Spurious solutions in mixed finite element method for Maxwell’s equations: Dispersion analysis and new basis functions

The finite element method is a well known computational technique used to obtain numerical solutions to boundary-value problems including Maxwell’s equations. This paper first presents a brief description of the mathematical structure, based on the De Rham diagram, to discretize Maxwell’s equations. Then it uses a numerical dispersion analysis of the mixed finite element method with both electric and magnetic fields as unknowns to evaluate the presence of spurious solutions for different basis functions. These unwanted spurious solutions appear when the same order of element is used for electric and magnetic fields, while the system is free of spurious modes when different orders of elements are employed for electric and magnetic fields. In this work, finite elements in both frequency and time domain are studied, and the effects of these spurious solutions in both domains are analyzed in one- and three-dimensional cases.

► We perform dispersion analysis for different choices of edge elements in the finite-element solution of time-domain Maxwell’s equations.
► The same-order edge elements for the electric and magnetic fields produce spurious modes.
► We propose basis functions with different orders of edge elements for electric and magnetic fields. These elements are free of spurious modes.
► Both 1-D and 3-D basis functions have been implemented and numerical results support the above conclusions.
► Such new basis functions are important for discontinuous Galerkin methods for Maxwell’s equations.