  The mixed spectral element method (MSEM) is applied to solve the waveguide
problem with Bloch periodic boundary condition (BPBC). Based on the BPBC for
the original Helmholtz equation and the periodic boundary condition (PBC) for
the equivalent but modified Helmholtz equation, two equivalent mixed
variational formulations are applied for the MSEM. Unlike the traditional
finite element method and SEM, both these mixed SEM schemes are completely free
of spurious modes because of their use of the Gauss' law and the
curl-conforming vector basis functions structured by the Gauss-Legendre-Lobatto
(GLL) points. A simple implementation method is used to deal with the BPBC and
the PBC for the mixed variational formulations so that both schemes can save
computational costs over the traditional methods. Several numerical results are
also provided to verify that both schemes are free of spurious modes and have
high accuracy with the propagation constants.
