In this paper, a difference finite element (DFE) method is presented for the 3D Poisson equation on non-uniform meshes by using the
P
1
−
P
1
P_1-P_1
-conforming element. This new method consists of combining the finite difference discretization based on the
P
1
P_1
-element in the
z
z
-direction with the finite element discretization based on the
P
1
P_1
-element in the
(
x
,
y
)
(x,y)
-plane. First, under the regularity assumption of
u
∈
H
3
(
Ω
)
∩
H
0
1
(
Ω
)
u\in H^3(\Omega )\cap H^1_0(\Omega )
and
∂
z
z
f
∈
L
2
(
(
0
,
L
3
)
;
\partial _{zz}f\in L^2((0, L_3);
H
−
1
(
ω
)
)
H^{-1}(\omega ))
, the
H
1
H^1
-superconvergence of the discrete solution
u
τ
u_\tau
in the
z
z
-direction to the first-order interpolation function
I
τ
u
I_\tau u
is obtained, and the
H
1
H^1
-superconvergence of the second-order interpolation function
I
2
τ
2
u
τ
I^2_{2\tau } u_\tau
in the
z
z
-direction to
u
u
is then provided. Moreover, the
H
1
H^1
-superconvergence of the DFE solution
u
h
u_h
to the
H
1
H^1
-projection
R
h
u
τ
R_hu_\tau
of
u
τ
u_\tau
is deduced and the
H
1
H^1
-superconvergence of the second-order interpolation function
I
2
τ
2
I
2
h
2
u
h
I^2_{2\tau }I^2_{2h} u_h
to
u
u
in the
(
(
x
,
y
)
,
z
)
((x,y),z)
-space is also established. Finally, numerical tests are presented to show the
H
1
H^1
-superconvergence results of the DFE method for the 3D Poisson equation under the above regularity assumption.