In [Z. Cai, T. Manteuffel, and S. F. McCormick, SIAM J. Numer. Anal., 34 (1997), pp. 425-454], an L-2-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 (H-1)(d) and H (div) boolean AND H (curl)-type, where d = 2 or 3 is the dimension. By directly approximating H (div) boolean AND 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 L-2 and H-1 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. [References: 8]
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