Anisotropic mesh refinement for discontinuous Galerkin methods in two-dimensional aerodynamic flow simulations

Abstract

We derive and implement two types of anisotropic indicators which can be used within an anisotropic refinement algorithm for 2nd but also for higher order discontinuous Galerkin discretizations. Whereas the first type of indicator employs the possible inter-element discontinuities of the discrete functions the second type of indicator estimates the approximation error in terms of 2nd but possibly also higher order derivatives. We implement a simple extension of these indicators to systems of equations which performs similar to the so-called metric intersection used to combine the metric information of several solution components and is applicable to higher order discretizations as well. The anisotropic indicators are incorporated into an adaptive refinement algorithm which uses state-of-the-art residual-based or adjoint-based indicators for goal-oriented refinement to select the elements to be refined whereas the anisotropic indicators determine which anisotropic case the selected elements shall be refined with. We demonstrate the performance of the anisotropic refinement algorithm for sub-, trans- and supersonic, inviscid and viscous compressible flows around a NACA0012 airfoil.