We consider the enhancement of accuracy, by means of a simple post-processing technique, for finite element approximations to transient hyperbolic equations. The post-processing is a convolution with a kernel whose support has measure of order one in the case of arbitrary unstructured meshes; if the mesh is locally translation invariant, the support of the kernel is a cube whose edges are of size of the order of
Δ
x
\Delta x
only. For example, when polynomials of degree
k
k
are used in the discontinuous Galerkin (DG) method, and the exact solution is globally smooth, the DG method is of order
k
+
1
/
2
k+1/2
in the
L
2
L^2
-norm, whereas the post-processed approximation is of order
2
k
+
1
2k+1
; if the exact solution is in
L
2
L^2
only, in which case no order of convergence is available for the DG method, the post-processed approximation converges with order
k
+
1
/
2
k+1/2
in
L
2
(
Ω
0
)
L^2(\Omega _0)
, where
Ω
0
\Omega _0
is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented.