Finite Element Methods for Maxwell's Equations (1910.10069v1)

Abstract: We survey finite element methods for approximating the time harmonic Maxwell equations. We concentrate on comparing error estimates for problems with spatially varying coefficients. For the conforming edge finite element methods, such estimates allow, at least, piecewise smooth coefficients. But for Discontinuous Galerkin (DG) methods, the state of the art of error analysis is less advanced (we consider three DG families of methods: Interior Penalty type, Hybridizable DG, and Trefftz type methods). Nevertheless, DG methods offer significant potential advantages compared to conforming methods.

• The paper presents an in-depth investigation of discontinuous Galerkin methods, demonstrating their enhanced flexibility over traditional conforming approaches in simulating Maxwell's equations.
• It details various DG variants, including IPDG, HDG, and UWVF, and outlines their computational benefits alongside challenges in stability and error convergence.
• The study emphasizes the theoretical and practical implications for electromagnetic simulations and encourages further research into advanced solver techniques.

Finite Element Methods for Maxwell's Equations: An Overview

The finite element method (FEM) has long been a critical tool for numerically solving complex systems of equations, such as Maxwell's equations that govern electromagnetic wave propagation. This paper provides an in-depth exploration of various FEM techniques applied to the time-harmonic Maxwell's equations, focusing on both conforming and discontinuous Galerkin (DG) methods. While the historical and mathematical foundations of these methods are discussed, the emphasis is primarily on DG methods, particularly their advantages, the mathematical challenges they present, and the potential benefits over conforming methods.

Conforming Methods

The paper initially discusses the conforming edge elements, which are traditionally used to discretize Maxwell's equations. These elements are well-understood mathematically and provide a straightforward approximation method by ensuring conformity in the functional space required by Maxwell's equations. However, they have some limitations, especially in handling media with spatially varying coefficients. While conforming methods serve as a reliable standard, they sometimes lack the flexibility desired in complex simulations involving non-homogeneous materials or intricate geometries.

Discontinuous Galerkin Methods

Discontinuous Galerkin methods exhibit significant flexibility and offer advantages over conforming methods, particularly in computational mechanics for large-scale problems. The paper evaluates three primary DG method categories: Interior Penalty (IP), Hybridizable Discontinuous Galerkin (HDG), and Trefftz methods, including the Ultra Weak Variational Formulation (UWVF).

1. Interior Penalty DG Method (IPDG): The IPDG method for Maxwell's equations is advantageous because it is inherently flexible with local polynomial degrees and mesh configurations. The method alleviates some difficulties faced by conforming methods related to element-wise conservation properties and adaptability to complex geometries and materials. However, it typically involves more degrees of freedom (DOFs) and requires fine-tuning of penalty parameters for optimal performance.
2. Hybridizable DG Method (HDG): HDG methods reduce the global problem's size significantly by condensing local element DOFs and focusing on interface DOFs. This feature allows for efficient computation and solves some scalability issues faced by traditional DG methods. The paper's analysis highlights these methods' theoretical and practical merits, although the full potential for error convergence and computational benefits remain active areas of research, especially in three-dimensional domains.
3. Trefftz and Ultra Weak Variational Formulation (UWVF): UWVF treats the solution as a Trefftz method, solving problems with solutions directly tailored to Maxwell's equations. This technique enables the accurate representation of waves within each element, reducing dispersion error especially for high-frequency domain problems. Moreover, the ability to work with general element shapes broadens its applicability. While computationally intensive, UWVF offers promising potential for wave problems and complex applications in engineering and physics.

Theoretical and Practical Implications

The exploration of DG methods reveals both theoretical and applied dimensions. Theoretically, these methods require robust error analysis and stability considerations, especially as they differ significantly in foundation and application from conforming methods. Practically, they offer immense computational benefits, particularly relevant in modern simulations requiring adaptability and detail, such as those present in electronic design and optical simulations.

Future Directions

Future research may focus on developing fast solution methods for the linear systems resulting from these methods, potentially exploring hybridization techniques or advanced multigrid solvers. Additionally, enhancing the mathematical framework around these methods—especially concerning their performance at high frequencies—remains a crucial endeavor. Continued innovation in this domain holds the promise of ushering in superior tools for computational electromagnetics, thus broadening the horizon of practical engineering applications.