A numerical study of the pollution error and DPG adaptivity for long waveguide simulations

TL;DR

This paper investigates pollution errors in high-frequency waveguide simulations using DPG finite element methods, highlighting their diffusive effects, and explores adaptive refinement and load balancing for accurate, large-scale 3D electromagnetic problems.

Contribution

It provides a detailed numerical analysis of pollution effects in long waveguide simulations with DPG methods and demonstrates adaptive strategies and load balancing for large-scale problems.

Findings

Pollution causes energy loss, less phase error in DPG methods.

Adaptive refinement improves accuracy in multi-mode waveguides.

Load balancing is crucial for large parallel simulations.

Abstract

High-frequency wave propagation has many important applications in acoustics, elastodynamics, and electromagnetics. Unfortunately, the finite element discretization for these problems suffers 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 (2011). 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.