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.