TY - JOUR
T1 - Anisotropic $hp$-Adaptive Discontinuous Galerkin Finite Element Methods for Compressible Fluid Flows
AU - S. Giani & P. Houston
JO - International Journal of Numerical Analysis and Modeling
VL - 4
SP - 928
EP - 949
PY - 2012
DA - 2012/09
SN - 9
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/ijnam/666.html
KW - Discontinuous Galerkin methods, a posteriori error estimation, adaptivity, anisotropic $hp$-refinement, compressible flows.
AB - <p style="text-align: justify;">In this article we consider the construction of general isotropic and anisotropic adaptive mesh refinement strategies, as well as $hp$–mesh refinement techniques, for the numerical
approximation of the compressible Euler and Navier–Stokes equations. To discretize the latter
system of conservation laws, we exploit the (adjoint consistent) symmetric version of the interior
penalty discontinuous Galerkin finite element method. The a posteriori error indicators are derived
based on employing the dual-weighted-residual approach in order to control 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 adjoint
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 efficient error estimation. Numerical experiments demonstrating
the performance of the proposed adaptive algorithms will be presented.</p>