FIRST-ORDER SYSTEM LEAST-SQUARES (FOSLS) FOR CONVECTION-DIFFUSION PROBLEMS - NUMERICAL RESULTS

The focus of this paper is on planar linear convection-diffusion probl ems, to which we apply a special form of first-order system least squa res (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 vari able, scaled by certain exponential functions. The convection-diffusio n equation is then recast as a minimization principle for a functional corresponding to a sum of weighted L-2 norms of the resulting first-o rder system. Discretization is accomplished by a Rayleigh{Ritz method based on standard finite element subspaces, and the resulting linear s ystems are solved by basic multigrid algorithms. The main goal here is to obtain optimal discretization accuracy and solver speed that is es sentially 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 H-1 nor m. Included in our study is a multilevel adaptive refinement method ba sed on locally uniform composite grids and local error estimates based on the functional itself.