Civil Engineering Reference
In-Depth Information
§ 6. Nonlinear Problems
Multigrid methods are also very useful for the numerical solution of nonlinear
differential equations. We need only make some changes in the multigrid method
for linear equations. These changes are typical for the efficient treatment of non-
linear problems. However, there is one essential idea involved which we might
not otherwise encounter. We have to correct the right-hand side of the nonlinear
equation on the coarse grid in order to compensate for the error which arises in
moving between grids.
As an example of an important nonlinear differential equation, consider the
Navier-Stokes equation
−
u
+
Re
(u
∇
)u
−
grad
p
=
f
in
,
div
u
=
0in
,
(
6
.
1
)
u
=
u
0
on
∂.
If we drop the quadratic term in the first equation, we get the Stokes problem
(III.6.1). Another typical nonlinear differential equation is
−
u
=
e
λu
in
,
(
6
.
2
)
=
u
0 n
∂.
It arises in the analysis of explosive processes. The parameter
λ
specifies the
relation between the reaction heat and the diffusion constant. - Nonlinear boundary
conditions are also of interest, in particular for problems in (nonlinear) elasticity.
We write a nonlinear boundary-value problem as an equation of the form
L
(u)
=
0. Suppose that for each
=
0
,
1
,...,
max
, the discretization at level
leads to the nonlinear equation
L
(u
)
=
0
(
6
.
3
)
with
N
:
dim
S
unknowns. In the sequel it is often more convenient to consider
the formally more general equation
=
L
(u
)
=
f
(
6
.
4
)
N
.
Within the framework of multigrid methods, there are two fundamentally
different approaches:
1. The
multigrid Newton method
(MGNM), which solves the linearized equation
using the multigrid method.
2. The
nonlinear multigrid method
(NMGM), which applies the multigrid method
directly to the given nonlinear equation.
with given
f
∈ R
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