
-
Previous Article
A $ C^1 $ Petrov-Galerkin method and Gauss collocation method for 1D general elliptic problems and superconvergence
- DCDS-B Home
- This Issue
-
Next Article
Bifurcations in periodic integrodifference equations in $ C(\Omega) $ I: Analytical results and applications
An adaptive finite element DtN method for the three-dimensional acoustic scattering problem
1. | School of Mathematical Sciences, Zhejiang University, Hangzhou 310027, China |
2. | Department of Mathematics, Purdue University, West Lafayette, IN 47907, USA |
This paper is concerned with a numerical solution of the acoustic scattering by a bounded impenetrable obstacle in three dimensions. The obstacle scattering problem is formulated as a boundary value problem in a bounded domain by using a Dirichlet-to-Neumann (DtN) operator. An a posteriori error estimate is derived for the finite element method with the truncated DtN operator. The a posteriori error estimate consists of the finite element approximation error and the truncation error of the DtN operator, where the latter is shown to decay exponentially with respect to the truncation parameter. Based on the a posteriori error estimate, an adaptive finite element method is developed for the obstacle scattering problem. The truncation parameter is determined by the truncation error of the DtN operator and the mesh elements for local refinement are marked through the finite element approximation error. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.
References:
[1] |
I. Babuška and A. Aziz, Survey Lectures on mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by A. Aziz, Academic Press, New York, 1972.
![]() |
[2] |
G. Bao, R. Delgadillo, G. Hu, D. Liu and S. Luo, Modeling and computation of nano-optics, CSIAM Trans. Appl. Math.. Google Scholar |
[3] |
G. Bao, Y. Gao and P. Li,
Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.
doi: 10.1007/s00205-018-1228-2. |
[4] |
G. Bao, G. Hu and D. Liu,
An h-adaptive finite element solver for the calculations of the electronic structures, J. Comput. Phys., 231 (2012), 4967-4979.
doi: 10.1016/j.jcp.2012.04.002. |
[5] |
G. Bao, P. Li and H. Wu,
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures, Math. Comp., 79 (2010), 1-34.
doi: 10.1090/S0025-5718-09-02257-1. |
[6] |
G. Bao and H. Wu,
Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121-2143.
doi: 10.1137/040604315. |
[7] |
A. Bayliss and E. Turkel,
Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33 (1980), 707-725.
doi: 10.1002/cpa.3160330603. |
[8] |
J.-P. Berenger,
A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.
doi: 10.1006/jcph.1994.1159. |
[9] |
Z. Chen and X. Liu,
An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.
doi: 10.1137/040610337. |
[10] |
Z. Chen and H. Wu,
An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.
doi: 10.1137/S0036142902400901. |
[11] |
F. Collino and P. Monk,
The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090.
doi: 10.1137/S1064827596301406. |
[12] |
D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983. |
[13] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition, Springer, Berlin, New York, 1998.
doi: 10.1007/978-3-662-03537-5. |
[14] |
B. Engquist and A. Majda,
Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.
doi: 10.2307/2005997. |
[15] |
Q. Fang, D. P. Nicholls and J. Shen,
A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering, J. Comput. Phys., 224 (2007), 1145-1169.
doi: 10.1016/j.jcp.2006.11.018. |
[16] |
C. Geuzaine and J.-F. Remacle,
GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), 1309-1331.
doi: 10.1002/nme.2579. |
[17] |
M. J. Grote and J. B. Keller,
On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243.
doi: 10.1006/jcph.1995.1210. |
[18] |
M. J. Grote and C. Kirsch,
Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201 (2004), 630-650.
doi: 10.1016/j.jcp.2004.06.012. |
[19] |
T. Hagstrom,
Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.
doi: 10.1017/S0962492900002890. |
[20] |
I. Harari and T. J. R. Hughes,
Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97 (1992), 103-124.
doi: 10.1016/0045-7825(92)90109-W. |
[21] |
D. Jerison and C. Kenig,
Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. Math., 121 (1985), 463-488.
doi: 10.2307/1971205. |
[22] |
X. Jiang, P. Li, J. Lv and W. Zheng,
An adaptive finite element method for the wave scattering with transparent boundary condition, J. Sci. Comput., 72 (2017), 936-956.
doi: 10.1007/s10915-017-0382-2. |
[23] |
X. Jiang, P. Li and W. Zheng,
Numerical solution of acoustic scattering by an adaptive DtN finite element method, Commun. Comput. Phys., 13 (2013), 1227-1244.
doi: 10.4208/cicp.301011.270412a. |
[24] |
J. Jin, The Finite Element Method in Electromagnetics, Second edition. New York, 2002. |
[25] |
A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-11086-8. |
[26] |
P. Li and X. Yuan, Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112722.
doi: 10.1016/j.cma.2019.112722. |
[27] |
Y. Li, W. Zheng and X. Zhu,
A CIP-FEM for high-frequency scattering problem with the truncated DtN boundary condition, CSIAM Trans. Appl. Math., 1 (2020), 530-560.
doi: 10.4208/csiam-am.2020-0025. |
[28] |
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001.![]() ![]() |
[29] |
J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. Google Scholar |
[30] |
A. H. Schatz,
An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), 959-962.
doi: 10.2307/2005357. |
[31] |
E. Turkel and A. Yefet,
Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), 533-557.
doi: 10.1016/S0168-9274(98)00026-9. |
[32] |
Z. Wang, G. Bao, J. Li, P. Li and H. Wu,
An adaptive finite element method for the diffraction grating problem with transparent boundary condition, SIAM J. Numer. Anal., 53 (2015), 1585-1607.
doi: 10.1137/140969907. |
[33] |
X. Yuan, G. Bao and P. Li,
An adaptive finite element DtN method for the open cavity scattering problems, CSIAM Trans. Appl. Math., 1 (2020), 316-345.
doi: 10.4208/csiam-am.2020-0013. |
show all references
References:
[1] |
I. Babuška and A. Aziz, Survey Lectures on mathematical foundations of the finite element method, in The Mathematical Foundations of the Finite Element Method with Application to the Partial Differential Equations, ed. by A. Aziz, Academic Press, New York, 1972.
![]() |
[2] |
G. Bao, R. Delgadillo, G. Hu, D. Liu and S. Luo, Modeling and computation of nano-optics, CSIAM Trans. Appl. Math.. Google Scholar |
[3] |
G. Bao, Y. Gao and P. Li,
Time-domain analysis of an acoustic-elastic interaction problem, Arch. Ration. Mech. Anal., 229 (2018), 835-884.
doi: 10.1007/s00205-018-1228-2. |
[4] |
G. Bao, G. Hu and D. Liu,
An h-adaptive finite element solver for the calculations of the electronic structures, J. Comput. Phys., 231 (2012), 4967-4979.
doi: 10.1016/j.jcp.2012.04.002. |
[5] |
G. Bao, P. Li and H. Wu,
An adaptive edge element method with perfectly matched absorbing layers for wave scattering by periodic structures, Math. Comp., 79 (2010), 1-34.
doi: 10.1090/S0025-5718-09-02257-1. |
[6] |
G. Bao and H. Wu,
Convergence analysis of the perfectly matched layer problems for time-harmonic Maxwell's equations, SIAM J. Numer. Anal., 43 (2005), 2121-2143.
doi: 10.1137/040604315. |
[7] |
A. Bayliss and E. Turkel,
Radiation boundary conditions for wave-like equations, Comm. Pure Appl. Math., 33 (1980), 707-725.
doi: 10.1002/cpa.3160330603. |
[8] |
J.-P. Berenger,
A perfectly matched layer for the absorption of electromagnetic waves, J. Comput. Phys., 114 (1994), 185-200.
doi: 10.1006/jcph.1994.1159. |
[9] |
Z. Chen and X. Liu,
An adaptive perfectly matched layer technique for time-harmonic scattering problems, SIAM J. Numer. Anal., 43 (2005), 645-671.
doi: 10.1137/040610337. |
[10] |
Z. Chen and H. Wu,
An adaptive finite element method with perfectly matched absorbing layers for the wave scattering by periodic structures, SIAM J. Numer. Anal., 41 (2003), 799-826.
doi: 10.1137/S0036142902400901. |
[11] |
F. Collino and P. Monk,
The perfectly matched layer in curvilinear coordinates, SIAM J. Sci. Comput., 19 (1998), 2061-2090.
doi: 10.1137/S1064827596301406. |
[12] |
D. L. Colton and R. Kress, Integral Equation Methods in Scattering Theory, John Wiley & Sons, New York, 1983. |
[13] |
D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory, Second Edition, Springer, Berlin, New York, 1998.
doi: 10.1007/978-3-662-03537-5. |
[14] |
B. Engquist and A. Majda,
Absorbing boundary conditions for the numerical simulation of waves, Math. Comp., 31 (1977), 629-651.
doi: 10.2307/2005997. |
[15] |
Q. Fang, D. P. Nicholls and J. Shen,
A stable, high-order method for three-dimensional, bounded-obstacle, acoustic scattering, J. Comput. Phys., 224 (2007), 1145-1169.
doi: 10.1016/j.jcp.2006.11.018. |
[16] |
C. Geuzaine and J.-F. Remacle,
GMSH: A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities, Internat. J. Numer. Methods Engrg., 79 (2009), 1309-1331.
doi: 10.1002/nme.2579. |
[17] |
M. J. Grote and J. B. Keller,
On nonreflecting boundary conditions, J. Comput. Phys., 122 (1995), 231-243.
doi: 10.1006/jcph.1995.1210. |
[18] |
M. J. Grote and C. Kirsch,
Dirichlet-to-Neumann boundary conditions for multiple scattering problems, J. Comput. Phys., 201 (2004), 630-650.
doi: 10.1016/j.jcp.2004.06.012. |
[19] |
T. Hagstrom,
Radiation boundary conditions for the numerical simulation of waves, Acta Numerica, 8 (1999), 47-106.
doi: 10.1017/S0962492900002890. |
[20] |
I. Harari and T. J. R. Hughes,
Analysis of continuous formulations underlying the computation of time-harmonic acoustics in exterior domains, Comput. Methods Appl. Mech. Engrg., 97 (1992), 103-124.
doi: 10.1016/0045-7825(92)90109-W. |
[21] |
D. Jerison and C. Kenig,
Unique continuation and absence of positive eigenvalues for Schrodinger operators, Ann. Math., 121 (1985), 463-488.
doi: 10.2307/1971205. |
[22] |
X. Jiang, P. Li, J. Lv and W. Zheng,
An adaptive finite element method for the wave scattering with transparent boundary condition, J. Sci. Comput., 72 (2017), 936-956.
doi: 10.1007/s10915-017-0382-2. |
[23] |
X. Jiang, P. Li and W. Zheng,
Numerical solution of acoustic scattering by an adaptive DtN finite element method, Commun. Comput. Phys., 13 (2013), 1227-1244.
doi: 10.4208/cicp.301011.270412a. |
[24] |
J. Jin, The Finite Element Method in Electromagnetics, Second edition. New York, 2002. |
[25] |
A. Kirsch and F. Hettlich, The Mathematical Theory of Time-Harmonic Maxwell's Equations, Springer International Publishing, 2015.
doi: 10.1007/978-3-319-11086-8. |
[26] |
P. Li and X. Yuan, Convergence of an adaptive finite element DtN method for the elastic wave scattering by periodic structures, Comput. Methods Appl. Mech. Engrg., 360 (2020), 112722.
doi: 10.1016/j.cma.2019.112722. |
[27] |
Y. Li, W. Zheng and X. Zhu,
A CIP-FEM for high-frequency scattering problem with the truncated DtN boundary condition, CSIAM Trans. Appl. Math., 1 (2020), 530-560.
doi: 10.4208/csiam-am.2020-0025. |
[28] |
P. Monk, Finite Element Methods for Maxwell's Equations, Oxford University Press, New York, 2003.
doi: 10.1093/acprof:oso/9780198508885.001.0001.![]() ![]() |
[29] |
J.-C. Nédélec, Acoustic and Electromagnetic Equations Integral Representations for Harmonic Problems, Springer-Verlag, New York, 2001. Google Scholar |
[30] |
A. H. Schatz,
An observation concerning Ritz–Galerkin methods with indefinite bilinear forms, Math. Comp., 28 (1974), 959-962.
doi: 10.2307/2005357. |
[31] |
E. Turkel and A. Yefet,
Absorbing PML boundary layers for wave-like equations, Appl. Numer. Math., 27 (1998), 533-557.
doi: 10.1016/S0168-9274(98)00026-9. |
[32] |
Z. Wang, G. Bao, J. Li, P. Li and H. Wu,
An adaptive finite element method for the diffraction grating problem with transparent boundary condition, SIAM J. Numer. Anal., 53 (2015), 1585-1607.
doi: 10.1137/140969907. |
[33] |
X. Yuan, G. Bao and P. Li,
An adaptive finite element DtN method for the open cavity scattering problems, CSIAM Trans. Appl. Math., 1 (2020), 316-345.
doi: 10.4208/csiam-am.2020-0013. |






1 | Given a tolerance |
2 | Choose |
3 | Construct an initial tetrahedral partition |
4 | While |
5 | mark |
6 | solve the discrete problem on the |
7 | compute the corresponding error estimators; |
8 | End while. |
1 | Given a tolerance |
2 | Choose |
3 | Construct an initial tetrahedral partition |
4 | While |
5 | mark |
6 | solve the discrete problem on the |
7 | compute the corresponding error estimators; |
8 | End while. |
[1] |
Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 |
[2] |
Murat Uzunca, Ayşe Sarıaydın-Filibelioǧlu. Adaptive discontinuous galerkin finite elements for advective Allen-Cahn equation. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 269-281. doi: 10.3934/naco.2020025 |
[3] |
Qiang Guo, Dong Liang. An adaptive wavelet method and its analysis for parabolic equations. Numerical Algebra, Control & Optimization, 2013, 3 (2) : 327-345. doi: 10.3934/naco.2013.3.327 |
[4] |
Armin Lechleiter, Tobias Rienmüller. Factorization method for the inverse Stokes problem. Inverse Problems & Imaging, 2013, 7 (4) : 1271-1293. doi: 10.3934/ipi.2013.7.1271 |
[5] |
Yizhuo Wang, Shangjiang Guo. A SIS reaction-diffusion model with a free boundary condition and nonhomogeneous coefficients. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1627-1652. doi: 10.3934/dcdsb.2018223 |
[6] |
Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1757-1778. doi: 10.3934/dcdss.2020453 |
[7] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[8] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[9] |
Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028 |
[10] |
Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems - A, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25 |
[11] |
Mikhail Gilman, Semyon Tsynkov. Statistical characterization of scattering delay in synthetic aperture radar imaging. Inverse Problems & Imaging, 2020, 14 (3) : 511-533. doi: 10.3934/ipi.2020024 |
[12] |
Y. Latushkin, B. Layton. The optimal gap condition for invariant manifolds. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 233-268. doi: 10.3934/dcds.1999.5.233 |
[13] |
Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024 |
[14] |
Arseny Egorov. Morse coding for a Fuchsian group of finite covolume. Journal of Modern Dynamics, 2009, 3 (4) : 637-646. doi: 10.3934/jmd.2009.3.637 |
[15] |
Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321 |
[16] |
Carlos Gutierrez, Nguyen Van Chau. A remark on an eigenvalue condition for the global injectivity of differentiable maps of $R^2$. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 397-402. doi: 10.3934/dcds.2007.17.397 |
[17] |
Mansour Shrahili, Ravi Shanker Dubey, Ahmed Shafay. Inclusion of fading memory to Banister model of changes in physical condition. Discrete & Continuous Dynamical Systems - S, 2020, 13 (3) : 881-888. doi: 10.3934/dcdss.2020051 |
[18] |
Enkhbat Rentsen, Battur Gompil. Generalized Nash equilibrium problem based on malfatti's problem. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 209-220. doi: 10.3934/naco.2020022 |
[19] |
Alexandr Mikhaylov, Victor Mikhaylov. Dynamic inverse problem for Jacobi matrices. Inverse Problems & Imaging, 2019, 13 (3) : 431-447. doi: 10.3934/ipi.2019021 |
[20] |
Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182 |
2019 Impact Factor: 1.27
Tools
Metrics
Other articles
by authors
[Back to Top]