Superclosenes error estimates for the div least-squares finite element method on elliptic problems

Abstract: In this paper we provide some error estimates for the div least-squares finite element method on elliptic problems. The main contribution is presenting a complete error analysis, which improves the current \emph{state-of-the-art} results. The error estimates for both the scalar and the flux variables are established by specially designed dual arguments with the help of two projections: elliptic projection and H(div) projection, which are crucial to supercloseness estimates. In most cases, $H^3$ regularity is omitted to get the optimal convergence rate for vector and scalar unknowns, and most of our results require a lower regularity for the vector variable than the scalar. Moreover, a series of supercloseness results are proved, which are \emph{never seen} in the previous work of least-squares finite element methods.