Eliminating spurious solutions of dielectric waveguides: computational performance of the reduced integration penalty method

A major drawback to the use of the finite element method (FEM) for solving general dielectric waveguide problems is the appearance of the so-called spurious modes. These are numerical solutions that satisfy Maxwell´s curl equation but not the divergence equation for the (E or H) field associated to the vector variational expression which must be used when the dielectrics are inhomogeneous and/or anisotropic. Two different approaches for imposing solenoidality to the vector fields, thus eliminating the spurious solutions, are treated here. The first one consists in a modification of the penalty method, whose efficacy has been attested in a large number of publications. The second approach considered employs first order nodal (Lagrangean) basis functions for the longitudinal component of the magnetic field, which must be everywhere continuous, and edge elements for the transverse component, which may present normal discontinuity on material (dielectric) interfaces. This simple and efficient method furnishes very good results with the use of low order elements. Following the software implementation of the two approaches, their computational performances for different types of dielectric waveguides were thoroughly compared, considering the accuracy of computed eigenvalues, mesh densities, computation times and storage demands.