Civil Engineering Reference
In-Depth Information
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Fig. 9.
A uniform triangulation of a rectangle
According to Cea's lemma, the accuracy of a numerical solution depends
essentially on choosing function spaces which are capable of approximating the
solution
u
well. For polynomials, the order of approximation is determined by the
smoothness of the solution. However, for boundary-value problems, the smooth-
ness of the solution typically decreases as we approach the boundary. Thus, it
doesn't make much sense to use polynomials that are defined on the whole do-
main and to insist on a high accuracy by forcing the degree of the polynomials
to be high. As we shall see in §§6 and 7, it makes more sense to use piecewise
polynomials, and to achieve the desired accuracy by making the associated parti-
tion of
sufficiently fine. The so-called
h-p-methods
combine refinements of the
partitions and an increase of the degree of the polynomials; see Schwab [1998].
Model Problem
4.3 Example
(Courant [1943]). Suppose we want to solve the Poisson equation in
the unit square (or in a general domain which can be triangulated with congruent
triangles):
in
=
(
0
,
1
)
2
,
−
u
=
f
u
=
0 n
∂.
Suppose we partition
¯
with a uniform triangulation of mesh size
h
, as shown in
Fig. 9. Choose
={
v
∈
C(
¯
)
;
v
is linear in every triangle and
v
=
S
h
:
0on
∂
}
.
(
4
.
8
)
In every triangle,
v
∈
S
h
has the form
v(x,y)
=
a
+
bx
+
cy
, and is uniquely
defined by its values at the three vertices of the triangle. Thus, dim
S
h
=
N
=
number of interior mesh points. Globally,
v
is determined by its values at the
N
grid points
(x
j
,y
j
)
. Now choose a basis
N
i
=
{
ψ
i
}
with
1
ψ
i
(x
j
,y
j
)
=
δ
ij
.
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