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Error estimation and adaptive mesh refinement for aerodynamic flows

Hartmann, Ralf and Houston, Paul (2010) Error estimation and adaptive mesh refinement for aerodynamic flows. In: ,Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium. ISBN 13 978-2-930389-98-2.



This lecture course covers the theory of so-called duality-based a posteriori error estimation of DG finite element methods. In particular, we formulate consistent and adjoint consistent DG methods for the numerical approximation of both the compressible Euler and Navier-Stokes equations; in the latter case, the viscous terms are discretized based on employing an interior penalty method. By exploiting a duality argument, adjoint-based a posteriori error indicators will be established. Moreover, application of these computable bounds within automatic adaptive finite element algorithms will be developed. Here, a variety of isotropic and anisotropic adaptive strategies, as well as hp-mesh refinement will be investigated. The outline of these notes is as follows. In Section~2 we give an introduction to the adjoint-based a posteriori error estimation and mesh refinement for linear problems, and their subsequent exploitation within an automatic adaptive finite element algorithms. Then, in Section~3 we introduce both the compressible Euler and Navier-Stokes equations and formulate DG numerical methods for their discretization. In particular, here we will be concerned with the derivation of so-called adjoint consistent methods, which ensure the optimal approximation of target functionals of the underlying solution. Section~4 is devoted to the derivation of adjoint-based a posteriori error bounds for the computed error in a given target functional of interest. Moreover, extensions to the case when there are multiple quantities of interest will be considered. The practical performance of these a posteriori error estimates within adaptive finite element algorithms will be studied through a series of numerical experiments. In Section~5 we consider the generalization of the above ideas to the case when anisotropic mesh refinement is permitted. In this setting, we derive both a priori and a posteriori error bounds for the DG approximation of linear functionals of the underlying analytical solution. The a priori analysis is fully explicit in terms of the anisotropy of the underlying computational mesh. Further, we introduce an anisotropic refinement algorithm, based on choosing the most competitive subdivision of a given element from a series of trial (Cartesian) refinements. The extension of these ideas to general anisotropic hp-version DG finite element methods is undertaken in Section~6. Finally, Section~7 is devoted to the application of goal-oriented adaptive finite element algorithms to complex aerodynamic flows, including three dimensional laminar flows as well as two and three dimensional turbulent flows.

Item URL in elib:https://elib.dlr.de/61976/
Document Type:Contribution to a Collection
Title:Error estimation and adaptive mesh refinement for aerodynamic flows
AuthorsInstitution or Email of AuthorsAuthors ORCID iD
Houston, PaulUniversity of NottinghamUNSPECIFIED
Open Access:Yes
Gold Open Access:No
In ISI Web of Science:No
Deconinck, HermanVKI, Belgium
Publisher:,Von Karman Institute for Fluid Dynamics, Rhode Saint Genese, Belgium
ISBN:13 978-2-930389-98-2
Keywords:Discontinuous Galerkin methods, error estimation, adjoint-based mesh refinement, anisotropic mesh refinement, hp refinement
HGF - Research field:Aeronautics, Space and Transport
HGF - Program:Aeronautics
HGF - Program Themes:Aircraft Research (old)
DLR - Research area:Aeronautics
DLR - Program:L AR - Aircraft Research
DLR - Research theme (Project):L - Concepts & Integration (old)
Location: Braunschweig
Institutes and Institutions:Institute of Aerodynamics and Flow Technology > CASE
Deposited By: Hartmann, Dr.rer.nat. Ralf
Deposited On:05 Jan 2010 10:17
Last Modified:31 Jul 2019 19:26

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