First-Order System Least Squares (FOSLS) for Convection-Diffusion Problems: Numerical Results

The focus of this paper is on planar linear convection-diffusion problems, to which we apply a special form of first-order system least squares (FOSLS [Cai et al., SIAM J. Numer. Anal., 31 (1994), pp. 1785--1799; Cai, Manteuffel, and McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425--454]). This we do by introducing the gradient of the primary variable, scaled by certain exponential functions. The convection-diffusion equation is then recast as a minimization principle for a functional corresponding to a sum of weighted L2 norms of the resulting first-order system. Discretization is accomplished by a Rayleigh--Ritz method based on standard finite element subspaces, and the resulting linear systems are solved by basic multigrid algorithms. The main goal here is to obtain optimal discretization accuracy and solver speed that is essentially uniform in the size of convection. Our results show that the FOSLS approach achieves this goal in general when the performance is measured either by the functional or by an equivalent weighted H1 norm. Included in our study is a multilevel adaptive refinement method based on locally uniform composite grids and local error estimates based on the functional itself.