Lorentz-Invariant Meshless Vector Basis Function for Translational Motion of Coordinates in Computational Electromagnetics

Laws of physics remain unchanged under translational motion of coordinates. To guarantee the above postulate in electromagnetics, Lorenz gauge eliminates the additional terms generated in the wave equation of magnetic vector potential during translational motion. When it comes to computational electromagnetics, nonetheless, Coulomb gauge is still preferred to represent the divergence of the magnetic vector potential; the vector basis functions involved in the computation of magnetic vector potential are thus divergence-free. There is, however, an immediate consequence that we shall consider here. These vector basis functions cannot incorporate any kinematic transformation of the system of coordinates. The solution achieved by them is, therefore, invalid under translational motion of the system of coordinates as a whole. Less attention has been paid to this side of computational electromagnetics, as the problems that we solve do not usually undergo any kinematic transformation. The new meshless vector basis function presented in this article is Lorentz-invariant. The solution achieved by it is, therefore, valid under translational motion. Even in local problems, the solution achieved by the newly-introduced Lorentz-invariant vector basis function demonstrates more accuracy and efficiency with respect to the solution achieved by the divergence-free vector basis functions in meshless method.