Domain decomposition methods for harmonic wave-propagation problems. The Borg theorem for vectorial Hill equation. (Méthodes de décomposition de domaine pour les problèmes de propagation d’ondes en régime harmonique. Le théorème de Borg pour l’équation de Hill vectorielle.) (French. English summary) Zbl 0849.65085

Rocquencourt: Institut National de Recherche en Informatique et en Automatique (INRIA). vi, 233 p. (Univ. de Paris IX, 1991) (1991).
Summary: The two parts of this dissertation are independent of each other. In Part I we study an iterative nonoverlapping domain decomposition method, with Robin-type transmission conditions, which solves harmonic wave propagation problems, such as the Helmholtz problem or the harmonic Maxwell equations. Theoretically, we prove the convergence with energy estimates, explore the rate of convergence in special cases with spectral techniques, and examine some changes in the method. Numerically, we discretize the method with the help of mixed hybrid finite elements, and show some numerical results in three-dimensional domains for both the Helmholtz problem and the harmonic Maxwell equations. The second part deals with an inverse spectral problem: the Borg theorem for the vectorial Hill equation. We prove this result with the help of an estimate on the first derivative of the eigenvalues with respect to the parameter of ‘pseudo-periodicity’. This estimate seems to be new.


65N55 Multigrid methods; domain decomposition for boundary value problems involving PDEs
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
34B30 Special ordinary differential equations (Mathieu, Hill, Bessel, etc.)
34A55 Inverse problems involving ordinary differential equations
35Q60 PDEs in connection with optics and electromagnetic theory
78A40 Waves and radiation in optics and electromagnetic theory
65L15 Numerical solution of eigenvalue problems involving ordinary differential equations