Superconvergence of least-squares methods for a coupled system of elliptic equations

We present a numerical method for solving a coupled system of elliptic partial differential equations (PDEs). Our method is based on the least-squares (LS) approach. We develop ellipticity estimates and error bounds for the method. The main idea of the error estimates is the establishment of supercloseness of the LS solutions, and solutions of the mixed finite element methods and Ritz projections. Using the supercloseness property, we obtain $L_2$-norm error estimates, and the error estimates for each quantity of interest show different convergence behaviors depending on the choice of the approximation spaces. Moreover, we present maximum norm error estimates and construct asymptotically exact a posteriori error estimators under mild conditions. Application to optimal control problems is briefly considered.