In this talk, we study the weak formulation of the Poisson problem on closed Lipschitz manifolds. Lipschitz manifolds do not admit tangent spaces everywhere and the definition of the Laplace-Beltrami operator is more technical than on classical differentiable manifolds (Gesztesy2011). They however arise naturally after the triangulation of a smooth surface for computer vision or simulation purposes. We derive Stokes' and Green's theorems as well as a Poincaré's inequality on Lipschitz manifolds. The existence and uniqueness of weak solutions of the Poisson problem are given in this new framework for both the continuous and discrete problems. As an example of application, numerical results are given for the Poisson problem on the boundary of the unit cube.

For the coding of the script, we use Python language and the open-source Linux based library (SOLVERLAB) which is very practical for the manipulation of large matrices, vectors, meshes and fields. It (SOLVERLAB) can handle finite element and finite volume discretizations, read general 1D, 2D and 3D geometries and meshes generated by SALOME.

Key words :Lipschitz manifold, Laplace-Beltrami operator, Finite element method, Elliptic equation, closed surfaces