Error estimation and adaptive mesh refinement for aerodynamic flows

Kurzfassung

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.