An example of explicit implementation strategy and preconditioning for the high order edge finite elements applied to the time-harmonic Maxwell's equations. (1st March 2018)

• Record Type: Journal Article
• Authors: Bonazzoli, Marcella; Dolean, Victorita; Hecht, Frédéric; Rapetti, Francesca
• Is Part Of: Computers & mathematics with applications. Volume 75:issue 5(2018)
• Page Start: 1498
• Page End: 1514
• Publication Date: 2018-03-01
• Subjects: High order finite elements -- Edge elements -- Schwarz preconditioners -- Time-harmonic Maxwell's equations -- FreeFem++; Electronic data processing -- Periodicals; Mathematics -- Data processing -- Periodicals; 510.28541
• DOI: 10.1016/j.camwa.2017.11.013
• Languages: English
• ISSNs: 0898-1221

Abstract

Abstract: In this paper we focus on high order finite element approximations of the electric field combined with suitable preconditioners, to solve the time-harmonic Maxwell's equations in waveguide configurations. The implementation of high order curl-conforming finite elements is quite delicate, especially in the three-dimensional case. Here, we explicitly describe an implementation strategy, which has been embedded in the open source finite element software FreeFem++ (http://www.freefem.org/ff++/ ). In particular, we use the inverse of a generalized Vandermonde matrix to build basis functions in duality with the degrees of freedom, resulting in an easy-to-use but powerful interpolation operator. We carefully address the problem of applying the same Vandermonde matrix to possibly differently oriented tetrahedra of the mesh over the computational domain. We investigate the preconditioning for Maxwell's equations in the time-harmonic regime, which is an underdeveloped issue in the literature, particularly for high order discretizations. In the numerical experiments, we study the effect of varying several parameters on the spectrum of the matrix preconditioned with overlapping Schwarz methods, both for 2d and 3d waveguide configurations.