1,313 research outputs found
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
The discontinuous Galerkin method for fractional degenerate convection-diffusion equations
We propose and study discontinuous Galerkin methods for strongly degenerate
convection-diffusion equations perturbed by a fractional diffusion (L\'evy)
operator. We prove various stability estimates along with convergence results
toward properly defined (entropy) solutions of linear and nonlinear equations.
Finally, the qualitative behavior of solutions of such equations are
illustrated through numerical experiments
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