main point of investigation will be the Navier-Stokes equations for incompressible flow
(Claude Navier, 1785–1836, and George Stokes, 1819–1903). A mathematically rigorous
derivation from fundamental physical principles and conservation laws can be found in
[Fei93].
We denote by u the velocity of the fluid, p the static pressure, ρ the density of the
fluid, µ its viscosity and f some outer force. Then the instationary flow of incompressible
Newtonian fluids in a domain G (where G is an open, connected subset of R d with Lipschitz
continuous boundary ∂G) is governed by
ρ ∂ u − µ∆u + ρ(u · ∇)u + ∇p
∂t
= f (2.1a)
div u = 0. (2.1b)
Equation (2.1a) expresses Newton’s law of motion, (2.1b) the conservation of mass.
The underlying physical assumption for these equations to hold are incompressibility
and Stokes’ hypothesis for the stress tensor
for incompressible Newtonian fluids.
T(u, p) = −pI + µ ( ∇u + ∇u T ) , (2.2)
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