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

Mixed Spectral Element Method for the Waveguide Problem with Bloch Periodic Boundary