Numerical methods for the resolution of surface PDE. Application to embryogenegis

Keywords :

• mixed finite elements
• curvature computation
• embryogenegis

We develop a novel finite element approach for linear elasticity or Stokes-type PDEs set on a closed surface of $mathbb{R}^3$. The surface we consider is described as the zero of a sufficiently smooth level-set function. The problem can be written as the minimisation of an energy function over a constrained velocity field. Constraints are of two different types: (i) the velocity field is tangential to the surface, (ii) the surface is inextensible. This second constraint is equivalent to surface incompressibility of the velocity field. We address this problem in two different ways: a penalty method and a mixed method involving two Lagrange multipliers. This latter method allows us to solve the limiting case of incompressible surface flow, for which we present a novel theoretical and numerical analysis. Error estimates for the discrete solution are given and numerical tests shows the optimality of the estimates. For this purpose, several approaches for the numerical computation of the normal and curvature of the surface are proposed. The implementation relies on the Rheolef open-source finite element library. We present numerical simulations for a biological application: the morphogenesis of Drosophila embryos, during which tangential flows of a cell monolayer take place with a low surface-area variation. This phenomenon is known as germ-band extension.

Directors:

• Mr Pierre Saramito (Chargé de recherche - CNRS )
• Mr Jocelyn Etienne (Chargé de recherche - CNRS )

Raporteurs:

• Mr Sébastien Martin (Professeur - Université Paris Descartes )
• Mr Franck Pigeonneau (Ingénieur de recherche - Saint-Gobain recherches )

Examinators:

• Mr François Graner (Directeur de recherche - CNRS )
• Mr Stéphane Labbé (Professeur - Université Grenoble Alpes )