Maximum norm error estimates for Div least-squares method for Darcy flows

Abstract

Least-squares finite element methods for second order elliptic partial differential equations such as Darcy flows are considered. While there has been a significant progress in terms of obtaining error estimates for the methods, the estimates are essentially based on $L_2$-norm of the error. In this paper, we provide maximum norm error estimates for the primary variable using a smoothed Green's function introduced in [33] and maximum norm error for the dual variables by taking advantage of the fact that least-squares solutions are higher-order perturbations of Galerkin solutions [8].