Civil Engineering Reference
In-Depth Information
. Whenever
a
is an
H
m
-elliptic bilinear
where
A
ik
:
=
a(ψ
k
,ψ
i
)
and
b
i
:
=
, ψ
i
form, the matrix
A
is positive definite:
z
Az
=
z
i
A
ik
z
k
i,k
a
k
z
k
ψ
k
,
i
z
i
ψ
i
=
=
a(u
h
,u
h
)
2
≥
α
u
h
m
,
and so
z
Az >
0 for
z
=
0. Here we have made use of the bijective mapping
N
−→
S
h
which is defined by (4.3). Without explicitly referring to this canonical
mapping, in the sequel we will identify the function space
S
h
with
R
N
.
In engineering sciences, and in particular if the problem comes from contin-
uum mechanics, the matrix
A
is called the
stiffness matrix
or
system matrix
.
R
Methods.
There are several related methods:
Rayleigh-Ritz Method
: Here the minimum of
J
is sought in the space
S
h
. Instead
of the basis-free derivation via (4.2), usually one finds
u
h
as in (4.3) by solving
the equation
(∂/∂z
i
)J (
k
z
k
ψ
k
)
=
0.
Galerkin Method
: The weak equation (4.2) is solved for problems where the bilin-
ear form is not necessarily symmetric. If the weak equations arise from a variational
problem with a positive quadratic form, then often the term
Ritz-Galerkin Method
is used.
Petrov-Galerkin Method
: Here we seek
u
h
∈
S
h
with
a(u
h
,v)
=
, v
for all
v
∈
T
h
,
where the two
N
-dimensional spaces
S
h
and
T
h
need not be the same. The choice
of a space of test functions which is different from
S
h
is particularly useful for
problems with singularities.
As we saw in §§2 and 3, the boundary conditions determine whether a prob-
lem should be formulated in
H
m
()
or in
H
0
()
. For the purposes of a unified
notation, in the following we always suppose
V
⊂
H
m
()
, and that the bilinear
form
a
is always
V
-elliptic, i.e.,
2
m
a(v, v)
≥
α
v
and
|
a(u, v)
|≤
C
u
m
v
m
for all
u, v
∈
V,
where 0
<α
≤
C
. The norm
·
m
is thus equivalent to the energy norm (2.14),
which we use to get our first error bounds. - In addition, let
∈
V
with
|
, v
| ≤
·
v
m
for
v
∈
V
. Here
is the (dual) norm of
.
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