A numerical study of the pollution error and DPG adaptivity for long waveguide simulations
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High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffer from significant numerical pollution errors that increase with the wavenumber. It is critical to control these errors to obtain a stable and accurate method. We study the effect of pollution for very long waveguide problems in the context of robust discontinuous Petrov-Galerkin (DPG) finite element discretizations. Our numerical experiments show that the pollution primarily has a diffusive effect causing energy loss in the DPG method while phase errors appear less significant. We report results for 3D vectorial time-harmonic Maxwell problems in waveguides with more than 8000 wavelengths. Our results corroborate previous analysis for the Galerkin discretization of the Helmholtz operator by Melenk and Sauter ("Wavenumber explicit convergence analysis for Galerkin discretizations of the Helmholtz equation". In: SIAM J. Numer. Anal. 49.3 (2011), pp. 1210-1243). Additionally, we discuss adaptive refinement strategies for multi-mode fiber waveguides where the propagating transverse modes must be resolved sufficiently. Our study shows the applicability of the DPG error indicator to this class of problems. Finally, we illustrate the importance of load balancing in these simulations for distributed-memory parallel computing.
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