This manuscript describes some numerical and mathematical aspects of incompressible multiphase flows simulations with a diffuse interface Cahn-Hillliard/Navier-Stokes model (interfaces have a small but a positive thickness). The space discretisation is performed thanks to a Galerkin formulation and the finite elements method. The presence of different scales in the system (interfaces have a very small thickness compared to the characteristic lengths of the domain) suggests the use of a local adaptive refinement method. The algorithm, that we introduced, allows to implicitly handle the non conformities of the generated meshes to produce conformal finite elements approximation spaces. It consists in refining basis functions instead of cells. The refinement of a basis function is made possible by the conceptual existence of a nested sequence of uniformly refined grids from which “parent-child” relationships are deduced, linking the basis functions of two consecutive refinement levels.
Moreover, we show how this method can be exploited to build multigrid preconditioners. From a composite finite elements approximation space, it is indeed possible to rebuild, by “coarsening”, a sequence of auxiliairy nested spaces which allows to enter in the abstract multigrid framework. Concerning the time discretization, we begin by the study of the Cahn-Hilliard system. A semi-implicit scheme is proposed to remedy to convergence failures of the Newton method used to solve this (non linear) system. It guarantees the decrease of the discrete free energy ensuring the stability of the scheme. We show existence and convergence of discrete solutions towards the weak solution of the system. We then continue this study by providing an inconditionnaly stable time discretization of the complete Cahn-Hilliard/Navier-Stokes model. An important point is that this discretization does not strongly couple the Cahn-Hilliard and Navier-Stokes systems allowing to independently solve the two systems in each time step. We show the existence of discrete solutions and, in the case where the three fluids have the same densities, we show their convergence towards weak solutions. We study, to finish this part, different issues linked to the use of the incremental projection method. Finally, the last part presents several examples of numerical simulations, diphasic and triphasic, in two and three dimensions.