Localized pointwise error estimates for direct flux approximation

Pointwise error estimates for the first-order div least-squares (LS) finite element method for second-order elliptic partial differential equations are presented. Direct flux approximation is considered as an important advantage of the LS method. However, there are no known pointwise error estimates for the direct flux approximation. In this paper, we provide optimal pointwise estimates which show local dependence of the error at a point and weak dependence of the global norm. As an elementary consequence of these estimates, we provide an asymptotic error expansion inequality. The inequality has applications to super-convergence and a posteriori estimates.