This thesis is devoted to the study of some numerical and mathematical aspects of incompressible multiphase flows simulations with a diffuse interface Cahn-Hillliard/Navier-Stokes model. The space discretisation is performed thanks to the finite elements method. The presence of different scales in the system 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. Moreover, we show that this method can be exploited to build multigrid preconditioners. Concerning the time discretization, we begin by the study of the Cahn-Hilliard system. A semi-implicit scheme, which ensure the decrease of the discrete energy, is proposed to remedy to convergence failures of the Newton method used to solve this (non linear) system. 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, 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.
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