A multigrid method with local adaptive refinement, application to a ternary Cahn-Hilliard model.

Franck Boyer (1) , Céline Lapuerta (2) , Sébastian Minjeaud (2) , Bruno Piar (2)

1 LATP - Laboratoire d'Analyse, Topologie, Probabilités
2 IRSN/DPAM/SEMIC/LIMSI - Laboratoire d'études de l'Incendie et de développement de Méthodes pour la Simulation et les Incertitudes

The aim of this communication is to describe some numerical aspects linked to incompressible three-phase flow simulations, thanks to Cahn-Hilliard type model. The numerical capture of transfer phenomenon in the neighborhood of the interface require a mesh thickness which become crippling in the case where it is applied to the whole computational domain. This suggests the use of a local refinement method which allows to dynamically focus on problematic areas. The notion of refinement pattern, introduced for Lagrange finite elements, allows to build a conceptual hierarchy of nested conformal approximation spaces which is then used to implement the so-called CHARMS local refinement methods. Properties of these methods are proved ensuring in particular the conformity of approximation spaces at every time of simulations. Furthermore, the multilevel structure obtained by this method, is used to construct multigrid preconditioners. Finally, after a validation on a model problem, the performance of the whole method is illustrated on an example of a liquid lens spreading between two stratified fluids.

Mots clés: multiphase flows; Multilevel finite elements; multigrid solver; Cahn-Hilliard equation

Domaines: Equations aux dérivées partielles [math.AP]; Analyse numérique [math.NA]

Franck Boyer, Céline Lapuerta, Sébastian Minjeaud, Bruno Piar. A multigrid method with local adaptive refinement, application to a ternary Cahn-Hilliard model.. 2008. ⟨hal-00307186v1⟩