We consider the Darcy problem in an axisymmetric three-dimensional domain with data which are axisymmetric. The solution satisfies a system of equations in the meridian domain. We propose a discretization of this problem in the case of an axisymmetric solution. This discretization relies on a backward Euler's scheme for the time variable and finite elements for the space variables. We prove a priori error estimates and a posteriori error estimates both for the time steps and the meshes and we present some numerical experiments which are in good agreement with the analysis