Section: New Results

Control of approximation errors

Participants : Frédéric Alauzet [GAMMA team, INRIA-Rocquencourt] , Estelle Mbinky [GAMMA team, INRIA-Rocquencourt] , Olivier Allain [Lemma] , Anca Belme, Alexandre Carabias, Hubert Alcin, Alain Dervieux.

This is a joint research between three INRIA teams GAMMA (Rocquencourt), TROPICS, PUMAS and Lemma company. Roughly speaking, GAMMA brings mesh and approximation expertise, TROPICS contributes to adjoint methods, and CFD applications are developed in the context of PUMAS and Lemma.

The resolution of the optimum problem using the innovative approach of an AD-generated adjoint can be used in a slightly different context than the optimal shape design, namely the mesh adaptation. This will be possible if we can map the mesh adaptation problem into a differentiable optimal control problem. To this end, we have introduced a new methodology that consists in stating the mesh adaptation problem in a purely functional form: the mesh is reduced to a continuous property of the computational domain: the continuous metric. We minimize a continuous model of the error resulting from that metric. Thus the problem of searching an adapted mesh is transformed into the search of an optimal metric.

In 2011, a work on goal-oriented mesh adaptation for unsteady Euler flows has been extended, with further analysis, and a paper has been written and submitted to a journal. Its extension to the compressible Navier-Stokes model has been developed, [11] and a paper is being written. A further extension to Large Eddy Simulation is started. The method is being extended to a third-order approximation, the Vertex-CENO. This approximation was defined during a collaboration between university of Montpellier, IMM-Moscow and Tropics. A more accurate version has been studied by Alexandre Carabias and presented in Honom-Trento. a new theory involving error estimates and criteria has been developed by Gamma and Tropics. The extension of the multiscale adaptation method is considered by Estelle Mbinky at Rocquencourt. The extension of the goal-oriented method is considered by Alexandre Carabias at Sophia. Anisotropic mesh adaptation allows for better convergence to continuous solutions, and in particular more accurate a posteriori error estimates and correctors. The synergy between correctors and mesh adaptation is currently analysed and is the subject of a joint contribution (Gamma and Tropics) for the FP7 CARDINA proposal (nov. 2011).