The purpose of this thesis is the study, from the numerical simulation point of view, of the aluminum electrolysis process. Navier-Stokes equations for the computation of a two fluids flow with free interface are coupled with Maxwell equations describing the electric current repartition and the magnetic induction field in an electrolysis reduction cell. The emphasis is set on an efficient method for the computation of the magnetic induction in an unbounded domain. The algorithm is based on a Schwarz domain decomposition method and on the Poisson integral representation formula for harmonic functions. The partial differential equations that rule the phenomena are discretized in space and time and implemented in an existing numerical simulation software. This code is then tested on an academic test case and also in a more realistic situation. The key parts of the mathematical model are emphasized. Finally the time-evolution model is compared with another approach, dealing with stationary situations and their linear stability.