Explicit implementation strategy of high order edge finite elements and Schwarz preconditioning for the time-harmonic Maxwell's equations

TL;DR

This paper presents an explicit implementation strategy for high order edge finite elements in 3D Maxwell's equations, integrated into FreeFem++, and investigates effective Schwarz preconditioning techniques for waveguide problems.

Contribution

It introduces a practical implementation method for high order curl-conforming finite elements using Vandermonde matrices and explores preconditioning strategies for Maxwell's equations in waveguides.

Findings

Vandermonde matrix-based basis simplifies high order element implementation.

Preconditioning with Schwarz methods improves solver performance.

Parameter variations significantly affect the spectrum and convergence.

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 a basis of generators 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.