Numerical solutions for Helmholtz equation with stochastic interface based on PML method. (English) Zbl 1479.78012

Summary: In this paper, the stochastic interface for diffraction grating is considered and the model is formulated as the Helmholtz interface problems (HIPs). In order to have more accuracy simulation, PML boundary is used to describe the stochastic interface. Then we develop shape-Taylor expansion for the solution of HIPs, through perturbation method, we obtain the approximate simulations of second and third order. Error estimation and efficient computation of solutions by low-rank approximation are given. Finally, we illustrate these results with numerical simulations.


78A45 Diffraction, scattering
78M10 Finite element, Galerkin and related methods applied to problems in optics and electromagnetic theory
78M31 Monte Carlo methods applied to problems in optics and electromagnetic theory
65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs
65N50 Mesh generation, refinement, and adaptive methods for boundary value problems involving PDEs
65C05 Monte Carlo methods
65N15 Error bounds for boundary value problems involving PDEs
35A15 Variational methods applied to PDEs
35B25 Singular perturbations in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
35Q60 PDEs in connection with optics and electromagnetic theory
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI


[1] Wood, R. W., On a remarkable case of uneven distribution of light in a diffraction grating spetrum, Phil. Mag., 4, 399-402 (1902)
[2] Meecham, W. C., Variational method for the calculation of the distribution of energy reflected from a periodic surface, J. Appl. Phys., 27, 361-367 (1956) · Zbl 0075.10601
[3] Mu, L.; Wang, J. P.; Ye, X., Weak Galerkin finite element methods on polytopal meshes, Int. J. Numer. Anal. Model., 12, 31-53 (2015) · Zbl 1332.65172
[4] Nedelec, J. C.; Starling, F., Integral equation methods in a quasi-periodic diffraction problem for the time-harmonic Maxwell’s equations, SIAM J. Math. Anal., 22, 1679-1701 (1991) · Zbl 0756.35004
[5] Xiu, D. B.; Karniadakis, G. E., Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187, 137-167 (2003) · Zbl 1047.76111
[6] Ikuno, H.; Yasuura, K., Improved point-matching method with application to scattering from a periodic surface, IEEE Trans. Antennas Propag., 21, 657-662 (1973)
[7] Zhou, W.; Wu, H., An adaptive finite element method for the diffraction grating problem with PML and few-mode DtN truncations, J. Sci. Comput., 76, 1813-1838 (2018) · Zbl 1396.78007
[8] Wang, Z.; Bao, G.; Li, J.; Li, P.; Wu, H., An adaptive finite element method for the diffraction grating problem with transparent boundary condition, SIAM J. Numer. Anal., 53, 1585-1607 (2015) · Zbl 1328.65249
[9] Xia, Z.; Du, K., A tensor product finite element method for the diffraction grating problem with transparent boundary conditions, Comput. Math. Appl., 73, 628-639 (2017) · Zbl 1372.78018
[10] Zheng, E.; Ma, F.; Zhang, D., A least-squares non-polynomial finite element method for solving the polygonal-line grating problem, J. Math. Anal. Appl., 397, 550-560 (2013) · Zbl 1364.65260
[11] Wang, Z.; Zhang, Y., A posterior error estimates for the nonlinear grating problem with transparent boundary condition, Numer. Methods Partial Differential Equations, 31, 1101-1118 (2015) · Zbl 1331.78029
[12] Bao, G.; Cao, Y.; Hao, Y.; Zhang, K., A robust numerical method for the random interface grating problem via shape calculus, weak Galerkin method, and low-rank approximation, J. Sci. Comput., 77, 419-442 (2018) · Zbl 1405.78012
[13] Hu, G.; Qu, F.; Zhang, B., A linear sampling method for inverse problems of diffraction gratings of mixed type, Math. Methods Appl. Sci., 35, 1047-1066 (2012) · Zbl 1251.35188
[14] Armeanu, A. M.; Edee, K.; Granet, G.; Schiavone, P., The lamellar diffraction grating problem: A spectral method based on spline expansion, Sci. Stud. Res. Ser. Math. Inform., 19, 37-46 (2009) · Zbl 1240.78026
[15] Chen, Z.; Wu, H., An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41, 799-826 (2004) · Zbl 1049.78018
[16] Delfour, M. C.; Zolesio, J. P., Shapes and Geometries — Analysis, Differential Calculus, and Optimization (2001), SIAM · Zbl 1002.49029
[17] Pironneau, O., Optimal Shape Design for Elliptic Systems (1984), Springer: Springer New York · Zbl 0496.93029
[18] Hettlich, F.; Rundell, W., The determination of a discontinuity in a conductivity from a single boundary measurement, Inverse Problems, 14, 67-82 (1998) · Zbl 0894.35126
[19] Hettlich, F.; Rundell, W., Identification of a discontinuous source in the heat equation, Inverse Problems, 17, 1465-1482 (2001) · Zbl 0986.35129
[20] Ito, K.; Kunisch, K.; Li, Z., Level-set function approach to an inverse interface problem, Inverse Problems, 17, 1225-1242 (2001) · Zbl 0986.35130
[21] Sokolowski, J.; Zolesio, J. P., Introduction to Shape Optimization: Shape Sensitivity Analysis (1992), Springer-Verlag · Zbl 0761.73003
[22] Harbrecht, H.; Li, J., First order second moment analysis for stochastic interface problems based on low-rank approximation, ESAIM Math. Model. Numer. Anal., 47, 1533-1552 (2013) · Zbl 1297.65009
[23] Harbrecht, H., A finite element method for elliptic problems with stochastic input data, Appl. Numer. Math., 60, 227-244 (2010) · Zbl 1237.65009
[24] Harbrecht, H.; Schneider, R.; Schwab, C., Sparse second moment analysis for elliptic problems in stochastic domains, Numer. Math., 109, 385-414 (2008) · Zbl 1146.65007
[25] Chen, Z.; Zou, J., Finite element methods and their convergence for elliptic and parabolic interface problems, Numer. Math., 79, 175-202 (1998) · Zbl 0909.65085
[26] Li, J.; Melenk, J. M.; Wohlmuth, B.; Zou, J., Optimal a priori estimates for higher order finite elements for elliptic interface problems, Appl. Numer. Math., 60, 19-37 (2010) · Zbl 1208.65168
[27] Bramble, J. H.; King, J. T., A finite element method for interface problems with smooth boundaries and interfaces, Adv. Comput. Math., 6, 109-138 (1996) · Zbl 0868.65081
[28] Barrett, J. W.; Elliott, C. M., Fitted and unfitted finite-element methods for elliptic equations with interfaces, IMA J. Numer. Anal., 7, 283-300 (1987) · Zbl 0629.65118
[29] Harbrecht, H.; Peters, M.; Schneider, R., On the low-rank approximation by the pivoted Cholesky decomposition, Appl. Numer. Math., 62, 428-440 (2012) · Zbl 1244.65042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.