A higher-order finite element method with an explicit higher-order time integration scheme

Paper in proceeding, 2007

We formulate and analyze a FEM that exploits higher-order approximations of the fields for both space and time. An explicit leap-frog type scheme is used to update Faraday's and Ampére's law, and we present a stability and error analysis for this time-stepping scheme. Our formulation exploits a matrix free representation of Maxwell's equations, which reduces the memory requirements significantly. The scheme is tested by solving eigenvalue problems in a cubic cavity with metal walls. The tests demonstrate that the method yields exponential convergence with respect to spatial order p and polynomial convergence, h2p, with respect to the cell size h. Analogous convergence is achieved for the time-stepping scheme. The method reduces to the standard FDTD method for lowest order of approximation with respect to space and time. The technique presented is useful for computational studies of reverberation chambers and similar applications.