Mathematical analysis of a finite element method without spurious solutions for computation of dielectric waveguides

Abstract

Summary. In this paper we propose a finite element method to compute dielectric waveguides which does not present spurious solutions. In addition to a theoretical analysis showing the good stability properties of this method, we give numerical results for some classical test problems.

Spurious-free approximations of electromagnetic eigenproblems by means of Nedelec-type elements

“…A complete definition of the Nedelec's edge element families [29,30] can be carried out by considering the following sets of moments of a function u ∈ W 1,s (K) 3 , s > 2 [20], for the space R l , l > 0:…”

“…Actually Kikuchi [25] did not consider the case of mixed boundary conditions, but his proof does work even in that case if the hypothesis ∃s > 1/2 such that V 1 and H ⊂ H s (Ω) 3 , and ∃C > 0 such that v 1 s,Ω ≤ C v 1 curl,Ω ∀v 1 ∈ V 1 are still satisfied. These regularity assumptions are not satisfied for general mixed boundary conditions, but are satisfied when mixed boundary conditions arise from symmetry exploitation (see Sect.…”

“…, which can be extended to L 2 (Ω) 3 . As π h preserves both constant values on the connected components of Γ τ and constant jumps across…”

Mathematical analysis of electromagnetic open Waveguides

“…)> we shall use the following 5,1995 If u is an eigenfunction of A e (/?) associated to the eigenvalue c~_ ƒ?% we see that w 3 = 0 and rot u = 0.…”

“…The electromagnetic waveguides 506 P. JOLY, Ch. POIRIER have applications in various domains of physics (electronic components, optical fibers, integrated opties...) and have been already abundantly studied, for instance by D. Marcuse [22] who is the référence in the physical literature, or by A. Bamberger and A. S. Bonnet [2], A. S. Bonnet [7], R. Djellouli [13], N. Gmati [17], A. Bermudez and D. G. Pedreira [5], F. Kikuchi [20], concerning the mathematical or numerical studies.…”