Perfectly Matched Layers on Cubic Domains for Pauli’s Equations

Halpern, Laurence; Rauch, Jeffrey B.

Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 33 (2024) no. 2, pp. 361-403

This article proves the wellposedness of the boundary value problem that arises when PML algorithms are applied to Pauli’s equations with a three dimensional rectangle as computational domain. The absorptions are positive near the boundary and zero far from the boundary so are always $x$-dependent. At the flat parts of the boundary of the rectangle, the natural absorbing boundary conditions are imposed. The difficulty addressed is the analysis of the resulting variable coefficient problem on the rectangular solid with its edges and corners. The Laplace transform is analysed. We derive an additional boundary condition that is automatically satisfied and yields a coercive Helmholtz boundary value problem on smoothed boundaries with uniform estimates justifying the limit of vanishing smoothing.

Uniqueness is reduced by an analyticity argument to our uniqueness theorem for symmetric hyperbolic problems in domains with trihedral corners [16]. This yields the first stability proof with $x$-dependent absorptions on a domain whose boundary is not smooth.

Cet article établit que le problème aux limites associé aux algorithmes PML appliqués au système de Pauli dans un domaine parallélépipédique rectangle est bien posé. Les absorptions sont des fonctions positives de la variable d’espace, strictement positives au voisinage de la frontière et nulles loin de la frontière. Les conditions aux limites absorbantes naturelles sont imposées sur les faces du parallélépipède. Les difficultés sont de deux ordres : les coefficients sont variables, et la frontière du domaine contient faces, arêtes et coins. Pour l’analyse, nous utilisons la transformée de Laplace en temps du système. Nous dérivons une condition aux limites supplémentaire et obtenons un problème aux limites de Helmholtz, coercif sur un domaine régularisé. Nous obtenons sur ce problème des estimations uniformes qui justifient le passage à la limite vers le problème avec coin. Enfin un argument d’analyticité permet d’utiliser le résultat d’unicité établi dans [16] pour les problèmes symétriques hyperboliques dans les domaines contenant un coin triédral. Ce travail fournit la première preuve de stabilité pour un problème à absorption variable dans un domaine dont la frontière n’est pas régulière.

Classification : 35F46, 35J25, 35L20, 35L53, 35Q40, 65N12
Keywords: Hyperbolic boundary value problem, PML, trihedral corner, dissipative boundary conditions, PML, Bérenger, Pauli system, Laplace transform, holomorphy.

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