Study of spurious solutions of finite-element method in the three-component magnetic field formulation for dielectric waveguide problems

The cause of spurious solutions, which are included in the earlier finite-element formulation of the three magnetic field components, is discussed. To eliminate these spurious solutions, a new functional is proposed. In the finite-element analysis using this new functional, earlier spurious solutions do not appear but new ones appear. Reasons are given for these phenomena. The new spurious solutions are equivalent to the TE modes of 'hollow' waveguides and their appearance is limited to the 'fast-wave' region. Therefore, if one is interested only in the solutions in the 'slow-wave' region which correspond to the surface-wave modes of dielectric waveguides, the appearance of new spurious solutions is not a serious problem. The finite-element program based on the new functional is coincident with that based on a Galerkin procedure. This fact proves the validity of both finite-element programs.