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Superconvergence of a discontinuous Galerkin method for fractional diffusion and wave equations
We consider an initial-boundary value problem for
, that is, for a fractional
diffusion () or wave () equation. A numerical solution
is found by applying a piecewise-linear, discontinuous Galerkin method in time
combined with a piecewise-linear, conforming finite element method in space.
The time mesh is graded appropriately near , but the spatial mesh is
quasiuniform. Previously, we proved that the error, measured in the spatial
-norm, is of order , uniformly in , where
is the maximum time step, is the maximum diameter of the spatial finite
elements, and . Here,
we generalize a known result for the classical heat equation (i.e., the case
) by showing that at each time level the solution is
superconvergent with respect to : the error is of order
. Moreover, a simple postprocessing step
employing Lagrange interpolation yields a superconvergent approximation for any
. Numerical experiments indicate that our theoretical error bound is
pessimistic if . Ignoring logarithmic factors, we observe that the
error in the DG solution at , and after postprocessing at all , is of
order .Comment: 24 pages, 2 figure
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