A Finite Element Technique for Approximately Solving for the Electromagnetic Field Within a Cavity Resonator

This paper develops a four space-time dimensional finite element approach for approximately calculating the electromagnetic and Dirac fields within a 3-D cavity resonator taking into account the interaction between the two fields. The idea is to partition the four-dimensional space-time region consisting of the 3-D cavity and the finite time duration over which the fields are to be determined into four-dimensional simplices. Each simplex is defined by five vertex points in four-dimensional space-time and the fields within each such simplex are expressed as affine linear functions of the four simplex coordinates so that the values of the field at the vertices of the simplex coincides with the field value at that vertex. We then substitute these affine linear functions into the action functional of the electromagnetic field interacting with the Dirac field thereby yielding a quadratic function of the simplex vertex field values corresponding to the free field action plus cubic function of the vertex fields corresponding to the interaction action between the Dirac current and the electromagnetic field. By summing up this action over all the simplices into which we have partitioned the space-time region, we obtain a quadratic-cubic function of the field values at all the vertices of the different simplices. The problem of approximately calculating the fields within the cavity then amounts to minimizing this function of a finite number of complex variables corresponding to the field values at the vertices of the simplices. This optimization can be carried out using for example a gradient search algorithm or using perturbation theory for obtaining solutions to a system of nonlinear algebraic equations when the nonlinearity is small.