In
L
2
(
R
d
;
C
n
)
L_2(\mathbb {R}^d;\mathbb {C}^n)
, a strongly elliptic differential operator
A
ε
{\mathcal {A}}_\varepsilon
of order
2
p
2p
is studied. Its coefficients are periodic and depend on
x
/
ε
\mathbf {x}/\varepsilon
. The resolvent
(
A
ε
+
I
)
−
1
({\mathcal {A}}_\varepsilon +I)^{-1}
is approximated in the operator norm on
L
2
(
R
d
;
C
n
)
L_2(\mathbb {R}^d;\mathbb {C}^n)
:
(
A
ε
+
I
)
−
1
=
(
A
0
+
I
)
−
1
+
∑
j
=
1
2
p
−
1
ε
j
K
j
,
ε
+
O
(
ε
2
p
)
.
\begin{equation*} ({\mathcal {A}}_\varepsilon +I)^{-1} = ({\mathcal {A}}^0+I)^{-1} + \sum _{j=1}^{2p-1} \varepsilon ^{j} {\mathcal {K}}_{j,\varepsilon } + O(\varepsilon ^{2p}). \end{equation*}
Here
A
0
{\mathcal {A}}^0
is an effective operator with constant coefficients, and
K
j
,
ε
{\mathcal {K}}_{j,\varepsilon }
,
j
=
1
,
…
,
2
p
−
1
j=1,\dots ,2p-1
, are appropriate correctors.