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HighOrder/hpAdaptive Discontinuous Galerkin Finite Element
Methods for Compressible Fluid Flows

2012, ESCO 2012, Pilsen 

Abstract:
We present an overview of some recent developments concerning the a posteriori error analysis of h
and hpversion finite element approximations to compressible fluid flows for a specific discretisation
scheme: the hpversion of the discontinuous Galerkin finite element method. This method is capable
of exploiting both local polynomialdegreevariation (prefinement) and local mesh subdivision (h
refinement), thereby offering greater flexibility and efficiency than numerical techniques which only
incorporate hrefinement or prefinement in isolation.
We shall be particularly concerned with the derivation of a posteriori bounds on the error in
certain output functionals of the solution of practical interest; relevant examples include the lift and
drag coefficients for a body immersed into a fluid, the local mean value of the field or its flux through
the outflow boundary of the computational domain, and the pointwise evaluation of a component of
the solution.
By employing a duality argument we derive socalled 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 userdefined tolerance. Our adaptive finite element algorithm decides automatically between isotropic and
anisotropic refinement either in hrefine or in prefine. The performance of the resulting hprefinement
algorithm is demonstrated through a series of numerical experiments.
This research has been funded by the EU under the ADIGMA project.

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