The goal of this work is to use anisotropic adaptive finite elements for the numerical simulation of aluminium electrolysis. The anisotropic adaptive criteria are based on a posteriori error estimates derived for simplified problems. First, we consider an elliptic problem with smooth, but strongly varying coefficient. We then study the steady Stokes equation and finally the nonlinear p-Laplace problem. In particular, we focus on continuous piecewise linear finite elements with possibly large aspect ratio. Whenever possible, error estimates are proven to be equivalent to the numerical error with constants independent from the mesh aspect ratio. Numerical experiments confirming these predictions are presented. Adaptive strategies based on the derived error estimates are proposed. Numerical experiments show sharpness of the estimates on adapted meshes. For a given accuracy the computational time is reduced. Finally, numerical simulations of the aluminium electrolysis process, using adaptive meshes, are considered. The fluid-flow problem without and with gas is studied. Numerical results confirming a reduction of the computational time are presented.