A Delta-Regularization Finite Element Method for a Double Curl Problem with Divergence-Free Constraint

To deal with the divergence-free constraint in a double curl problem, ${\rm curl\,} \mu^{-1} {\rm curl\,} u=f$ and ${\rm div\,} \varepsilon u=0$ in $\Omega$, where $\mu$ and $\varepsilon$ represent the physical properties of the materials occupying $\Omega$, we develop a $\delta$-regularization method, ${\rm curl\,} \mu^{-1} {\rm curl\,} u_\delta +\delta \varepsilon u_\delta=f$, to completely ignore the divergence-free constraint ${\rm div\,} \varepsilon u=0$. We show that $u_\delta$ converges to $u$ in $H({\rm curl\,};\Omega)$ norm as $\delta\rightarrow 0$. The edge finite element method is then analyzed for solving $u_\delta$. With the finite element solution $u_{\delta,h}$, a quasioptimal error bound in the $H({\rm curl\,};\Omega)$ norm is obtained between $u$ and $u_{\delta,h}$, including a uniform (with respect to $\delta$) stability of $u_{\delta,h}$ in the $H({\rm curl\,};\Omega)$ norm. All the theoretical analysis is done in a general setting, where $\mu$ and $\varepsilon$ may be discontinuous, an...