Adaptive Discontinuous Galerkin Finite Element Methods for the Compressible Euler Equations

R. Hartmann and P. Houston

Abstract:

In this paper a recently developed approach for the design of adaptive discontinuous Galerkin finite element approximations is applied to physically relevant problems arising in inviscid compressible fluid flows governed by the Euler equations of gas dynamics. In particular, we employ so--called weighted or Type I a posteriori error bounds to drive adaptive finite element algorithms for the estimation of the error measured in terms of general linear and nonlinear target functionals of the solution; typical examples considered here include the point evaluation of a component of the solution vector, and the drag and lift coefficients of a body immersed in an inviscid fluid. 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. Indeed, the superiority of the proposed approach over standard mesh refinement algorithms which employ ad hoc error indicators will be illustrated by a series of numerical experiments; here, we consider transonic flow through a nozzle, as well as subsonic, transonic and supersonic flows around different airfoil geometries.