Civil Engineering Reference
In-Depth Information
§ 7. Finite Elements for the Stokes Problem
In the study of convergence for saddle point problems we assumed that the fi-
nite element spaces for velocities and pressure satisfy the inf-sup condition. This
raises the question of whether this condition is only needed to get a complete
mathematical theory, or whether it plays an essential role in practice.
The answer to this question is given by a well-known finite element method
for which the Brezzi condition is violated. Although instabilities had been ob-
served in computations with this element in fluid mechanics, attempts to explain
its instable behavior and to overcome it in a simple way mostly proved to be unsat-
isfactory. The Brezzi condition turned out to be the appropriate mathematical tool
for understanding and removing this instability, and it also provided the essential
breakthrough in practice. There are very few areas
7
where the mathematical theory
is of as great importance for the development of algorithms as in fluid mechanics.
After discussing the instable element mentioned above, we present two com-
monly used stable elements and another one which is easier to implement. There
is also a nonconforming divergence-free element which allows the elimination of
the pressure.
An Instable Element
In the Stokes equation (6.1),
u
and grad
p
are the terms with derivatives of
highest order for the velocity and pressure, respectively. Thus, the orders of the
differential operators differ by 1. This suggests the rule of thumb: the degree of the
polynomials used to approximate the velocities should be one larger than for the
approximation of the pressure. However, this “rule” is not sufficient to guarantee
stability - as we shall see.
Because of its simplicity, the so-called
Q
1
-
P
0
element has been popular for a
long time. It is a rectangular element which uses bilinear functions for the velocity
and piecewise constants for the pressure:
={
v
∈
C
0
(
¯
)
2
X
h
:
;
v
|
T
∈
Q
1
i.e., bilinear for
T
∈
T
h
}
,
M
h
:
={
q
∈
L
2
,
0
()
;
q
|
T
∈
P
0
for
T
∈
T
h
}
.
7
There are two comparable situations where purely mathematical considerations have
played a major role in the development of methods for differential equations. The approxi-
mation properties of the exponential function show that to solve stiff differential equations,
we need to use implicit methods. (In particular, parabolic differential equations lead to
stiff systems.) For hyperbolic equations, we need to enforce the Courant-Levy condition
in order to correctly model the domain of dependence in the discretization.
Search WWH ::
Custom Search