Séance Séminaire

Séminaire ACSIOM

Computation of Navier-Stokes on Riemannian manifolds annulé suite à la situation liée au Covid 19

Killing vector fields are important in differential geometry because their flows generate isometries on Riemannian manifolds. Equations for Killing fields is an overdetermined system of PDEs which can be hard to solve explicitly. This problem can be reduced to a symmetric eigenvalue problem where Killing fields are generated by the eigenvectors corresponding to zero eigenvalue. The method itself is valid in any dimension, but numerical results are computed only in two-dimensional case. To solve numerically this problem,we used finite element method. On a manifold one must use in general several coordinate systems to describe the problem, and the technical difficulty is then how to patch these coordinate systems together. We propose to solve this problem on the sphere with several local coordinate systems. This method of constructing operators on manifolds can also be used to study other PDE systems. The study of Killing fields is important because they appear in Navier-Stokes equations on compact manifolds when the appropriated operator for the Laplacian is used in the equations. Then we propose several numerical methods for solving Navier-Stokes equations on manifolds where we can see that given an initial condition, the vector field converges to a Killing field. Examples will be given for the standard torus and the sphere. As far as we know, these results are not well known.