https://hal-ensta-paris.archives-ouvertes.fr/hal-00849577Brandsmeier, HolgerHolgerBrandsmeierETH-SAM - Seminar for Applied MathematicsSchmidt, KerstenKerstenSchmidtETH-SAM - Seminar for Applied MathematicsPOEMS - Propagation des Ondes : Étude Mathématique et Simulation - Inria Saclay - Ile de France - Inria - Institut National de Recherche en Informatique et en Automatique - UMA - Unité de Mathématiques Appliquées - ENSTA Paris - École Nationale Supérieure de Techniques Avancées - CNRS - Centre National de la Recherche ScientifiqueHCM - Hausdorff Center for Mathematics - Rheinische Friedrich-Wilhelms-Universität BonnSchwab, ChristophChristophSchwabETH-SAM - Seminar for Applied MathematicsA multiscale hp-FEM for 2D photonic crystal bandsHAL CCSD2011[MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA][MATH.MATH-MP] Mathematics [math]/Mathematical Physics [math-ph]Arnoux, Aurélien2013-08-30 15:27:072022-05-11 12:06:042013-08-30 15:27:07enJournal articles10.1016/j.jcp.2010.09.0181A multiscale generalised hp-finite element method (MSFEM) for time harmonic wave propagation in bands of locally periodic media of large, but finite extent, e.g., photonic crystal (PhC) bands, is presented. The method distinguishes itself by its size robustness, i.e., to achieve a prescribed error its computational effort does not depend on the number of periods. The proposed method shows this property for general incident fields, including plane waves incident at a certain angle to the infinite crystal surface, and at frequencies in and outside of the bandgap of the PhC. The proposed MSFEM is based on a precomputed problem adapted multiscale basis. This basis incorporates a set of complex Bloch modes, the eigenfunctions of the infinite PhC, which are modulated by macroscopic piecewise polynomials on a macroscopic FE mesh. The multiscale basis is shown to be efficient for finite PhC bands of any size, provided that boundary effects are resolved with a simple macroscopic boundary layer mesh. The MSFEM, constructed by combing the multiscale basis inside the crystal with some exterior discretisation, is a special case of the generalised finite element method (g-FEM). For the rapid evaluation of the matrix entries we introduce a size robust algorithm for integrals of quasi-periodic micro functions and polynomial macro functions. Size robustness of the present MSFEM in both, the number of basis functions and the computation time, is verified in extensive numerical experiments. © 2010 Elsevier Inc.