First-Order System Least-Squares for Elliptic Problems with Robin Boundary Conditions

This paper studies a least-squares approach to scalar second-order elliptic partial differential equations with Robin boundary conditions. As in several earlier papers by Bramble, Lazarov, and Pasciak [ Math. Comp., 66 (1997), pp. 935--955]; 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, 1727--1741]; Chang [ SIAM J. Numer Anal., 29 (1992), pp. 452--461]; and Jiang and Povinelli [ Comput. Methods Appl. Mech. Engrg., 102 (1993), pp. 199--212], the scalar equation is first recast into a first-order system by introducing velocity or flux variables, where some norm is then applied to the residual of this system to create a least-squares functional. We show here that a least-squares principle based on an L2 norm produces optimal discretization error estimates in the H1 norm for all variables in the system. We further show that a least-squares principle based on a combination of L2 and H -1 norms yields optimal discretization error in a corresponding combination of H1 and L2 norms for the variables. Finally, a regularity result for the L2 least-squares formulation and numerical examples for both formulations are given.