Journal of the
Korean Mathematical Society JKMS

We propose and analyze a mixed finite volume method for the two-dimensional time-harmonic Maxwell's equations which simultaneously approximates the vector field $\boldsymbol{u}$ and the scalar function $\xi = \mu^{-1}\operatorname{curl}\boldsymbol{u}$. The method chooses the lowest-order N\'{e}d\'{e}lec edge element for $\boldsymbol{u}$ and the $P1$ Crouzeix--Raviart nonconforming element for $\xi$ on triangular meshes. It is shown that the method is reduced to a modified $P1$ nonconforming FEM for $\xi$ or a modified edge element method for $\boldsymbol{u}$ by eliminating the discrete variable of $\boldsymbol{u}$ or $\xi$. After solving the reduced method, the eliminated discrete variable can be recovered from the other one via a simple local formula. Using this feature, we also derive optimal a priori error estimates under weak regularity assumptions and show that the approximation to $\xi$ has a higher-order of convergence in the $L^2$ norm than the one obtained by direct differentiation of the approximation to $\boldsymbol{u}$ when the exact solution is sufficiently smooth.

Keywords: Maxwell's equations, mixed finite volume method, N\'{e}d\'{e}lec edge element, $P1$ nonconforming finite element

MSC numbers: Primary 65N08, 65N30, 65N15

Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2021R1F1A1050243).