Numerical Study of Mixed Least-Squares Finite Element Formulations for Transient Advection-Diffusion Equations

Abstract

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 results in the mean square sense in space for the primal unknown and its gradient, together with the flux variable and its divergence, and in appropriate senses in time applying to this pair of fields are given. Numerical experiments illustrate the performance of the scheme, while allowing to check the optimality of the convergence results.