Sharp $L_2$-Norm Error Estimates for First-Order div Least-Squares Methods

A theoretical analysis of a first-order div least-squares finite element method is presented. Our main interest is providing $L_2$-norm error estimates for the primary function $u$ and the flux $\mbox{\boldmath$$}=-\mathcal{A}\nabla u$. While there has been significant progress in the error estimates for the flux in $L_2$-norm, these estimates have drawbacks, such as requiring smooth solutions or that the error estimate for $\|\mbox{\boldmath$$}-\mbox{\boldmath$$}_h\|_0$ is coupled with the error estimate for $\|u-u_h\|_1$, where $(u_h,\mbox{\boldmath$$}_h)$ is the least squares approximate solution for $(u,\mbox{\boldmath$$}=-\mathcal{A}\nabla u)$. In this paper, with the minimum regularity assumption $u\in H^{1+\alpha}$, $\alpha>1/2$, we separate the error estimate for $\|\mbox{\boldmath$$}-\mbox{\boldmath$$}_h\|_0$ from $\|u-u_h\|_1$ and establish that $\|\mbox{\boldmath$$}-\mbox{\boldmath$$}_h\|_0$ is bounded by the best approximation for $\mbox{\boldmath$$}$ in the finite element space and $\|u-u_h\|_0$. Then, we proceed to obtain two new estimates for $\|u-u_h\|_0$. The first estimate is useful to understand how the accuracy of $\|u-u_h\|_0$ affects the accuracy of $\|\mbox{\boldmath$$}-\mbox{\boldmath$$}_h\|_0$ and the other sheds new light on the convergence behavior of $\|u-u_h\|_0$. Our analysis does not require smooth solutions and the domain is allowed to be nonconvex. The resulting error estimates are valid for both smooth and low regularity solutions and show that optimal rates of convergence can be achieved by choosing proper finite element spaces such as the Raviart-Thomas spaces for the flux $\mbox{\boldmath$$}$.