[Paper Review] An Adaptive Finite Element DtN Method for Maxwell's Equations in Biperiodic Structures

Xue Jiang, Peijun Li+4 more|arXiv (Cornell University)|Nov 29, 2018

Electromagnetic Simulation and Numerical Methods30 references4 citations

TL;DR

This paper presents an adaptive finite element method with a Dirichlet-to-Neumann (DtN) transparent boundary condition for solving Maxwell's equations in three-dimensional biperiodic grating structures. By deriving a posteriori error estimates that account for both finite element discretization and DtN operator truncation errors, the method adaptively adjusts both the mesh and the truncation parameter, achieving quasi-optimal convergence rates and outperforming traditional PML-based approaches in computational efficiency and accuracy.

ABSTRACT

Consider the diffraction of an electromagnetic plane wave by a biperiodic structure where the wave propagation is governed by the three-dimensional Maxwell equations. Based on transparent boundary condition, the grating problem is formulated into a boundary value problem in a bounded domain. Using a duality argument technique, we derive an a posteriori error estimate for the finite element method with the truncation of the nonlocal Dirichlet-to-Neumann (DtN) boundary operator. The a posteriori error consists of both the finite element approximation error and the truncation error of boundary operator which decays exponentially with respect to the truncation parameter. An adaptive finite element algorithm is developed with error controlled by the a posterior error estimate, which determines the truncation parameter through the truncation error and adjusts the mesh through the finite element approximation error. Numerical experiments are presented to demonstrate the competitive behavior of the proposed adaptive method.

Motivation & Objective

• To develop an adaptive finite element method for solving time-harmonic Maxwell's equations in 3D biperiodic grating structures.
• To address the challenge of truncating the nonlocal DtN boundary operator in unbounded computational domains.
• To design an adaptive algorithm that simultaneously controls finite element error and DtN truncation error.
• To provide a viable alternative to the PML method with reduced computational domain size and improved efficiency.
• To validate the method through numerical experiments demonstrating quasi-optimal convergence and competitive performance.

Proposed method

• The problem is formulated as a boundary value problem in a bounded domain using a transparent boundary condition derived from the DtN operator.
• The nonlocal DtN operator is truncated to a finite sum using a parameter $N$, with the truncation error decaying exponentially with $N$.
• A duality argument technique is employed to derive a reliable and efficient a posteriori error estimate combining finite element and truncation errors.
• The a posteriori error estimate guides adaptive refinement: the mesh is refined based on finite element error, and the truncation parameter $N$ is adjusted based on truncation error.
• The adaptive algorithm dynamically balances both error components to achieve optimal convergence rates.
• Numerical solutions are computed using Nédélec edge finite elements for the curl-conforming discretization of the vector field.

Experimental results

Research questions

• RQ1 Can a posteriori error estimation be derived for the combined finite element and DtN truncation errors in 3D biperiodic Maxwell problems?
• RQ2 How can adaptive mesh refinement and adaptive DtN truncation be simultaneously controlled to achieve optimal convergence?
• RQ3 What is the performance of the adaptive DtN method compared to the adaptive PML method in terms of accuracy and computational cost?
• RQ4 Can the method achieve quasi-optimal convergence rates $\|\mathbf{E} - \mathbf{E}_h^N\|_{H(\mathrm{curl},\Omega)} = O(N_k^{-1/3})$?
• RQ5 Is the DtN method a viable alternative to PML for wave propagation problems in unbounded domains?

Key findings

• The proposed adaptive DtN method achieves quasi-optimal convergence rate $\|\mathbf{E} - \mathbf{E}_h^N\|_{H(\mathrm{curl},\Omega)} = O(N_k^{-1/3})$ with respect to the number of degrees of freedom $N_k$, as confirmed by numerical experiments.
• The a posteriori error estimate $\eta_h$ accurately captures the true error and guides effective mesh and truncation parameter adaptation.
• For the flat plate example, the adaptive DtN method achieves comparable accuracy to the adaptive PML method with fewer degrees of freedom and a smaller computational domain.
• In the checkerboard grating example, the method successfully captures complex scattering patterns with high accuracy, and the error estimate remains reliable across adaptive refinements.
• The DtN method reduces computational cost by eliminating the need for an artificial PML layer, allowing the computational domain to be placed closer to the scatterer.
• The method is competitive with the adaptive PML method and offers a viable alternative, especially in cases where PML may not be applicable.