This thesis concerns the analytical and practical aspects of applying the Closest Point Method to solve elliptic partial differential equations (PDEs) on smooth surfaces and domains with smooth boundaries. A new numerical scheme is proposed to solve surface elliptic PDEs and a novel geometric multigrid solver is constructed to solve the resulting linear system. The method is also applied to coupled bulk-surface problems. A new embedding equation in a narrow band surrounding the surface is formulated so that it agrees with the original surface PDE on the surface and has a unique solution which is constant along the normals to the surface. The embedding equation is then discretized using standard finite difference scheme and barycentric Lagrange interpolation. The resulting scheme has 2nd-order accuracy in practice and is provably 2nd-order convergent for curves without boundary embedded in ℝ². To apply the method to solve elliptic equations on surfaces and domains with boundaries, the "ghost" point approach is adopted to handle Dirichlet, Neumann and Robin boundary conditions. A systematic method is proposed to represent values of ghost points by values of interior points according to boundary conditions. A novel geometric multigrid method based on the closest point representation of the surface is constructed to solve the resulting large sparse linear systems. Multigrid solvers are designed for surfaces with or without boundaries and domains with smooth boundaries. Numerical results indicate that the convergence rate of the multigrid solver stays roughly the same as we refine the mesh, as is desired of a multigrid algorithm. Finally the above methods are combined to solve coupled bulk-surface PDEs with some applications to biology.

Nous développons une nouvelle approche éléments finis pour des équations aux dérivées partielles elliptiques de type élasticité linéaire ou Stokes sur une surface fermée de R3. La surface considérée est décrite par le zéro d'une fonction de niveau assez régulière. Le problème se ramène à la minimisation d'une fonctionnelle énergie pour le champ de vitesse sous contraintes. Les contraintes sont de deux types : (i) la vitesse est tangentielle à la surface, (ii) la surface est inextensible. Cette deuxième contrainte équivaut à l'incompressibilité surfacique du champ de vitesse. Nous abordons ce problème de deux façons : la pénalisation et l'introduction de deux multiplicateurs de Lagrange. Cette dernière méthode a l'avantage de traiter le cas de la limite incompressible d'un écoulement en surface dont nous présentons pour la première fois l'analyse théorique et numérique. Nous montrons des estimations d'erreurs sur la solution discrète et les tests numériques confirment l'optimalité des ces estimations. Pour cela, nous proposons plusieurs approches pour le calcul numérique de la normale et la courbure de la surface. L'implémentation utilise la librairie libre d'éléments finis Rheolef. Nous présentons aussi des résultats de simulations numériques pour une application en biologie : la morphogenèse de l'embryon de la drosophile, durant laquelle des déformations tangentielles d'une monocouche de cellules avec une faible variation d'aire. Ce phénomène est connu sous le nom de l'extension de la bande germinale. / 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 areof 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 thisproblem 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 andnumerical 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, duringwhich tangential flows of a cell monolayer take place with a low surface-area variation. This phenomenon is known as germ-band extension.