Recent work on the Maxwell-Dirac equations is reviewed. The system we are interested in is the full MaxwellDirac equations: Dirac field with minimally coupled electromagnetic field and Maxwell field with Dirac current as source. Work on the static equations (the Dirac current is purely timelike in some Lorentz frame) is examined and two interesting theorems which apply in this case are discusse...

We investigate some simple finite element discretizations for the axisymmetric Laplace equation and the azimuthal component of the axisymmetric Maxwell equations as well as multigrid algorithms for these discretizations. Our analysis is targeted at simple model problems and our main result is that the standard V-cycle with point smoothing converges at a rate independent of the number of unknown...

We present and analyze exact solutions of the Einstein–Maxwell and Einstein–Maxwell– Dilaton equations that describe static pairs of oppositely charged extremal black holes, i.e., black diholes. The holes are suspended in equilibrium in an external magnetic field. We comment as well on the relation of these solutions to brane–antibrane configurations in string and M-theory.

The generalized Maxwell equations are considered which include an additional gradient term. Such equations describe massless particles possessing spins one and zero. We find and investigate the matrix formulation of the first order of equations under consideration. All the linearly independent solutions of the equations for a free particle are obtained in terms of the projection matrix-dyads (d...

We present a parallelized algorithm for solving the time-dependent Vlasov-Maxwell system of equations in the four-dimensional phase space (two spatial and velocity dimensions). One Vlasov equation is solved for every particle species, from which charge and current densities are calculated for the Maxwell equations. The parallelization is divided into two different layers. For the first layer, e...

We study linear time dispersive and dissipative systems. Very often such systems are not conservative and the standard spectral theory can not be applied. We develop a mathematically consistent framework allowing (i) to constructively determine if a given time dispersive system can be extended to a conservative one; (ii) to construct that very conservative system – which we show is essentially ...

A few years ago, Costabel and Dauge proposed a variational setting, which allows one to solve numerically the time-harmonic Maxwell equations in 3D geometries with the help of a continuous approximation of the electromagnetic field. In this paper, we investigate how their framework can be adapted to compute the solution to the time-dependent Maxwell equations. In addition, we propose some exten...