The Discrete First-Order System Least Squares: The Second-Order Elliptic Boundary Value Problem

In [Z. Cai, T. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425–454], an L2-norm version of first-order system least squares (FOSLS) was developed for scalar second-order elliptic partial differential equations. A limitation of this approach is the requirement of sufficient smoothness of the original problem, which is used for the equivalence of spaces between (H1)d and H(div ) ∩ H(curl)-type, where d = 2 or 3 is the dimension. By directly approximating H(div ) ∩H(curl)-type space based on the Helmholtz decomposition, this paper develops a discrete FOSLS approach in two dimensions. Under general assumptions, we establish error estimates in the L2 and H1 norms for the vector and scalar variables, respectively. Such error estimates are optimal with respect to the required regularity of the solution. A preconditioner for the algebraic system arising from this approach is also considered.