Elementos finitos em formulação mista de mínimos quadrados para a simulação da convecção-difusão em regime transiente

Abstract

A mixed least-squares finite element scheme designed for solving the transient convection-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 both simplicial and non-simplicial geometry, in any space dimension. The scheme is also convergent in space in the mean-square sense as far as the primal unknown field, its gradient, the flux variable and its divergence are concerned, and in time in an appropriate sense for each one of these four fields. Numerical results certify that the scheme performs well for any Péclet number, thereby allowing to confirm theoretical predictions, at least in the case where there is no narrow boundary layer. In the latter case however the method fails to produce reliable results. The technique is also compared with three existing methods to solve the convection-diffusion equations in the transient case. These include two recent ones proposed by the first author and collaborators.