DIV first-order system LL∗ (FOSLL∗) for second-order elliptic partial differential equations

The first-order system LL∗ (FOSLL∗) approach for general second-order elliptic partial differential equations was proposed and analyzed in [Z. Cai et al., SIAM J. Numer. Anal., 39 (2001), pp. 1418-1445], in order to retain the full efficiency of the L² norm first-order system least-squares (FOSLS) approach while exhibiting the generality of the inverse-norm FOSLS approach. The FOSLL∗ approach of Cai et al. was applied to the div-curl system with added slack variables, and hence it is quite complicated. In this paper, we apply the FOSLL∗ approach to the div system and establish its well-posedness. For the corresponding finite element approximation, we obtain a quasi-optimal a priori error bound under the same regularity assumption as the standard Galerkin method, but without the restriction to sufficiently small mesh size. Unlike the FOSLS approach, the FOSLL∗ approach does not have a free a posteriori error estimator. We then propose an explicit residual error estimator and establish its reliability and efficiency bounds.