Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations

Abstract

In this paper a recently developed approach for the design of adaptive discontinuous Galerkin finite element methods is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the Euler equations of gas dynamics. In particular, we employ (weighted) type I a posteriori bounds to drive adaptive finite element algorithms for the estimation of the error measured in terms of general target functionals of the solution; these error estimates involve the product of the finite element residuals with local weighting terms, involving the solution of a certain dual problem that must be numerically approximated. This general approach leads to the design of economical finite element meshes specifically tailored to the computation of the target functional of interest, as well as providing reliable and efficient error estimation. The superiority of the proposed approach over mesh refinement algorithms which employ standard unweighted (type II) error indicators, which do not require the solution of an auxiliary problem, are illustrated by a series of numerical experiments; here, we consider transonic flow through a nozzle, as well as subsonic and supersonic flows around different airfoil geometries.