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High-Order/hp-Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows – 2009, BAMC 09, Nottingham 

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We present an overview of some recent developments concerning the a posteriori error analysis of h– and hp–version finite element approximations to compressible fluid flows. 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.