A second order finite element method with mass lumping for Maxwell's equations on tetrahedra

Abstract: We consider the numerical approximation of Maxwell's equations in time domain by a second order $H(curl)$ conforming finite element approximation. In order to enable the efficient application of explicit time stepping schemes, we utilize a mass-lumping strategy resulting from numerical integration in conjunction with the finite element spaces introduced by Elmkies and Joly. We prove that this method is second order accurate if the true solution is divergence free, but the order of accuracy reduces to one in the general case. We then propose a modification of the finite element space, which yields second order accuracy in the general case.