The application of the \(hp\)-finite element method to electromagnetic problems
This is a review paper on the application of hp-finite element method to electromagnetic problems. The authors reviewed the works and the main development of finite element methods for electromagnetic scattering problems and electromagnetic eigenvalue problems. This paper addressed many issues in the numerical solution of time-harmonic scattering problems and eigenvalue problems. The main topics discussed in this paper include the following items. 1. The authors reviewed several PDE models of Maxwell's equations with different boundary conditions and weak formulations. 2. The authors reviewed the H(curl)-conforming edge element methods and their historical development. Considerable references to edge elment methods, which particularly include the works due to Whitney, Nédélec, Bossavit, Mur and De Hoop, Hiptmair, Demkowicz, Ainsworth, Ledge, and so on, are mentioned in the development of numerical Maxwell's equations. 3. The hp-edge element approximations: The paper presents the explicit forms of hierarchical basis functions of both \(H^1\)-conforming high-order nodal element spaces and H(curl)-conforming high-order edge element spaces in 2D and 3D. The main efforts are devoted to the computational aspect of finite element methods. The authors discussed the affine transformation and the covariant mappings from the reference element to physical elements, numerical integrations in implementations, and the technique of static condensation which removes in the element-level the interior unknowns of high-order elements from the global stiffness matrix. 4. Computation of time-harmonic scattering problems: The authors reviewed the recent works on deriving the discrete dispersion relationship and on reducing the dispersion by hp-finite element methods and modified finite element methods. Moreover, a brief review of the application of the perfectly matched layer method to electromagnetic scattering problems is presented for the computation of the far-field pattern. The ``goal-oriented'' error estimation and mesh adaption are also discussed for the computation of the scattering width. 5. Electromagnetic eigenvalue problems: The difficulty arising in computing the eigenvalues of Maxwell's equations is mainly due to the large kernel of the curl-operator which contains the gradients of all differentiable scalar functions. This usually deteriorates the divergence-free property of eigenfunctions and produces spurious eigenmodes. Works and open issues on this topic are reviewed in more detail for 2D and 3D problems. This paper is a good literature for the mathematicians who work on electromagnetic scattering and eigenvalue problems.
scientific article; zbMATH DE number 5575518
scientific article