Numerical analysis of a least-squares finite element method for the time-dependent advection–diffusion equation

A mixed finite element scheme designed for solving the time-dependent advection–diffusion equations expressed in terms of both the primal unknown and its flux, incorporating or not a reaction term, is studied. Once a time discretization of the Crank–Nicholson type is performed, the resulting system of equations allows for a stable approximation of both fields, by means of classical Lagrange continuous piecewise polynomial functions of arbitrary degree, in any space dimension. Convergence in the norm of H1×H(div) in space and in appropriate senses in time applying to this pair of fields is demonstrated.