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## Section: New Results

### Control of approximation errors

Participants : Gautier Brèthes, Eléonore Gauci, Alain Dervieux, Adrien Loseille [Gamma team, Inria-Rocquencourt] , Frederic Alauzet [Gamma team, Inria-Rocquencourt] , Stephen Wornom, Olivier Allain [Lemma] , Anca Belme [University Paris VI] .

A study of an interesting combination of Full Multigrid (FMG) and Anisotropic mesh Adaptation (AA) started last year, with the beginning of the thesis of Gautier Brèthes. FMG is one of the (very) few algorithm giving $N$ results by consuming $kN$ floats. Anisotropic adaptation produces approximation errors less than $\epsilon$ with $N={\epsilon }^{-\frac{1}{dim}}$ nodes, this for smooth and non-smooth solution fields. Anisotropic adaptative FMG may produce approximation errors less than $\epsilon$ by consuming $k{\epsilon }^{-\frac{1}{dim}}$ floats. Moreover, theory and experiments show that FMG works better when combined with AA. A first AA-FMG platform has been developed. It combines several mesh-adaptation modules developed by Gamma and Distene. It is used for testing new adaptation criteria.

Third-order mesh adaptation was the main topic of last year in error control. The scheme is the ENO finite-volume formulation with quadratic reconstruction. An article describing our results for 2D applications is being written. A 3D version is developed in the Aironum CFD platform. A cooperation with Lemma is also running, with Eléonore Gauci, to apply the scheme to fluid-gas interfaces. Further studies of mesh adaptation for viscous flows are ongoing and an article in collaboration with Gamma3 and University Paris VI (Anca Belme) is being written.

An important novelty in mesh adaptation is the norm-oriented AA method. The method relies on the definition of ad hoc correctors. It has been developed in the academic FMG platform for elliptic problems. Another version is developed by Gamma, in collaboration with Ecuador, for the compressible flow models. The purpose is to devise a composite algorithm in which an approximation error norm can be specified by the user. The introduction of the norm-oriented idea considerably amplifies the impact of adjoint-based AA. The applied mathematician and the engineer now have methods when faced to mesh adaptation for the simulation of a complex PDE system, since they can specify which error norm level they wish, and for which norm. Eléonore Gauci starts a thesis, co-advised by Alain Dervieux and Frédéric Alauzet, on the norm-oriented criteria for CFD and coupled CSM-CFD systems. She also works on a new version of the mesh adaptive CFD demonstrator of Gamma3. This new version improves the resolution of curved features. A cooperation is also starting between Gautier Brèthes and Thierry Coupez (Ecole Centrale de Nantes) on discrete metrics.

These studies are supported by an European FP7 project UMRIDA which deals with the application of AA to approximation error modelling and control, and by ANR project MAIDESC coordinated by Ecuador, which deals with meshes for interfaces, third-order accuracy, meshes for boundary layers, and curved meshes.