Civil Engineering Reference
In-Depth Information
2
;
x
2
+
y
2
<
4.4 Lemma.
Suppose is contained in the disk B
R
(
0
)
:
={
(x, y)
∈ R
R
2
}
. Let V be the solution of the equation
L
h
V
=
1
in
h
,
(
4
.
3
)
V
=
0
on ∂
h
,
where L
h
is the standard five-point stencil. Then
1
4
(R
2
x
i
−
y
i
).
0
≤
V(x
i
,y
i
)
≤
−
(
4
.
4
)
−
y
2
)
, and set
W
i
=
w(x
i
,y
i
).
Since
w
is a polynomial of second degree, the derivatives of higher order which
were dropped when forming the difference star vanish. Hence we have
(L
h
W)
i
=
Lw(x
i
,y
i
)
=
1
4
(R
2
−
x
2
Proof.
Consider the function
w(x,y)
:
=
0on
∂
. The discrete comparison principle
implies that
V
≤
W
, while the minimum principle implies
V
≥
1. Moreover,
W
≥
0, and (4.4) is
proved.
The essential fact about (4.4) is that it provides a bound which is independent
of
h
. - This lemma can be extended to any elliptic differential equation for which
the finite difference approximation is exact for polynomials of degree 2. In this
case the factor
1
4
in (4.4) is replaced by a number which depends on the constant
of ellipticity.
4.5 Convergence Theorem.
Suppose the solution of the Poisson equation is a C
2
function, and that the derivatives u
xx
and u
yy
are uniformly continuous in . Then
the approximations obtained using the five-point stencil converge to the solution.
In particular
max
z
h
|
U
h
(z)
−
u(z)
|→
0
as h
→
0
.
(
4
.
5
)
∈
Proof.
By the Taylor expansion at the point
(x
i
,y
i
)
,
L
h
u(x
i
,y
i
)
=−
u
xx
(ξ
i
,y
i
)
−
u
yy
(x
i
,η
i
),
where
ξ
i
and
η
i
are certain numbers. Because of the uniform continuity, the local
discretization error max
i
|
r
i
|
tends to 0. It now follows from (4.2) and Lemma 4.4
that
R
2
4
max
|
η
i
|≤
max
|
r
i
|
,
(
4
.
6
)
which gives the convergence assertion.
O
(h
2
)
estimates for the global
error, provided
u
is in
C
3
(
¯
)
or in
C
4
(
¯
)
, respectively.
Analogously, using (4.6), we can get
O
(h)
or
Limits of the Convergence Theory
The hypotheses on the derivatives required for the above convergence theorem are
often too restrictive.
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