Abstract
Discrete complexes have attracted the attention of numerical scientists because of the many advantages they offer for the discretization of partial differential equations.They are particularly interesting for the creation of structure preserving methodssuch as methods imposing discrete solutions to be exactly divergence-free.In this thesis, we explore the possible applications of these complexes to incompressible fluids.We first focused on the De Rham complex with minimal regularity.Although it is widely used for electromagnetism, it has been rarely considered for fluids.We wished to study in more depth the possibility of using this complex with fluids.We construct a scheme for the Navier-Stokes equations and analyze it by demonstratingconvergence results, error estimates and especially the preservation of the structure with an exact conservation of some quantities.However, the minimal regularity imposes constraints, and in particular limits the applicable boundary conditions.In order to overcome these problems, we design a new discretization of the Stokes complex (which is another complex with higher regularity suitable for Stokes or Navier-Stokes equations) and study its properties.This discrete complex is based on a hybrid and fully discrete method which allows, in addition to take advantage of the complex properties, to use any polyhedral mesh, that is not necessarily conformal nor simplicial.