A Yee-like finite-element scheme for Maxwell's equations on unstructured grids.

The article focuses on the development and analysis of a novel finite element scheme for efficiently solving time-dependent Maxwell's equations on unstructured grids, extending the classical Yee scheme to more general settings including inhomogeneous lossy media. The proposed method reduces the degrees of freedom to one per edge for most edges while maintaining a sparse inverse mass matrix, enabling explicit time-stepping without solving linear systems. A rigorous convergence analysis is provided, demonstrating first-order accuracy under assumptions on mesh regularity and material parameter continuity, particularly requiring two degrees of freedom at edges with discontinuous conductivity to preserve convergence rates. Numerical experiments in two and three dimensions validate the theoretical results, showing that the reduced scheme achieves comparable accuracy and computational efficiency to standard finite element methods, with benefits in memory usage and time stepping. The work also discusses the algebraic structure of the scheme, its relation to existing Yee-like methods, and highlights challenges and remedies for extending Yee schemes to unstructured grids with discontinuous parameters.