Problems involving the solution of partial differential equations over surfaces appear in many engineering and scientific applications. Some of those applications include crystal growth, fluid mechanics and computer graphics. Many times analytic solutions to such problems are not available. Numerical algorithms, such as Finite Element Methods, are used in practice to find approximate solutions in those cases. In this work we present L2 and pointwise a posteriori error estimates for Adaptive Surface Finite Elements solving the Laplace-Beltrami equation −△Γ u = f . The two sources of errors for Surface Finite Elements are a Galerkin error, and a geometric error that comes from replacing the original surface by a computational mesh. A posteriori error estimates on flat domains only have a Galerkin component. We use residual type error estimators to measure the Galerkin error. The geometric component of our error estimate becomes zero if we consider flat domains, but otherwise has the same order as the residual one. This is different from the available energy norm based error estimates on surfaces, where the importance of the geometric components diminishes asymptotically as the mesh is refined. We use our results to implement an Adaptive Surface Finite Element Method. An important tool for proving a posteriori error bounds for non smooth functions is the Scott-Zhang interpolant. A refined version of a standard Scott-Zhang interpolation bound is also proved during our analysis. This local version only requires the interpolated function to be in a Sobolev space defined over an element T instead of an element patch containing T. In the last section we extend our elliptic results to get estimates for the surface heat equation ut − △Γ u = f using the elliptic reconstruction technique.

In this paper, a mixed boundary value problem for the Laplace-Beltrami operator is considered for spherical domains in $R^3$, i.e. for domains on the unit sphere. These domains are parametrized by spherical coordinates (\varphi, \theta), such that functions on the unit sphere are considered as functions in these coordinates. Careful investigation leads to the introduction of a proper finite element space corresponding to an isotropic triangulation of the underlying domain on the unit sphere. Error estimates are proven for a Clément-type interpolation operator, where appropriate, weighted norms are used. The estimates are applied to the deduction of a reliable and efficient residual error estimator for the Laplace-Beltrami operator.

The solutions to certain elliptic boundary value problems have singularities with a typical structure near polyhedral corners. This structure can be exploited to devise an eigenvalue problem whose solution can be used to quantify the singularities of the given boundary value problem. It is necessary to parametrize a ball centered at the corner. There are different possibilities for a suitable parametrization; from the numerical point of view, spherical coordinates are not necessarily the best choice. This is why we do not specify a parametrization in this paper but present all results in a rather general form. We derive the eigenvalue problems that are associated with the Laplace and the linear elasticity problems and show interesting spectral properties. Finally, we discuss the necessity of widely accepted symmetry properties of the elasticity tensor. We show in an example that some of these properties are not only dispensable, but even invalid, although claimed in many standard books on linear elasticity.

The Laplace-Beltrami operator corresponds to the Laplace operator on curved surfaces. In this paper, we consider an eigenvalue problem for the Laplace-Beltrami operator on subdomains of the unit sphere in $\R^3$. We develop a residual a posteriori error estimator for the eigenpairs and derive a reliable estimate for the eigenvalues. A global parametrization of the spherical domains and a carefully chosen finite element discretization allows us to use an approach similar to the one for the two-dimensional case. In order to assure results in the quality of those for plane domains, weighted norms and an adapted Clément-type interpolation operator have to be introduced.

Dans tout ce qui suit, nous considérons une variété riemannienne compacte de dimension au moins égale à 2. A tout domaine (suffisamment régulier) $\Omega$, on peut associer la première valeur propre $\lambda_\Omega$ de l'opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous dirons qu'un domaine $\Omega$ est extrémal (sous entendu, pour la première valeur propre de l'opérateur de Laplace-Beltrami) si $\Omega$ est un point critique de la fonctionnelle $\Omega \rightarrow \lambda_\Omega$ sous une contrainte de volume $Vol (\Omega) = c_0$. Autrement dit, $\Omega$ est extrémal si, pour toute famille régulière $\{\Omega_t\}_{t \in (-t_0,t_0)}$ de domaines de volume constant, telle que $\Omega_0 = \Omega$, la dérivée de la fonction $t \rightarrow \lambda_{\Omega_t}$ en $0$ est nulle. Rappelons que les domaines extrémaux sont caractérisés par le fait que la fonction propre, associée à la première valeur propre sur le domaine avec condition de Dirichlet au bord, a une donnée de Neumann constante au bord. Ce résultat a été démontré par A. El Soufi et S. Ilias en 2007. Les domaines extrémaux sont donc des domaines sur lesquels peut être résolu un problème elliptique surdéterminé. L'objectif principal de cette thèse est la construction de domaines extrémaux pour la première valeur propre de l'opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous donnons des résultats d'existence de domaines extrémaux dans le cas de petits volumes ou bien dans le cas de volumes proches du volume de la variété. Nos résultats permettent ainsi de donner de nouveaux exemples non triviaux de domaines extrémaux. Le premier résultat que nous avons obtenu affirme que si une variété admet un point critique non dégénéré de la courbure scalaire, alors pour tout volume petit il existe un domaine extrémal qui peut être construit en perturbant une boule géodésique centrée en ce point critique non dégénéré de la courbure scalaire. La méthode que nous utilisons pour construire ces domaines extrémaux revient à étudier l'opérateur (non linéaire) qui à un domaine associe la donnée de Neumann de la première fonction propre de l'opérateur de Laplace-Beltrami sur le domaine. Il s'agit d'un opérateur (hautement non linéaire), nonlocal, elliptique d'ordre 1. Dans $\mathbb R^n \times \mathbb{R}/\, \mathbb{Z}$, le domaine cylindrique $B_r \times \mathbb{R}/\, \mathbb{Z}$, où $B_r$ est la boule de rayon $r >0$ dans $\mathbb{R}^{n}$, est un domaine extrémal. En étudiant le linéarisé de l'opérateur elliptique du premier ordre défini par le problème précédent et en utilisant un résultat de bifurcation, nous avons démontré l'existence de domaines extrémaux nontriviaux dans $\mathbb R^{n}\times \mathbb{R}/\, \mathbb{Z}$. Ces nouveaux domaines extrémaux sont proches de domaines cylindriques $B_r \times \mathbb{R}/ \mathbb{Z}$. S'ils sont invariants par rotation autour de l'axe vertical, ces domaines ne sont plus invariants par translations verticales. Ce deuxième résultat donne un contre-exemple à une conjecture de Berestycki, Caffarelli et Nirenberg énoncée en 1997. Pour de grands volumes la construction de domaines extrémaux est techniquement plus difficile et fait apparaître des phénomènes nouveaux. Dans ce cadre, nous avons dû distinguer deux cas selon que la première fonction propre $\phi_0$ de l'opérateur de Laplace-Beltrami sur la variété est constante ou non. Les résultats que nous avons obtenus sont les suivants : $\phi_0$ a des points critiques non dégénérés (donc en particulier n'est pas constante), alors pour tout volume assez proche du volume de la variété, il existe un domaine extrémal obtenu en perturbant le complément d'une boule géodésique centrée en un des points critiques non dégénérés de $\phi_0$. Si $\phi_0$ est constante et la variété admet des points critiques non dégénérés de la courbure scalaire, alors pour tout volume assez proche du volume de la variété il existe un domaine extrémal obtenu en perturbant le complément d'une boule géodésique centrée en un des points critiques non dégénérés de la courbure scalaire.

A mecânica celular jaz nas propriedades materiais da membrana plasmática, fundamentalmente uma bicamada fosfolipídica com espessura de dimensões moleculares. Além de forças elásticas, tal material bidimensional também experimenta tensões viscosas devido ao seu comportamento fluido (presumivelmente newtoniano) na direção tangencial. A despeito da notável concordância entre teoria e experimentos biofísicos sobre a geometria de membranas celulares, ainda não se faz presente um método computacional para simulação de sua (real) dinâmica viscosa governada pela lei de Boussinesq-Scriven. Assim sendo, introduzimos uma formulação variacional mista de três campos para escoamentos viscosos de superfícies fechadas curvas. Nela, o fluido circundante é levado em conta considerando-se uma restrição de volume interior, ao passo que uma restrição de área corresponde à inextensibilidade. As incógnitas são a velocidade, o vetor curvatura e a pressão superficial, todas estas interpoladas com elementos finitos lineares contínuos via estabilização baseada na projeção do gradiente de pressão. O método é semi-implícito e requer a solução de apenas um único sistema linear por passo de tempo. Outro ingrediente numérico proposto é uma força que mimetiza a ação de uma pinça óptica, permitindo interação virtual com a membrana, onde a qualidade e o refinamento de malha são mantidos por remalhagem adaptativa automática. Extensivos experimentos numéricos de dinâmica de relaxação são apresentados e comparados com soluções quasi-analíticas. É observada estabilidade temporal condicional com uma restrição de passo de tempo que escala como o quadrado do tamanho de malha. Reportamos a convergência e os limites de estabilidade de nosso método e sua habilidade em predizer corretamente o equilíbrio dinâmico de compridas e finas elongações cilíndricas (tethers) que surgem a partir de pinçamentos membranais. A dependência de forma membranal com respeito a uma velocidade imposta de pinçamento também é discutida, sendo que há um valor limiar de velocidade abaixo do qual um tether não se forma de início. Testes adicionais ilustram a robustez do método e a relevância dos efeitos viscosos membranais quando sob a ação de forças externas. Sem dúvida, ainda há um longo caminho a ser trilhado para o entendimento completo da mecânica celular (há de serem consideradas outras estruturas tais como citoesqueleto, canais iônicos, proteínas transmembranares, etc). O primeiro passo, porém, foi dado: a construção de um esquema numérico variacional capaz de simular a intrigante dinâmica das membranas celulares. / Cell mechanics lies on the material properties of the plasmatic membrane, fundamentally a two-molecule-thick phospholipid bilayer. Other than bending elastic forces, such a two-dimensional interfacial material also experiences viscous stresses due to its (presumably Newtonian) surface fluid tangential behaviour. Despite the remarkable agreement on membrane shapes between theory and biophysical experiments, there is no computational method for simulating its (actual) viscous dynamics governed by the Boussinesq- Scriven law. Accordingly, we introduce a mixed three-field variational formulation for viscous flows of closed curved surfaces. In it, the bulk fluid is taken into account by means of an enclosed-volume constraint, whereas an area constraint stands for the membranes inextensible character. The unknowns are the velocity, vector curvature and surface pressure fields, all of which are interpolated with linear continuous finite elements by means of a pressure-gradient-projection scheme. The method is semi-implicit and it requires the solution of a single linear system per time step. Another proposed ingredient is a numerical force that emulates the action of an optical tweezer, allowing for virtual interaction with the membrane, where mesh quality and refinement are maintained by adaptively remeshing. Extensive relaxation experiments are reported and compared with quasi-analytical solutions. Conditional time stability is observed, with a time step restriction that scales as the square of the mesh size. We discuss both convergence and the stability limits of our method, its ability to correctly predict the dynamical equilibrium of the tether due to tweezing. The dependence of the membrane shape on imposed tweezing velocities is also addressed. Basically, there is a threshold velocity value below which the tethers shape does not arise at first. Further tests illustrate the robustness of the method and show the significance of viscous effects on membranes deformation under external forces. Undoubtedly, there is still a long way to track toward the understanding of celullar mechanics (one still has to account other structures such as cytoskeleton, ion channels, transmembrane proteins, etc). The first step has given, though: the design of a numerically robust variational scheme capable of simulating the engrossing dynamics of fluid cell membranes.

Dans tout ce qui suit, nous considérons une variété riemannienne compacte de dimension au moins égale à 2. A tout domaine (suffisamment régulier) , on peut associer la première valeur propre ?Ù de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous dirons qu’un domaine est extrémal (sous entendu, pour la première valeur propre de l’opérateur de Laplace-Beltrami) si est un point critique de la fonctionnelle Ù? ?O sous une contrainte de volume V ol(Ù) = c0. Autrement dit, est extrémal si, pour toute famille régulière {Ot}te (-t0,t0) de domaines de volume constant, telle que Ù 0 = Ù, la dérivée de la fonction t ? ?Ot en 0 est nulle. Rappelons que les domaines extrémaux sont caractérisés par le fait que la fonction propre, associée à la première valeur propre sur le domaine avec condition de Dirichlet au bord, a une donnée de Neumann constante au bord. Ce résultat a été démontré par A. El Soufi et S. Ilias en 2007. Les domaines extrémaux sont donc des domaines sur lesquels peut être résolu un problème elliptique surdéterminé. L’objectif principal de cette thèse est la construction de domaines extrémaux pour la première valeur propre de l’opérateur de Laplace-Beltrami avec condition de Dirichlet au bord. Nous donnons des résultats d’existence de domaines extrémaux dans le cas de petits volumes ou bien dans le cas de volumes proches du volume de la variété. Nos résultats permettent ainsi de donner de nouveaux exemples non triviaux de domaines extrémaux. Le premier résultat que nous avons obtenu affirme que si une variété admet un point critique non dégénéré de la courbure scalaire, alors pour tout volume petit il existe un domaine extrémal qui peut être construit en perturbant une boule géodésique centrée en ce point critique non dégénéré de la courbure scalaire. La méthode que nous utilisons pour construire ces domaines extrémaux revient à étudier l’opérateur (non linéaire) qui à un domaine associe la donnée de Neumann de la première fonction propre de l’opérateur de Laplace-Beltrami sur le domaine. Il s’agit d’un opérateur (hautement non linéaire), nonlocal, elliptique d’ordre 1. Dans Rn × R/Z, le domaine cylindrique Br × R/Z, o`u Br est la boule de rayon r > 0 dans Rn, est un domaine extrémal. En étudiant le linéarisé de l’opérateur elliptique du premier ordre défini par le problème précédent et en utilisant un résultat de bifurcation, nous avons démontré l’existence de domaines extrémaux nontriviaux dans Rn × R/Z. Ces nouveaux domaines extrémaux sont proches de domaines cylindriques Br × R/Z. S’ils sont invariants par rotation autour de l’axe vertical, ces domaines ne sont plus invariants par translations verticales. Ce deuxi`eme r´esultat donne un contre-exemple à une conjecture de Berestycki, Caffarelli et Nirenberg énoncée en 1997. Pour de grands volumes la construction de domaines extrémaux est techniquement plus difficile et fait apparaître des phénomènes nouveaux. Dans ce cadre, nous avons dû distinguer deux cas selon que la première fonction propre Ø0 de l’opérateur de Laplace-Beltrami sur la variété est constante ou non. Les résultats que nous avons obtenus sont les suivants : 1. Si Ø0 a des points critiques non dégénérés (donc en particulier n’est pas constante), alors pour tout volume assez proche du volume de la variété, il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de Ø0. 2. Si Ø0 est constante et la variété admet des points critiques non dégénérés de la courbure scalaire, alors pour tout volume assez proche du volume de la variété il existe un domaine extrémal obtenu en perturbant le complément d’une boule géodésique centrée en un des points critiques non dégénérés de la courbure scalaire / In what follows, we will consider a compact Riemannian manifold whose dimension is at least 2. Let Ù be a (smooth enough) domain and ?O the first eigenvalue of the Laplace-Beltrami operator on Ù with 0 Dirichlet boundary condition. We say that Ù is extremal (for the first eigenvalue of the Laplace-Beltrami operator) if is a critical point for the functional Ù? ?O with respect to variations of the domain which preserve its volume. In other words, Ù is extremal if, for all smooth family of domains { Ù t}te(-t0,t0) whose volume is equal to a constant c0, and Ù 0 = Ù, the derivative of the function t ? ?Ot computed at t = 0 is equal to 0. We recall that an extremal domain is characterized by the fact that the eigenfunction associated to the first eigenvalue of the Laplace-Beltrami operator over the domain with 0 Dirichlet boundary condition, has constant Neumann data at the boundary. This result has been proved by A. El Soufi and S. Ilias in 2007. Extremal domains are then domains over which can be solved an elliptic overdeterminated problem. The main aim of this thesis is the construction of extremal domains for the first eigenvalue of the Laplace-Beltrami operator with 0 Dirichlet boundary condition. We give some existence results of extremal domains in the cases of small volume or volume closed to the volume of the manifold. Our results allow also to construct some new nontrivial exemples of extremal domains. The first result we obtained states that if the manifold has a nondegenerate critical point of the scalar curvature, then, given a fixed volume small enough, there exists an extremal domain that can be constructed by perturbation of a geodesic ball centered in that nondegenerated critical point of the scalar curvature. The methode used is based on the study of the operator that to a given domain associes the Neumann data of the first eigenfunction of the Laplace-Beltrami operator over the domain. It is a highly nonlinear, non local, elliptic first order operator. In Rn × R/Z, the circular-cylinder-type domain Br × R/Z, where Br is the ball of radius r > 0 in Rn, is an extremal domain. By studying the linearized of the elliptic first order operator defined in the previous problem, and using some bifurcation results, we prove the existence of nontrivial extremal domains in Rn × R/Z. Such extremal domains are closed to the circular-cylinder-type domains Br × R/Z. If they are invariant by rotation with respect to the vertical axe, they are not invariant by vertical translations. This second result gives a counterexemple to a conjecture of Berestycki, Caffarelli and Nirenberg stated in 1997. For big volumes the construction of extremal domains is technically more difficult and shows some new phenomena. In this context, we had to distinguish two cases, according to the fact that the first eigenfunction Ø0 of the Laplace-Beltrami operator over the manifold is constant or not. The results obtained are the following : 1. If Ø0 has a nondegenerated critical point (in particular it is not constant), then, given a fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerated critical point of Ø0. 2. If Ø0 is constant and the manifold has some nondegenerate critical points of the scalar curvature, then, for a given fixed volume closed to the volume of the manifold, there exists an extremal domain obtained by perturbation of the complement of a geodesic ball centered in a nondegenerate critical point of the scalar curvature

A mecânica celular jaz nas propriedades materiais da membrana plasmática, fundamentalmente uma bicamada fosfolipídica com espessura de dimensões moleculares. Além de forças elásticas, tal material bidimensional também experimenta tensões viscosas devido ao seu comportamento fluido (presumivelmente newtoniano) na direção tangencial. A despeito da notável concordância entre teoria e experimentos biofísicos sobre a geometria de membranas celulares, ainda não se faz presente um método computacional para simulação de sua (real) dinâmica viscosa governada pela lei de Boussinesq-Scriven. Assim sendo, introduzimos uma formulação variacional mista de três campos para escoamentos viscosos de superfícies fechadas curvas. Nela, o fluido circundante é levado em conta considerando-se uma restrição de volume interior, ao passo que uma restrição de área corresponde à inextensibilidade. As incógnitas são a velocidade, o vetor curvatura e a pressão superficial, todas estas interpoladas com elementos finitos lineares contínuos via estabilização baseada na projeção do gradiente de pressão. O método é semi-implícito e requer a solução de apenas um único sistema linear por passo de tempo. Outro ingrediente numérico proposto é uma força que mimetiza a ação de uma pinça óptica, permitindo interação virtual com a membrana, onde a qualidade e o refinamento de malha são mantidos por remalhagem adaptativa automática. Extensivos experimentos numéricos de dinâmica de relaxação são apresentados e comparados com soluções quasi-analíticas. É observada estabilidade temporal condicional com uma restrição de passo de tempo que escala como o quadrado do tamanho de malha. Reportamos a convergência e os limites de estabilidade de nosso método e sua habilidade em predizer corretamente o equilíbrio dinâmico de compridas e finas elongações cilíndricas (tethers) que surgem a partir de pinçamentos membranais. A dependência de forma membranal com respeito a uma velocidade imposta de pinçamento também é discutida, sendo que há um valor limiar de velocidade abaixo do qual um tether não se forma de início. Testes adicionais ilustram a robustez do método e a relevância dos efeitos viscosos membranais quando sob a ação de forças externas. Sem dúvida, ainda há um longo caminho a ser trilhado para o entendimento completo da mecânica celular (há de serem consideradas outras estruturas tais como citoesqueleto, canais iônicos, proteínas transmembranares, etc). O primeiro passo, porém, foi dado: a construção de um esquema numérico variacional capaz de simular a intrigante dinâmica das membranas celulares. / Cell mechanics lies on the material properties of the plasmatic membrane, fundamentally a two-molecule-thick phospholipid bilayer. Other than bending elastic forces, such a two-dimensional interfacial material also experiences viscous stresses due to its (presumably Newtonian) surface fluid tangential behaviour. Despite the remarkable agreement on membrane shapes between theory and biophysical experiments, there is no computational method for simulating its (actual) viscous dynamics governed by the Boussinesq- Scriven law. Accordingly, we introduce a mixed three-field variational formulation for viscous flows of closed curved surfaces. In it, the bulk fluid is taken into account by means of an enclosed-volume constraint, whereas an area constraint stands for the membranes inextensible character. The unknowns are the velocity, vector curvature and surface pressure fields, all of which are interpolated with linear continuous finite elements by means of a pressure-gradient-projection scheme. The method is semi-implicit and it requires the solution of a single linear system per time step. Another proposed ingredient is a numerical force that emulates the action of an optical tweezer, allowing for virtual interaction with the membrane, where mesh quality and refinement are maintained by adaptively remeshing. Extensive relaxation experiments are reported and compared with quasi-analytical solutions. Conditional time stability is observed, with a time step restriction that scales as the square of the mesh size. We discuss both convergence and the stability limits of our method, its ability to correctly predict the dynamical equilibrium of the tether due to tweezing. The dependence of the membrane shape on imposed tweezing velocities is also addressed. Basically, there is a threshold velocity value below which the tethers shape does not arise at first. Further tests illustrate the robustness of the method and show the significance of viscous effects on membranes deformation under external forces. Undoubtedly, there is still a long way to track toward the understanding of celullar mechanics (one still has to account other structures such as cytoskeleton, ion channels, transmembrane proteins, etc). The first step has given, though: the design of a numerically robust variational scheme capable of simulating the engrossing dynamics of fluid cell membranes.