Fully anisotropic goal-oriented mesh adaptation for 3D steady Euler equations
Adrien Loseille
Fréderic Alauzet
(INRIA Rocquencourt)
Alain Dervieux
(INRIA, BP93, 06902 Sophia-Antipolis, France)
Article in Journal of Computational Physics, 2010 (32 pages)
Abstract:
This paper studies the coupling between anisotropic mesh adaptation and
goal-oriented error estimate. The former is very well suited to the control of
the interpolation error. It is generally interpreted as a local geometric error
estimate. On the contrary, the latter is preferred when studying approximation
errors for PDEs. It generally involves non local error contributions.
Consequently, a full and strong coupling between both is hard to achieve
due to this apparent incompatibility. This paper shows how to achieve this
coupling in three steps.
First, a new a priori error estimate is proved in a formal framework
adapted to goal-oriented mesh adaptation for output functionals. This estimate
is based on a careful analysis of the contributions of the implicit error
and of the interpolation error. Second, it is applied to the set of steady
compressible Euler equations which are solved by a stabilized Galerkin finite
element discretization. A goal-oriented error estimation is derived. It involves
the interpolation error of the Euler fluxes weighted by the gradient of
the adjoint state associated with the observed functional. Third, rewritten
in the continuous mesh framework, the previous estimate is minimized on
the set of continuous meshes thanks to a calculus of variations. The optimal
continuous mesh is derived analytically. Thus, it can be used as a metric
tensor field to drive the mesh adaptation. From a numerical point of view,
this method is completely automatic, intrinsically anisotropic, and does not
depend on any a priori choice of variables to perform the adaptation.
3D examples of steady flows around supersonic and transsonic jets are
presented to validate the current approach and to demonstrate its efficiency.
Keywords:
Anisotropic unstructured mesh adaptation, goal-oriented mesh
adaptation, metric-based mesh adaptation, steady compressible Euler
equations, a priori error estimate, adjoint
Full text (pdf)
@article{LDA10a,
author = {Loseille, A. and Dervieux, A. and Alauzet, F.},
title = {Fully anisotropic goal-oriented mesh adaptation for {3D} steady Euler equations},
journal = {Journal of Computational Physics},
volume = {229},
pages = {2866-2897},
year = 2010
}