[1] Antoine, X., Arnold, A., Besse, C., Ehrhardt, M., Schädle, A., A review of transparent and artificial boundary conditions techniques for linear and nonlinear Schrödinger equations. Commun. Comput. Phys. 4:4 (2008), 729–796.
[3] Givoli, D., Numerical Methods for Problems in Infinite Domains. 1992, Elsevier.
[4] Givoli, D., Computational absorbing boundaries. Computational Acoustics of Noise Propagation in Fluids, 2008, Springer, Berlin, 145–166 (Chapter 5).
[5] Hagstrom, T., Radiation boundary conditions for the numerical simulation of waves. Acta Numer. 8 (1999), 47–106.
[6] Hagstrom, T., Radiation boundary conditions for Maxwell's equations: a review of accurate time-domain formulations. J. Comput. Math. 25 (2007), 305–336.
[7] Tsynkov, S.V., Numerical solution of problems on unbounded domains. A review. Appl. Numer. Math. 27:4 (1998), 465–532.
[8] Schmidt, K., Diaz, J., Heier, C., Non-conforming Galerkin finite element methods for local absorbing boundary conditions of higher order. Comput. Math. Appl. 70:9 (2015), 2252–2269.
[9] Antoine, X., Barucq, H., Bendali, A., Bayliss-Turkel-like radiation conditions on surfaces of arbitrary shape. J. Math. Anal. Appl. 229:1 (1999), 184–211.
[10] Givoli, D., High-order local non-reflecting boundary conditions: a review. Wave Motion 39:4 (2004), 319–326.
[11] Hagstrom, T., Givoli, D., Rabinovich, D., Bielak, J., The double absorbing boundary method. J. Comput. Phys. 259:0 (2014), 220–241.
[12] Appelö, D., Hagstrom, T., Kreiss, G., Perfectly matched layers for hyperbolic systems: General formulation, well-posedness, and stability. SIAM J. Appl. Math. 67:1 (2006), 1–23.
[13] Bérenger, J.-P., A perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 114:2 (1994), 185–200.
[14] Bérenger, J.-P., Perfectly Matched Layer (PML) for Computational Electromagnetics. 2007, Morgan & Claypool.
[15] Bermúdez, A., Hervella-Nieto, L., Prieto, A., Rodríguez, R., Perfectly matched layers for time-harmonic second order elliptic problems. Arch. Comput. Methods Eng. 17:1 (2010), 77–107.
[16] Katz, D., Thiele, E., Taflove, A., Validation and extension to three dimensions of the Bérenger PML absorbing boundary condition for FD-TD meshes. IEEE Microw. Guid. Wave Lett. 4:8 (1994), 268–270.
[17] Hu, F.Q., On absorbing boundary conditions for linearized Euler equations by a perfectly matched layer. J. Comput. Phys. 129:1 (1996), 201–219.
[18] Hu, F.Q., Development of PML absorbing boundary conditions for computational aeroacoustics: A progress review. Comput. & Fluids 37:4 (2008), 336–348.
[19] Hastings, F.D., Schneider, J.B., Broschat, S.L., Application of the perfectly matched layer (PML) absorbing boundary condition to elastic wave propagation. J. Acoust. Soc. Am., 100, 1996, 3061.
[20] Chew, W.C., Weedon, W.H., A 3D perfectly matched medium from modified Maxwell's equations with stretched coordinates. Microw. Opt. Technol. Lett. 7:13 (1994), 599–604.
[21] Rappaport, C., Interpreting and improving the PML absorbing boundary condition using anisotropic lossy mapping of space. IEEE Trans. Magn. 32:3 (1996), 968–974.
[22] Teixeira, F.L., Chew, W.C., Differential forms, metrics, and the reflectionless absorption of electromagnetic waves. J. Electromagn. Waves Appl. 13:5 (1999), 665–686.
[23] Lassas, M., Liukkonen, J., Somersalo, E., Complex Riemannian metric and absorbing boundary condition. J. Math. Pures Appl. 80:7 (2001), 739–768.
[24] Gedney, S.D., An anisotropic perfectly matched layer-absorbing medium for the truncation of FDTD lattices. IEEE Trans. Antennas and Propagation 44:12 (1996), 1630–1639.
[25] Sacks, Z.S., Kingsland, D.M., Lee, R., Lee, J.-F., A perfectly matched anisotropic absorber for use as an absorbing boundary condition. IEEE Trans. Antennas and Propagation 43:12 (1995), 1460–1463.
[26] Zhao, L., Cangellaris, A.C., GT-PML: Generalized theory of perfectly matched layers and its application to the reflectionless truncation of finite-difference time-domain grids. IEEE Trans. Microw. Theory Tech. 44:12 (1996), 2555–2563.
[27] Nataf, F., A new approach to perfectly matched layers for the linearized Euler system. J. Comput. Phys. 214:2 (2006), 757–772.
[28] Hu, F.Q., Li, X.D., Lin, D.K., Absorbing boundary conditions for nonlinear Euler and Navier-Stokes equations based on the perfectly matched layer technique. J. Comput. Phys. 227:9 (2008), 4398–4424.
[29] Lavelle, J.W., Thacker, W.C., A pretty good sponge: Dealing with open boundaries in limited-area ocean models. Ocean Modell. 20:3 (2008), 270–292.
[30] Navon, I.M., Neta, B., Hussaini, M.Y., A perfectly matched layer approach to the linearized shallow water equations models. Mon. Weather Rev. 132:6 (2004), 1369–1378.
[31] Bécache, E., Fauqueux, S., Joly, P., Stability of perfectly matched layers, group velocities and anisotropic waves. J. Comput. Phys. 188:2 (2003), 399–433.
[32] Chew, W.C., Liu, Q.H., Perfectly matched layers for elastodynamics: A new absorbing boundary condition. J. Comput. Acoust. 4:4 (1996), 341–359.
[33] Collino, F., Tsogka, C., Application of the perfectly matched absorbing layer model to the linear elastodynamic problem in anisotropic heterogeneous media. Geophysics 66:1 (2001), 294–307.
[34] Basu, U., Chopra, A.K., Perfectly matched layers for time-harmonic elastodynamics of unbounded domains: theory and finite-element implementation. Comput. Methods Appl. Mech. Engrg. 192:11–12 (2003), 1337–1375.
[35] Basu, U., Chopra, A.K., Perfectly matched layers for transient elastodynamics of unbounded domains. Internat. J. Numer. Methods Engrg. 59:8 (2004), 1039–1074.
[36] Meza-Fajardo, K., Papageorgiou, A., A nonconvolutional, split-field, perfectly matched layer for wave propagation in isotropic and anisotropicelastic media: Stability analysis. Bull. Seismol. Soc. Amer. 98:4 (2008), 1811–1836.
[37] Zheng, C., A perfectly matched layer approach to the nonlinear Schrödinger wave equations. J. Comput. Phys. 227:1 (2007), 537–556.
[38] Chew, W.C., Jin, J.M., Michielssen, E., Complex coordinate stretching as a generalized absorbing boundary condition. Microw. Opt. Technol. Lett. 15:6 (1997), 363–369.
[39] Teixeira, F.L., Chew, W.C., PML-FDTD in cylindrical and spherical grids. IEEE Microw. Guid. Wave Lett. 7:9 (1997), 285–287.
[40] Teixeira, F.L., Chew, W.C., Systematic derivation of anisotropic PML absorbing media in cylindrical and spherical coordinates. IEEE Microw. Guid. Wave Lett. 7:11 (1997), 371–373.
[41] Collino, F., Monk, P.B., The perfectly matched layer in curvilinear coordinates. SIAM J. Sci. Comput. 19 (1998), 2061–2090.
[42] Petropoulos, P.G., Reflectionless sponge layers as absorbing boundary conditions for the numerical solution of Maxwell equations in rectangular, cylindrical, and spherical coordinates. SIAM J. Appl. Math. 60:3 (2000), 1037–1058.
[43] Teixeira, F.L., Chew, W.C., Analytical derivation of a conformal perfectly matched absorber for electromagnetic waves. Microw. Opt. Technol. Lett. 17:4 (1998), 231–236.
[44] Teixeira, F.L., Chew, W.C., Complex space approach to perfectly matched layers: a review and some new developments. Int. J. Numer. Model.-Electron. Netw. Dev. Fields 13:5 (2000), 441–455.
[45] Lassas, M., Somersalo, E., Analysis of the PML equations in general convex geometry. Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 1183–1207.
[46] Zschiedrich, L., Klose, R., Schädle, A., Schmidt, F., A new finite element realization of the perfectly matched layer method for Helmholtz scattering problems on polygonal domains in two dimensions. J. Comput. Appl. Math. 188:1 (2006), 12–32.
[48] Roden, J.A., Gedney, S.D., Efficient implementation of the uniaxial-based PML media in three-dimensional nonorthogonal coordinates with the use of the FDTD technique. Microw. Opt. Technol. Lett. 14:2 (1997), 71–75.
[49] Teixeira, F., Hwang, K.-P., Chew, W., Jin, J.-M., Conformal PML-FDTD schemes for electromagnetic field simulations: A dynamic stability study. IEEE Trans. Antennas and Propagation 49:6 (2001), 902–907.
[50] Donderici, B., Teixeira, F.L., Conformal perfectly matched layer for the mixed finite element time-domain method. IEEE Trans. Antennas and Propagation 56:4 (2008), 1017–1026.
[51] Dosopoulos, S., Lee, J.-F., Interior penalty discontinuous Galerkin finite element method for the time-dependent first order Maxwell's equations. IEEE Trans. Antennas and Propagation 58:12 (2010), 4085–4090.
[53] Guddati, M.N., Lim, K.-W., Continued fraction absorbing boundary conditions for convex polygonal domains. Internat. J. Numer. Methods Engrg. 66:6 (2006), 949–977.
[54] Demaldent, E., Imperiale, S., Perfectly matched transmission problem with absorbing layers: Application to anisotropic acoustics in convex polygonal domains. Internat. J. Numer. Methods Engrg. 96:11 (2013), 689–711.
[55] Do Carmo, M.P., Differential Geometry of Curves and Surfaces. 1976, Prentice-hall, Englewood Cliffs.
[56] Teixeira, F.L., Chew, W.C., Unified analysis of perfectly matched layers using differential forms. Microw. Opt. Technol. Lett. 20:2 (1999), 124–126.
[57] Bérenger, J.-P., Three-dimensional perfectly matched layer for the absorption of electromagnetic waves. J. Comput. Phys. 127:2 (1996), 363–379.
[58] Rahmouni, A.N., An algebraic method to develop well-posed PML models: Absorbing layers, perfectly matched layers, linearized Euler equations. J. Comput. Phys. 197:1 (2004), 99–115.
[59] Chen, Z., Wu, X., Long-time stability and convergence of the uniaxial perfectly matched layer method for time-domain acoustic scattering problems. SIAM J. Numer. Anal. 50:5 (2012), 2632–2655.
[60] Abarbanel, S., Gottlieb, D., On the construction and analysis of absorbing layers in CEM. Appl. Numer. Math. 27:4 (1998), 331–340.
[61] Abarbanel, S., Gottlieb, D., A mathematical analysis of the PML method. J. Comput. Phys. 134:2 (1997), 357–363.
[62] Bécache, E., Joly, P., On the analysis of Bérenger's perfectly matched layers for Maxwell's equations. ESAIM Math. Model. Numer. Anal. 36:1 (2002), 87–119.
[63] Bécache, E., Prieto, A., Remarks on the stability of cartesian PMLs in corners. Appl. Numer. Math. 62:11 (2012), 1639–1653.
[64] Duru, K., The role of numerical boundary procedures in the stability of perfectly matched layers. SIAM J. Sci. Comput. 38:2 (2016), A1171–A1194.
[65] Halpern, L., Petit-Bergez, S., Rauch, J., The analysis of matched layers. Confluentes Math. 3:02 (2011), 159–236.
[66] Kaltenbacher, B., Kaltenbacher, M., Sim, I., A modified and stable version of a perfectly matched layer technique for the 3-D second order wave equation in time domain with an application to aeroacoustics. J. Comput. Phys. 235 (2013), 407–422.
[67] Halpern, L., Rauch, J., Hyperbolic Boundary Value Problems with Trihedral Corners AIMS Series in Applied Mathematics, 2016.
[68] Diaz, J., Joly, P., A time domain analysis of PML models in acoustics. Comput. Methods Appl. Mech. Engrg. 195:29 (2006), 3820–3853.
[69] Bécache, E., Petropoulos, P.G., Gedney, S.D., On the long-time behavior of unsplit perfectly matched layers. IEEE Trans. Antennas and Propagation 52:5 (2004), 1335–1342.
[70] Kuzuoglu, M., Mittra, R., Frequency dependence of the constitutive parameters of causal perfectly matched anisotropic absorbers. IEEE Microw. Guid. Wave Lett. 6:12 (1996), 447–449.
[71] Roden, J.A., Gedney, S.D., Convolutional PML (CPML): An efficient FDTD implementation of the CFS-PML for arbitrary media. Microw. Opt. Technol. Lett. 27:5 (2000), 334–338.
[72] Hesthaven, J.S., Warburton, T., Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications, Vol. 54. 2008, Springer-Verlag, New York.
[73] LeVeque, R.J., Finite Volume Methods for Hyperbolic Problems, Vol. 31. 2002, Cambridge University Press.
[74] Geuzaine, C., Remacle, J.-F., Gmsh: A 3-D finite element mesh generator with built-in pre-and post-processing facilities. Internat. J. Numer. Methods Engrg. 79:11 (2009), 1309–1331.
[75] Bermúdez, A., Hervella-Nieto, L., Prieto, A., Rodríguez, R., An exact bounded perfectly matched layer for time-harmonic scattering problems. SIAM J. Sci. Comput. 30:1 (2007), 312–338.
[76] Bermúdez, A., Hervella-Nieto, L., Prieto, A., Rodríguez, R., An optimal perfectly matched layer with unbounded absorbing function for time-harmonic acoustic scattering problems. J. Comput. Phys. 223:2 (2007), 469–488.
[77] Modave, A., Kameni, A., Lambrechts, J., Delhez, E., Pichon, L., Geuzaine, C., An optimum PML for scattering problems in the time domain. Eur. Phys. J. Appl. Phys., 64, 2013 11.
[78] Modave, A., Delhez, E., Geuzaine, C., Optimizing perfectly matched layers in discrete contexts. Internat. J. Numer. Methods Engrg. 99:6 (2014), 410–437.
[79] E.W. Weisstein, Ellipse. http://mathworld.wolfram.com/Ellipse.html, From MathWorld–A Wolfram Web Resource, 2015.
[80] E.W. Weisstein, Ellipsoid. http://mathworld.wolfram.com/Ellipsoid.html, From MathWorld–A Wolfram Web Resource, 2015.
