Local error estimates for least-squares finite element methods for first-order system

We present local energy type error estimates for first-order system div least-squares (LS) finite element methods. The estimate shows that the local energy norm error is bounded by the local best approximation and weaker norms which account for the pollution. The estimate is similar to the one for the standard Galerkin methods. However, our estimate needs to consider the effect of error of dual (flux) variables since LS methods approximate the primary and dual variables simultaneously. The effect of error of the dual variables is shown to be of higher order. Moreover, our estimate shows the convergence behavior when locally enriched approximation spaces are used in the area of interest. As an elementary consequence of the estimate, asymptotically exact a posteriori error estimator is constructed for the local area of interest under mild assumptions.