An Adaptive Anisotropic hp-Refinement Algorithm for the 2D Maxwell Eigenvalue Problem

Abstract

We present a novel adaptive mesh refinement algorithm for efficiently solving a continuous Galerkin formulation of the 2D Maxwell Eigenvalue Problem over quadrilateral discretizations. The algorithm harnesses a Refinement-by-Superposition framework with support for anisotropic hp-adaptivity. Following a brief summary of the underlying methodology, we develop a novel approach to directing and deploying intelligent fully anisotropic hp-refinements. Numerical examples targeting the Maxwell Eigenvalue problem verify the ability to achieve exponential rates of convergence on a challenging benchmark and demonstrate that including anisotropic refinement yields a significant enhancement of computational efficiency. The success of this algorithm, along with the simplicity of the underlying Refinement-by-Superposition approach, creates a promising path for implementing highly efficient yet lightweight finite element method (FEM) codes for a variety of applications in computational electromagnetics (CEM) and elsewhere.

FEM_2D: A Rust Package for 2D Finite Element Method Computations with Extensive Support for hp-refinement

“…FEM_2D aims to expose some features similar to those in other FEM libraries such as Deal.II (Arndt et al, 2021) and MFEM (Anderson et al, 2021) that are designed to be general purpose frameworks for implementing FEM codes. Although it is not nearly as feature-rich as these libraries, FEM_2D's hp-refinement API aims to provide the basic functionality needed to iteratively solve challenging 2D FEM problems as described in the associated work (Corrado et al, 2021(Corrado et al, , 2022Harmon et al, 2021). These features will be discussed in detail in the following sections.…”

“…Lastly, there are various methods for querying properties of individual elements and the relationships between them (adjacency/descendancy). Overall, the Mesh Refinement API is intended to provide enough abstraction and feature-richness to make the implementation of refinement algorithms, like those used in Corrado et al (2022), as straightforward and unencumbered as possible.…”