An optimal domain decomposition preconditioner for low-frequency time-harmonic Maxwell equations

Abstract

Abstract. The time-harmonic Maxwell equations are considered in the lowfrequency case. A finite element domain decomposition approach is proposed for the numerical approximation of the exact solution. This leads to an iteration-by-subdomain procedure, which is proven to converge. The rate of convergence turns out to be independent of the mesh size, showing that the preconditioner implicitly defined by the iterative procedure is optimal. For obtaining this convergence result it has been necessary to prove a regularity theorem for Dirichlet and Neumann harmonic fields.

Discrete compactness and the approximation of Maxwell's equations in $\mathbb{R}^3$

“…These moments are defined if u ∈ H r (Ω) and ∇ × u ∈ H r (Ω) for some r > 1/2, or if u ∈ H 1+ (Ω) for some > 0 (see [1]). Unfortunately the moments are not defined in general for u ∈ H 1 (Ω), which is a complicating factor in the analysis.…”

Domain Decomposition Algorithms for Time-Harmonic Maxwell Equations with Damping

“…The matrices ε and µ are assumed to be uniformly positive definite in Ω. (A matrix ζ(x) is uniformly positive definite in Ω if there exists a constant ζ * > 0 such that 3 l,m=1 ζ l,m (x)ξ l ξ m ≥ ζ * |ξ ξ ξ| 2 for almost all x ∈ Ω and for all ξ ξ ξ ∈ C 3 .) The conductivity σ is an uniformly positive definite matrix in a conductor and it is equal to 0 in an insulator.…”

“…Assuming J ∈ (L 2 (Ω)) 3 , and setting, for simplifying notation, F = iωJ, the weak formulation of the boundary value problem (2-3) reads:…”

“…In the last years some domain decomposition methods for the numerical solution of the time-harmonic Maxwell equations have been proposed (see [15,16,24,25], see also [3] and [4]). In [16] an iterative nonoverlapping domain decomposition method with Robin type transmission condition is given for the equation (2) in an insulator ( i.e., σ ≡ 0), with non-reflecting boundary conditions.…”

“…In [15] a new iteration scheme is introduced for the same problem modifying the interface conditions proposed in [16]. In [3] and [4] we study iterative domain decomposition methods for the low-frequency model (namely, the case where the term −ω 2 εE is neglected), both for the case of a conductor and for the case of the coupling insulator-conductor (eddy-current problem).…”