Supercloseness Error Estimates for the Div Least-Squares Finite Element Method on Elliptic Problems

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 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, H3 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 never seen in the previous work of least-squares finite element methods. © The Author(s), under exclusive licence to Springer Science+Business Media, LLC, part of Springer Nature 2025.