In this paper a simple turning point (
y
=
y
c
y = {y^c}
,
λ
=
λ
c
\lambda = {\lambda ^c}
) of the parameter-dependent Hammerstein equation
\[
y
(
t
)
=
f
(
t
)
+
λ
∫
a
b
k
(
t
,
s
)
g
(
s
,
y
(
s
)
)
d
s
,
t
∈
[
a
,
b
]
,
y(t) = f(t) + \lambda \int _a^b {k(t,s)g(s,y(s))\;ds,\quad t \in [a,b],}
\]
is approximated numerically in the following way. A simple turning point (
z
=
z
c
z = {z^c}
,
λ
=
λ
c
\lambda = {\lambda ^c}
) of an equivalent equation for
z
(
t
)
:=
λ
g
(
t
,
y
(
t
)
)
z(t):=\lambda g(t,y(t))
is computed first. This is done by solving a discretized version of a certain system of equations which has (
z
c
{z^c}
,
λ
c
{\lambda ^c}
) as part of an isolated solution. The particular discretization used here is standard piecewise polynomial collocation. Finally, an approximation to
y
c
{y^c}
is obtained by use of the (exact) equation
\[
y
(
t
)
=
f
(
t
)
+
∫
a
b
k
(
t
,
s
)
z
(
s
)
d
s
,
t
∈
[
a
,
b
]
.
y(t) = f(t) + \int _a^b {k(t,s)z(s)\;ds,\quad t \in [a,b].}
\]
The main result of the paper is that, under suitable conditions, the approximations to
y
c
{y^c}
and
λ
c
{\lambda ^c}
are both superconvergent, that is, they both converge to their respective exact values at a faster rate than the collocation approximation (of
z
c
{z^c}
) does to
z
c
{z^c}
.