The analysis of matched layers
Confluentes Mathematici, Volume 3 (2011) no. 2, pp. 159-236.

A systematic analysis of matched layers is undertaken with special attention to better understand the remarkable method of Bérenger. We prove that the Bérenger and closely related layers define well-posed transmission problems in great generality. When the Bérenger method or one of its close relatives is well-posed, perfect matching is proved. The proofs use the energy method, Fourier–Laplace transform, and real coordinate changes for Laplace transformed equations. It is proved that the loss of derivatives associated with the Bérenger method does not occur for elliptic generators. More generally, an essentially necessary and sufficient condition for loss of derivatives in Bérenger's method is proved. The sufficiency relies on the energy method with pseudodifferential multiplier. Amplifying and nonamplifying layers are identified by a geometric optics computation. Among the various flavors of Bérenger's algorithm for Maxwell's equations, our favorite choice leads to a strongly well-posed augmented system and is both perfect and nonamplifying in great generality. We construct by an extrapolation argument an alternative matched layer method which preserves the strong hyperbolicity of the original problem and though not perfectly matched has leading reflection coefficient equal to zero at all angles of incidence. Open problems are indicated throughout.

Published online:
DOI: 10.1142/S1793744211000291
Laurence Halpern 1; Sabrina Petit-Bergez 1; Jeffrey B. Rauch 1

1
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Laurence Halpern; Sabrina Petit-Bergez; Jeffrey B. Rauch. The analysis of matched layers. Confluentes Mathematici, Volume 3 (2011) no. 2, pp. 159-236. doi : 10.1142/S1793744211000291. https://cml.centre-mersenne.org/articles/10.1142/S1793744211000291/

[1] S. Abarbanel and D. Gottlieb, A mathematical analysis of the PML method, J. Com- put. Phys. 134 (1997) 357–363.

[2] S. Abarbanel and D. Gottlieb, On the construction and analysis of absorbing layers in cem, Appl. Numer. Math. 27 (1998) 331–340.

[3] D. Appelö, T. Hagström and G. Kreiss, Perfectly matched layers for hyperbolic sys- tems: General formulation, well-posedness and stability, SIAM J. Appl. Math. 67 (2006) 1–23.

[4] C. Bardos and J. Rauch, Maximal positive boundary value problems as limits of singular perturbation problems, Trans. Amer. Math. Soc. (1982) 377–408.

[5] E. Bécache, S. Fauqueux and P. Joly, Stability of perfectly matched layers, group velocities and anisotropic waves, J. Comput. Phys. 188 (2003) 399–433.

[6] M. D. Bronshtein, Smoothness of roots of polynomials depending on parameters, Sib. Mat. Zh. 20 (1979) 493–501, in Russian [English transl., Siberian Math. J. 20 (1980) 347–352].

[7] M. D. Bronshtein, The Cauchy problem for hyperbolic operators with characteristics of variable multiplicity, Trudy Moskov. Mat. Obshch. 41 (1980) 83–99, in Russian [English transl., Trans. Moscow. Math. Soc., No. 1 (1982) 87–103.

[8] J. Chazarain and A. Piriou, Introduction à la Théorie des Équations aux Dérivées Partielles Linéaires (Gauthier-Villars, 1981).

[9] W. C. Chew and W. H. Weedon, A 3d perfectly matched medium from modified Maxwell’s equations with stretched coordinates, IEEE Microwave and Optical Tech. Lett. 17 (1995) 599–604.

[10] J. Diaz and P. Joly, A time domain analysis of PML models in acoustics, Comput. Methods Appl. Mech. Engrg. 195 (2006) 3820–3853.

[11] R. Hersh, Mixed problems in several variables, J. Math. Mech. 12 (1963) 317–334.

[12] J. S. Hesthaven, On the analysis and construction of perfectly matched layers for the linearized Euler equations, J. Comput. Phys. 142 (1998) 129–147.

[13] L. Hörmander, The Analysis of Linear Partial Differential Operators. II. Differential Operators with Constant Coefficients (Springer-Verlag, 2005).

[14] F. Q Hu, On absorbing boundary conditions of linearized Euler equations by a per- fectly matched layer, J. Comput. Phys. 129 (1996) 201–219.

[15] F. Q Hu, A stable, perfectly matched layer for linearized Euler equations in unsplit physical variables, J. Comput. Phys. 173 (2001) 455–480.

[16] M. Israeli and S. A. Orszag, Approximation of radiation boundary conditions, J. Comput. Phys. 41 (1981) 115–135.

[17] J. L. Joly, G. Métivier and J. Rauch, Hyperbolic domains of determinacy and Hamilton–Jacobi equations, J. Hyp. Part. Differential Eqns. 2 (2005) 713–744.

[18] K. Kasahara, On weak well-posedness of mixed problems for hyperbolic systems, Publ. Res. Inst. Math. Sci. 6 (1970/71) 503–514.

[19] H.-O. Kreiss and J. Lorenz, Initial-Boundary Value Problems and the Navier–Stokes Equations (Academic Press, 1989).

[20] P. A. Mazet, S. Paintandre and A. Rahmouni, Interprétation dispersive du milieu PML de Bérenger, C. R. Math. Acad. Sci. Paris 327 (1998) 59–64.

[21] L. Métivier, Utilisation des équations Euler–PML en milieu hétérogène borné pour la résolution d’un problème inverse en géophysique, in ESAIM Proc, CANUM 2008, (EDP Sciences, 2009), pp. 156–170.

[22] J. Métral and O. Vacus, Caractère bien posé du problème de Cauchy pour le système de Bérenger, C. R. Math. Acad. Sci. Paris 328 (1999) 847–852.

[23] T. Nishitani, Energy inequality for non strictly hyperbolic operators in Gevrey class, J. Math. Kyoto Univ. 23 (1983) 739–773.

[24] T. Nishitani, Sur les équations hyperboliques à coefficients hölderiens en t et de classe Gevrey en x, Bull. Sci. Math. 107 (1983) 113–138.

[25] S. Petit-Bergez, Problèmes faiblement bien posés: Discrétisation et applications, Ph.D. thesis, Université Paris 13, 2006. http://tel.archives-ouvertes.fr/tel-00545794/fr/.

[26] P. G. Petropoulos, Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell’s equations in rectangular, cylindrical and spherical coordinates, SIAM J. Appl. Math. 60 (2000) 1037–1058.

[27] L. Zhao, P. G. Petropoulos and A. C. Cangellaris, A reflectionless sponge layer absorb- ing boundary condition for the solution of Maxwell’s equations with high-order stag- gered finite difference schemes, J. Comput. Phys. 139 (1998) 184–208.

[28] A. Rahmouni, Un modèle PML bien posé pour l’élastodynamique anisotrope, C. R. Math. Acad. Sci. Paris 338 (2004) 963–968.

[29] C. M. Rappaport, Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space, IEEE Trans. Magn. 32 (1996) 968–974.

[30] M. Taylor, Pseudodifferential Operators (Princeton Univ. Press, 1981).

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