Abstract:

We derive energy-norm a posteriori error bounds for an Euler
timestepping method combined with various spatial discontinuous
Galerkin schemes for linear parabolic problems. For accessibility, we
address first the spatially semidiscrete case, and then move to the
fully discrete scheme by introducing the implicit Euler timestepping.
All results are presented in an abstract setting and then illustrated
with particular applications. This enables the error bounds to hold
for a variety of discontinuous Galerkin methods (or other
non-conforming methods), provided that energy-norm a posteriori error
bounds for the corresponding elliptic problem are available. To
illustrate the method, we apply it to the interior penalty
discontinuous Galerkin method, which prompts the derivation of new a
posteriori error bounds. For the analysis of the time-dependent
problems we use the elliptic reconstruction technique and we deal with
the nonconforming part of the error by deriving appropriate computable
a posteriori bounds for it. This is joint work with Omar Lakkis
(Sussex, UK).