In this paper we consider the continuous piecewise linear finite element approximation of the following problem: Given
p
∈
(
1
,
∞
)
p \in (1,\infty )
, f, and g, find u such that
\[
−
∇
⋅
(
|
∇
u
|
p
−
2
∇
u
)
=
f
in
Ω
⊂
R
2
,
u
=
g
on
∂
Ω
.
- \nabla \cdot (|\nabla u{|^{p - 2}}\nabla u) = f\quad {\text {in}}\;\Omega \subset {\mathbb {R}^2},\quad u = g\quad {\text {on}}\;\partial \Omega .
\]
The finite element approximation is defined over
Ω
h
{\Omega ^h}
, a union of regular triangles, yielding a polygonal approximation to
Ω
\Omega
. For sufficiently regular solutions u, achievable for a subclass of data f, g, and
Ω
\Omega
, we prove optimal error bounds for this approximation in the norm
W
1
,
q
(
Ω
h
)
,
q
=
p
{W^{1,q}}({\Omega ^h}),q = p
for
p
>
2
p > 2
and
q
∈
[
1
,
2
]
q \in [1,2]
for
p
>
2
p > 2
, under the additional assumption that
Ω
h
⊆
Ω
{\Omega ^h} \subseteq \Omega
. Numerical results demonstrating these bounds are also presented.