Mixed Finite Volume Method for Two-dimensional Maxwell's Equations

Abstract

We propose and analyze a mixed finite volume method for the two-dimensional time-harmonic Maxwell's equations which simultaneously approximates the vector field $\bu$ and the scalar function $\xi = \mu^{-1}\curl\bu$. The method chooses the lowest-order N\'{e}d\'{e}lec edge element for $\bu$ 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 $\bu$ by eliminating the discrete variable of $\bu$ 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 $\bu$ when the exact solution is sufficiently smooth.