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

## TL;DR

This paper introduces a mixed spectral element method for waveguide problems with Bloch periodic boundary conditions, achieving high accuracy and eliminating spurious modes through a novel variational formulation.

## Contribution

The paper develops two new mixed spectral element schemes that are free of spurious modes and more computationally efficient for waveguide problems with Bloch periodic boundaries.

## Key findings

- Both schemes are free of spurious modes.
- They demonstrate high accuracy in computing propagation constants.
- The methods reduce computational costs compared to traditional approaches.

## Abstract

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.