High-Order/hp-Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows -
2009, BAMC 09, Nottingham
We present an overview of some recent developments concerning the
a posteriori error analysis
of h-- and hp--version finite element approximations to compressible
After highlighting some of the conceptual difficulties in error
control for problems of hyperbolic/nearly--hyperbolic type,
such as the lack of correlation between
the local error and the local finite element residual, we concentrate on
a specific discretisation scheme: the hp--version of the discontinuous Galerkin finite
element method. This method is capable of exploiting both local
polynomial--degree--variation (p--refinement) and local mesh subdivision
(h--refinement), thereby offering greater flexibility and efficiency
than numerical techniques which only incorporate h--refinement or
p--refinement in isolation.
By employing a duality argument we derive so--called weighted
or Type I a posteriori estimates which bound the error between
the true value of the prescribed functional, and the actual
computed value. In these error estimates, the
element residuals of the computed numerical solution are
multiplied by local weights involving the solution of a certain
dual or adjoint problem.
On the basis of the resulting a posteriori error bound,
we design and implement an adaptive finite element algorithm
to ensure reliable and efficient control of the error in the
computed functional with respect to a user--defined tolerance.
The performance of the resulting hp--refinement algorithm is
demonstrated through a series
of numerical experiments.
This research has been funded by the EU under the ADIGMA project.