Civil Engineering Reference
In-Depth Information
§ 6. Approximation Properties
In this section we give error bounds for finite element approximations. By Cea's
lemma, in the energy norm it suffices to know how well the solution can be ap-
proximated by elements in the corresponding finite element space
S
h
=
S
h
(
T
h
)
.
For general methods, a suitable framework is provided by the theory of
affine fam-
ilies
. We do not derive results for every individual element, but instead examine
a
reference element
, and use transformation formulas to carry the results over to
shape-regular grids.
We intend to provide error bounds in other norms besides the energy norm.
We will concentrate primarily on affine families of triangular elements. Clearly,
the error for an interpolation method provides an upper bound for the error of the
best
approximation. It turns out that we actually get the correct order of approximation
in this way. - We consider
C
0
elements, which according to Theorem 5.2, are not
contained in
H
m
()
for
m>
1, and so the higher Sobolev norms are not applicable.
As substitutes, we use certain
mesh-dependent norms
which are tailored to the
problem at hand. We do not use the symbols
·
m,h
for fixed norms,
but allow the norm to change from case to case. Often mesh-dependent norms are
broken norms
as in (6.1) or norms with weight factors
h
−
m
·
h
and
as in Problem III.1.9.
T
h
={
T
1
,T
2
,...,T
M
}
of
and
m
≥
6.1 Notation.
Given a partition
1, let
T
j
∈
T
h
v
2
v
m,h
:
=
m,T
j
.
(
6
.
1
)
v
m,h
=
v
m,
for
v
∈
H
m
()
.
Let
m
≥
Clearly,
2. By the Sobolev imbedding theorem (see Remark 3.4)
H
m
()
⊂
C
0
()
, i.e. every
v
∈
H
m
has a continuous representer. For every
v
∈
H
m
, there
exists a uniquely defined interpolant in
S
h
=
S
h
(
T
h
)
associated with the points in
5.6. We denote it by
I
h
v
. The goal of this section is to estimate
v
−
I
h
v
m,h
by
v
t,
for
m
≤
t.
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