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 -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 -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.
Accepted:
Published online:
Keywords: Hyperbolic boundary value problem, PML, trihedral corner, dissipative boundary conditions, PML, Bérenger, Pauli system, Laplace transform, holomorphy.
Laurence Halpern 1; Jeffrey B. Rauch 2
CC-BY 4.0
@article{AFST_2024_6_33_2_361_0,
author = {Laurence Halpern and Jeffrey B. Rauch},
title = {Perfectly {Matched} {Layers} on {Cubic} {Domains} for {Pauli{\textquoteright}s} {Equations}},
journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
pages = {361--403},
publisher = {Universit\'e Paul Sabatier, Toulouse},
volume = {Ser. 6, 33},
number = {2},
year = {2024},
doi = {10.5802/afst.1774},
language = {en},
url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1774/}
}
TY - JOUR AU - Laurence Halpern AU - Jeffrey B. Rauch TI - Perfectly Matched Layers on Cubic Domains for Pauli’s Equations JO - Annales de la Faculté des sciences de Toulouse : Mathématiques PY - 2024 SP - 361 EP - 403 VL - 33 IS - 2 PB - Université Paul Sabatier, Toulouse UR - https://afst.centre-mersenne.org/articles/10.5802/afst.1774/ DO - 10.5802/afst.1774 LA - en ID - AFST_2024_6_33_2_361_0 ER -
%0 Journal Article %A Laurence Halpern %A Jeffrey B. Rauch %T Perfectly Matched Layers on Cubic Domains for Pauli’s Equations %J Annales de la Faculté des sciences de Toulouse : Mathématiques %D 2024 %P 361-403 %V 33 %N 2 %I Université Paul Sabatier, Toulouse %U https://afst.centre-mersenne.org/articles/10.5802/afst.1774/ %R 10.5802/afst.1774 %G en %F AFST_2024_6_33_2_361_0
Laurence Halpern; Jeffrey B. Rauch. Perfectly Matched Layers on Cubic Domains for Pauli’s Equations. Annales de la Faculté des sciences de Toulouse : Mathématiques, Serie 6, Volume 33 (2024) no. 2, pp. 361-403. doi : 10.5802/afst.1774. https://afst.centre-mersenne.org/articles/10.5802/afst.1774/
[1] On the construction and analysis of absorbing layers in CEM, Appl. Numer. Math., Volume 27 (1998) no. 4, pp. 331-340 | DOI | Zbl
[2] Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I, Commun. Pure Appl. Math., Volume 12 (1959) no. 4, pp. 623-727 | DOI | Zbl
[3] Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions II, Commun. Pure Appl. Math., Volume 17 (1964) no. 1, pp. 35-92 | DOI | Zbl
[4] Perfectly matched layers for hyperbolic systems: General formulation, well-posedness and stability, SIAM J. Appl. Math., Volume 67 (2006) no. 1, pp. 1-23 | DOI | Zbl
[5] Energy decay and stability of a perfectly matched layer for the wave equation, J. Sci. Comput., Volume 81 (2019) no. 3, pp. 2237-2270 | DOI | Zbl
[6] Spectral properties of many-body Schrödinger operators with dilatation-analytic interactions, Commun. Math. Phys., Volume 22 (1971) no. 4, pp. 280-294 | DOI | Zbl
[7] On the analysis of Bérenger’s perfectly matched layers for Maxwell’s equations, M2AN, Math. Model. Numer. Anal., Volume 36 (2002) no. 1, pp. 87-119 | DOI | Numdam | Zbl
[8] Analysis of a Cartesian PML approximation to acoustic scattering problems in and , J. Comput. Appl. Math., Volume 247 (2013), pp. 209-230 | DOI | Zbl
[9] Convergence of the PML method for elastic wave scattering problems, Math. Comput., Volume 85 (2016) no. 302, pp. 2687-2714 | DOI | Zbl
[10] A 3D perfectly matched medium from modified Maxwell’s equations with stretched coordinates, IEEE Microwave Opt. Technol. Lett., Volume 17 (1994), pp. 599-604 | DOI
[11] The Perfectly Matched Layer in Curvilinear Coordinates, SIAM J. Sci. Comput., Volume 19 (1998) no. 6, pp. 2061-2090 | DOI | Zbl
[12] Transparent numerical boundary conditions for evolution equations: Derivation and stability analysis, Ann. Fac. Sci. Toulouse, Math., Volume 28 (2019) no. 2, pp. 259-327 | DOI | Numdam | Zbl
[13] A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Eng., Volume 195 (2006) no. 29-32, pp. 3820-3853 | DOI | Zbl
[14] Mathematical theory of scattering resonances, Graduate Studies in Mathematics, 200, American Mathematical Society, 2019 | DOI
[15] The analysis of matched layers, Confluentes Math., Volume 3 (2011) no. 2, pp. 159-236 | DOI | Numdam | Zbl
[16] Hyperbolic boundary value problems with trihedral corners, Discrete Contin. Dyn. Syst., Volume 36 (2016) no. 8, pp. 4403-4450 | DOI | MR | Zbl
[17] Notes on differential geometry, 3, van Nostrand; Princeton University Press, 1965
[18] Functional analysis and semi-groups, Colloquium Publications, 31, American Mathematical Society, 1996
[19] Perturbation theory for linear operators, Grundlehren der Mathematischen Wissenschaften, 132, Springer, 2013
[20] On the existence and convergence of the solution of PML equations, Computing, Volume 60 (1998) no. 3, pp. 229-241 | DOI
[21] Analysis of the PML equations in general convex geometry, Proc. R. Soc. Edinb., Sect. A, Math., Volume 131 (2001) no. 5, pp. 1183-1207 | DOI | Zbl
[22] Problèmes faiblement bien posés : discrétisation et applications, Ph. D. Thesis, Université Paris 13 (2006) (https://tel.archives-ouvertes.fr/tel-00545794/fr/)
[23] Hyperbolic Partial Differential Equations and Geometric Optics, Graduate Studies in Mathematics, 133, American Mathematical Society, 2012 | DOI
Cited by Sources: