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research
Enhanced accuracy by post-processing for finite element methods for hyperbolic equations
Authors
Bernardo Cockburn
Mitchell Luskin
Chi-Wang Shu
Endre Suli
Publication date
1 January 2001
Publisher
Mathematics of Computation
Abstract
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
Δ
x
only. For example, when polynomials of degree
k
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
k
+
1/2
in the
L
2
L^2
L
2
norm, whereas the post-processed approximation is of order
2
k
+
1
2k+1
2
k
+
1
; if the exact solution is in
L
2
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
k
+
1/2
in
L
2
(
Ω
0
)
L^2(\Omega_0)
L
2
(
Ω
0
​
)
where
Ω
0
\Omega_0
Ω
0
​
is a subdomain over which the exact solution is smooth. Numerical results displaying the sharpness of the estimates are presented
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