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Adjoint-Based Error Control for Target Functionals in a Discontinous-Galerkin Framework

Investigator: Jochen Schütz -- Advisors: Georg May, Sebastian Nölle

Outline

The last decades have seen a tremendous progress in the development and analysis of numerical methods for the solution of (nonlinear) convection-dominated problems. Solutions to these kinds of problems are seldomly regular in all the domain, but they develop singularities.
Finite Volume Methods were developed to capture these singularities. Unfortunately, it was very hard to extend them to high-order accuracy. On the other hand, classical Finite Element Methods could easily achieve high order, but they were not at all able to capture singularities. Cockburn and Shu combined the advantages of the two methods in the Discontinuous Galerkin Method. Though of industrial interest, commercial tools still mainly use low-order methods. This is due to the fact that high-order methods still haven't reached the maturity one knows from Finite Volume Methods concerning relaxation techniques, preconditioning etc...
We work on the improvement and analysis of an error indicator used for h/p refinement in a Discontinuous Galerkin Context for a hyperbolic problem. Unlike e.g. elliptic problems, the size of the Residual from the galerkin method is not a good measure for the error as with the hyperbolicity comes error-convection into play.
To circumvent this problem, we want to use a method introduced by Eriksson and Johnson and further developed by Sueli and Houston involving an adjoint problem (similar to techniques known from optimization). Instead of using the adjoint problem for (usually very pessimistic) a priori estimation, Hartmann and others calculated the adjoint solution and then got an a posteriori error estimator. This is what we want to further investigate.
The nonlinearities present in relevant applications render the calculation of the adjoint solution difficult with a lots of switches to tune. Best-practice methods are hoped to be achieved in the aeronautcial applications we have in mind.

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