Solution of problems of harmonic electromagnetic field simulation in regularized and mixed formulations

Abstract

Classic mixed and regularized variational formulations are considered in the paper for a complex vector Helmholtz equation. A comparative analysis is presented for these formulations and solvability issues are discussed. Preconditioned formulations of problems proposed here can be used for further solution of nite element analogues of continuous problems by iterative methods in Krylov's subspaces.The problem of simulation of electromagnetic elds harmonic in time (in the frequency domain) is urgent for many electrophysical applications, including electronics and electrical engineering, various antenna devices and microwave instruments. The Maxwell system of equations is reduced in this case to complex vector Helmholtz equations with respect to electric or magnetic elds E, H. In addition, there are approaches which involve the formulation of problems for auxiliary values, for example, for scalar and vector potentials. Many papers are focused on numerical solution of such problems with the use of nite element or nite volume methods, see, e.g., [6,11,12,14,16,19,22,26] and the references therein.The main di culties in the construction of e cient algorithms relate to the requirement of high-accuracy and divergence-free grid approximations of electromagnetic elds, because otherwise parasitic charges (having no physical sense) may appear [5,6]. Moreover, solving sparse systems of linear algebraic equations (SLAE) with non-Hermitian sign-inde nite matrices of a high order arising in this case is a hard computational problem even for modern supercomputers.This paper is focused on comparative analysis of two following mathematical formulations: the 'classic' Helmholtz equation for the electric eld E and a similar formulation with a Lagrangian multiplier. Section 1 contains a description of variational representations of the three-dimensional boundary value problems for a complex Helmholtz equation considered here. In Section 2 we consider mixed and regularized formulations of the problem and discuss their solvability. Section 3 presents the constructions of feasible preconditioning operators for the mixed and regularized formulations of the problem. In conclusion we discuss the results obtained in the paper. Formulations of boundary value problems for the Helmholtz equationLet the vectors of the intensity of the electric and magnetic elds have the form ℜ(Eae −iωt ), ℜ(Hae −iωt ), where t is the time, ω is the angular frequency, i is the imaginary unit, ℜ(x) is the real part of x, Ea and Ha are the complex amplitudes dependent on spatial coordinates only. In this case the system of Maxwell equations describing electromagnetic elds in the frequency domain in the absence of external volume electric charges and permeance is written in the following form:∇ × Ea = iωµHa, ∇ · (εrEa) = ∇ × Ha = −iωεEa + J, ∇ · (µrHa) = .(1.1) Brought to you by | Florida International University Libraries Authenticated Download Date | 5/31/15 11:37 AM