Bloch-periodic boundary conditions in classical and enriched FEM

In the attached paper (accepted for publication), we present enriched FE formulations to impose Bloch-periodic boundary conditions. Bloch-periodic BCs arise in the description of wave-like phenomena in periodic media: periodic composites, Schrodinger equation in quantum mechanics, photonic band-gap materials, etc. For a perspective, see the J-Club on elastodynamic bandgaps and metamaterials that was organized by Biswajit Banerjee

We consider the quantum-mechanical problem, which consists of the solution of the three-dimensional Schrodinger and Poisson equations subject to Bloch-periodic and periodic boundary conditions, respectively. In deriving the weak formulations, emphasis is placed on the imposition of value-periodic and derivative-periodic conditions. A simple means to impose value-periodicity (Bloch) via row and column addition and deletion operations on the Neumann matrices is proposed.  Various issues pertaining to the construction of suitable enrichment functions are discussed. Numerical examples for FE and partition-of-unity FE (higher-order serendipity brick elements) are presented for the Poisson equation (electrostatic potential) and the Schrodinger equation (energy eigenvalues) to assess convergence and accuracy.