We will construct a discrete model whose solution gives an accurate ap
proximation to the exact solution at a finite collection of selected points
(called nodes) in [0
,
1]. We take the nodes to be equally spaced and to in
clude the interval end points (boundary). So we choose a positive integer
N
and define the nodes or grid points
x
0
= 0
, x
1
=
h, x
2
= 2
h, . . . , x
N
=
Nh, x
N
+1
= 1
,
(10.64)
where
h
= 1
/
(
N
+ 1) is the
grid size
or node spacing. The nodes
x
1
, . . . , x
N
are called interior nodes, because they lie inside the interval [0
,
1], and the
nodes
x
0
and
x
N
+1
are called boundary nodes.
We now construct a discrete approximation to the ordinary differential
equation by replacing the second derivative with a second order finite differ
ence approximation. As we know,
u
00
(
x
j
) =
u
(
x
j
+1

2
u
(
x
j
) +
u
(
x
j

1
h
2
+
O
(
h
2
)
.
(10.65)
Neglecting the
O
(
h
2
) error and denoting the approximation of
u
(
x
j
) by
v
j
(i.e.
v
j
≈
u
(
x
j
))
f
j
=
f
(
x
j
) and
c
j
=
c
(
x
j
), for
j
= 1
, . . . , N
, then at each
interior node

v
j

1
+ 2
v
j
+
v
j
+1
h
2
+
c
j
v
j
=
f
j
,
j
= 1
,
2
, . . . , N
(10.66)
170
CHAPTER 10.
LINEAR SYSTEMS OF EQUATIONS I
and at the boundary nodes, applying (10.63), we have
v
0
=
v
N
+1
= 0
.
(10.67)
Thus, (10.66) is a linear system of
N
equations in
N
unknowns
v
1
, . . . , v
N
,
which we can write in matrix form as
1
h
2
2 +
c
1
h
2

1
0
· · ·
· · ·
0

1
2 +
c
2
h
2

1
.
.
.
.
.
.
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0
.
.
.
.
.
.
.
.
.

1
0
· · ·
0

1
2 +
c
N
h
2
v
1
v
2
.
.
.
.
.
.
.
.
.
.
.
.
v
N
=
f
1
f
2
.
.
.
.
.
.
.
.
.
.
.
.
f
N
.
(10.68)
The matrix, let us call it
A
, of this system is tridiagonal and symmetric.
A direct computation shows that for an arbitrary, nonzero, column vector
v
= [
v
1
, . . . , v
N
]
T
v
T
A
v
=
N
X
j
=1
"
v
j
+
c
j
v
j
h
2
+
c
j
v
2
j
#
>
0
,
∀
v
6
= 0
(10.69)
and therefore, since
c
j
≥
0 for all
j
,
A
is positive definite. Thus, there is a
unique solution to (10.68) and can be efficiently found with our tridiagonal
solver, Algorithm 5. Since the expected numerical error is
O
(
h
2
) =
O
(1
/
(
N
+
1)
2
), even a modest accuracy of
O
(10

4
) requires
N
≈
100.
10.6
A 2D BVP: Dirichlet Problem for the
Poisson’s Equation
We now look at a simple 2D BVP for an equation that is central to many
applications, namely Poisson’s equation. For concreteness here, we can think
of the equation as a model for small deformations
u
of a stretched, square
membrane fixed to a wire at its boundary and subject to a force density
10.6. A 2D BVP: DIRICHLET PROBLEM FOR THE POISSON’S EQUATION
171
f
. Denoting by Ω, and
∂
Ω, the unit square [0
,
1]
×
[0
,
1] and its boundary,
respectively, the BVP is to find
u
such that

Δ
u
(
x, y
) =
f
(
x, y
)
,
for (
x, y
)
∈
Ω
(10.70)
and
u
(
x, y
) = 0
.
for (
x, y
)
∈
∂
Ω
(10.71)
In (10.70), Δ
u
is the Laplacian of u, also denoted as
∇
2
u
, and is given by
Δ
u
=
∇
2
u
=
u
xx
+
u
yy
=
∂
2
u
∂x
2
+
∂
2
u
∂y
2
.
(10.72)
Equation (10.70) is Poisson’s equation (in 2D) and together with (10.71)
specify a (homogeneous) Dirichlet problem because the value of
u
is given at
the boundary.