doi: 10.3934/dcdsb.2020046

Discontinuous galerkin method for the helmholtz transmission problem in two-level homogeneous media

1. 

School of Mathematics and Statistics, Xi'an Jiaotong University, Xi'an, Shaanxi 710049, China

2. 

School of Mathematics and Statistics and Institute of Applied Mathematics, Henan University, Kaifeng 475004, China

* Corresponding author: Yinnian He

Received  January 2019 Revised  August 2019 Published  February 2020

Fund Project: The first and the third authors are supported by the Major Research and Development Program of China grant No.2016YFB0200901 and the NSF of China grant No.11771348. The second author is supported by the NSF of China under grant No. 11971150, and the NSF of Henan Province under grant No. 162300410031

In this paper, the discontinuous Galerkin (DG) method is developed and analyzed for solving the Helmholtz transmission problem (HTP) with the first order absorbing boundary condition in two-level homogeneous media. This whole domain is separated into two disjoint subdomains by an interface, where two types of transmission conditions are provided. The application of the DG method to the HTP gives the discrete formulation. A rigorous theoretical analysis demonstrates that the discrete formulation can retain absolute stability without any mesh constraint. We prove that the errors in $ H^{1} $ and $ L^{2} $ norms are bounded by $ C_{1}kh + C_{2}k^{4}h^{2} $ and $ C_{1}kh^{2} + C_{2}k^{3}h^{2} $, respectively, where $ C_1 $ and $ C_2 $ are positive constants independent of the wave number $ k $ and the mesh size $ h $. Numerical experiments are conducted to verify the accuracy of the theoretical results and the efficiency of the numerical method.

Citation: Qingjie Hu, Zhihao Ge, Yinnian He. Discontinuous galerkin method for the helmholtz transmission problem in two-level homogeneous media. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020046
References:
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D. N. ArnoldF. BrezziB. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

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[20]

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[21]

X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the {C}ahn-{H}illiard equation of phase transition, Math. Comp., 76 (2007), 1093-1117.  doi: 10.1090/S0025-5718-07-01985-0.  Google Scholar

[22]

X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896.  doi: 10.1137/080737538.  Google Scholar

[23]

X. Feng and Y. Xing, Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 82 (2013), 1269-1296.  doi: 10.1090/S0025-5718-2012-02652-4.  Google Scholar

[24]

H. GengT. Yin and L. Xu, A priori error estimates of the DtN-FEM for the transmission problem in acoustics, J. Comput. Appl. Math., 313 (2017), 1-17.  doi: 10.1016/j.cam.2016.09.004.  Google Scholar

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[26]

R. Hiptmair and P. Meury, Stabilized FEM-BEM coupling for Helmholtz transmission problems, SIAM J. Numer. Anal., 44 (2006), 2107-2130.  doi: 10.1137/050639958.  Google Scholar

[27]

G. C. Hsiao and L. Xu, A system of boundary integral equations for the transmission problem in acoustics, Appl. Numer. Math., 61 (2011), 1017-1029.  doi: 10.1016/j.apnum.2011.05.003.  Google Scholar

[28]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Ⅰ. The $h$-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37.  doi: 10.1016/0898-1221(95)00144-N.  Google Scholar

[29]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, 132, Springer-Verlag, New York, 1998. doi: 10.1007/b98828.  Google Scholar

[30]

G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar

[31]

C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp., 35 (1980), 1063-1079.  doi: 10.1090/S0025-5718-1980-0583487-9.  Google Scholar

[32]

R. Kittappa and R. E. Kleinman, Acoustic scattering by penetrable homogeneous objects, J. Mathematical Phys., 16 (1975), 421-432.  doi: 10.1063/1.522517.  Google Scholar

[33]

R. Kress and G. F. Roach, Transmission problems for the Helmholtz equation, J. Mathematical Phys., 19 (1978), 1433-1437.  doi: 10.1063/1.523808.  Google Scholar

[34]

A. Moiola and E. A. Spence, Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions, Math. Models Methods Appl. Sci., 29 (2019), 317-354.  doi: 10.1142/S0218202519500106.  Google Scholar

[35]

L. MuJ. WangX. Ye and S. Zhao, A numerical study on the weak Galerkin method for the Helmholtz equation, Commun. Comput. Phys., 15 (2014), 1461-1479.  doi: 10.4208/cicp.251112.211013a.  Google Scholar

[36]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978.  doi: 10.1137/060665737.  Google Scholar

[37]

T. Warburton and J. S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), 2765-2773.  doi: 10.1016/S0045-7825(03)00294-9.  Google Scholar

[38]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), 152-161.  doi: 10.1137/0715010.  Google Scholar

[39]

H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part Ⅰ: Linear version, IMA J. Numer. Anal., 34 (2014), 1266-1288.  doi: 10.1093/imanum/drt033.  Google Scholar

show all references

References:
[1]

M. AinsworthP. Monk and W. Muniz, Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation, J. Sci. Comput., 27 (2006), 5-40.  doi: 10.1007/s10915-005-9044-x.  Google Scholar

[2]

G. B. AlvarezA. F. D. LoulaE. G. Dutra do Carmo and F. A. Rochinha, A discontinuous finite element formulation for Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 195 (2006), 4018-4035.  doi: 10.1016/j.cma.2005.07.013.  Google Scholar

[3]

T. S. AngellR. E. Kleinman and F. Hettlich, The resistive and conductive problems for the exterior Helmholtz equation, SIAM J. Appl. Math., 50 (1990), 1607-1622.  doi: 10.1137/0150095.  Google Scholar

[4]

D. N. Arnold, An interior penalty finite element method with discontinuous elements, SIAM J. Numer. Anal., 19 (1982), 742-760.  doi: 10.1137/0719052.  Google Scholar

[5]

D. N. ArnoldF. BrezziB. Cockburn and L. D. Marini, Unified analysis of discontinuous Galerkin methods for elliptic problems, SIAM J. Numer. Anal., 39 (2002), 1749-1779.  doi: 10.1137/S0036142901384162.  Google Scholar

[6]

A. K. Aziz and A. Werschulz, On the numerical solutions of Helmholtz's equation by the finite element method, SIAM J. Numer. Anal., 17 (1980), 681-686.  doi: 10.1137/0717058.  Google Scholar

[7]

I. BabuškaF. IhlenburgE. T. Paik and S. A. Sauter, A generalized finite element method for solving the Helmholtz equation in two dimensions with minimal pollution, Comput. Methods Appl. Mech. Engrg., 128 (1995), 325-359.  doi: 10.1016/0045-7825(95)00890-X.  Google Scholar

[8]

G. A. Baker, Finite element methods for elliptic equations using nonconforming elements, Math. Comp., 31 (1977), 45-59.  doi: 10.1090/S0025-5718-1977-0431742-5.  Google Scholar

[9]

G. Bao, Finite element approximation of time harmonic waves in periodic structures, SIAM J. Numer. Anal., 32 (1995), 1155-1169.  doi: 10.1137/0732053.  Google Scholar

[10]

A. Ben AbdaF. Ben HassenJ. Leblond and M. Mahjoub, Sources recovery from boundary data: A model related to electroencephalography, Math. Comput. Modelling, 49 (2009), 2213-2223.  doi: 10.1016/j.mcm.2008.07.016.  Google Scholar

[11]

C. L. Chang, A least-squares finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 83 (1990), 1-7.  doi: 10.1016/0045-7825(90)90121-2.  Google Scholar

[12]

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.  Google Scholar

[13]

M. Cheney and B. Borden, Problems in synthetic-aperture radar imaging, Inverse Problems, 25 (2009), 18pp. doi: 10.1088/0266-5611/25/12/123005.  Google Scholar

[14]

B. Cockburn, G. E. Karniadakis and C.-W. Shu, Discontinuous Galerkin Methods. Theory, Computation and Applications, Lecture Notes in Computational Science and Engineering, 11, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-59721-3.  Google Scholar

[15]

B. Cockburn and C.-W. Shu, The local discontinuous Galerkin method for time-dependent convection-diffusion systems, SIAM J. Numer. Anal., 35 (1998), 2440-2463.  doi: 10.1137/S0036142997316712.  Google Scholar

[16]

D. Colton and P. Monk, The numerical solution of the three-dimensional inverse scattering problem for time harmonic acoustic waves, SIAM J. Sci. Statist. Comput., 8 (1987), 278-291.  doi: 10.1137/0908035.  Google Scholar

[17]

M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems, J. Math. Anal. Appl., 106 (1985), 367-413.  doi: 10.1016/0022-247X(85)90118-0.  Google Scholar

[18]

P. Cummings and X. Feng, Sharp regularity coefficient estimates for complex-valued acoustic and elastic Helmholtz equations, Math. Models Methods Appl. Sci., 16 (2006), 139-160.  doi: 10.1142/S021820250600108X.  Google Scholar

[19]

V. DolejšíM. Feistauer and V. Sobotíková, Analysis of the discontinuous Galerkin method for nonlinear convection-diffusion problems, Comput. Methods Appl. Mech. Engrg., 194 (2005), 2709-2733.  doi: 10.1016/j.cma.2004.07.017.  Google Scholar

[20]

B. Engquist and A. Majda, Radiation boundary conditions for acoustic and elastic wave calculations, Comm. Pure Appl. Math., 32 (1979), 314-358.  doi: 10.1002/cpa.3160320303.  Google Scholar

[21]

X. Feng and O. A. Karakashian, Fully discrete dynamic mesh discontinuous Galerkin methods for the {C}ahn-{H}illiard equation of phase transition, Math. Comp., 76 (2007), 1093-1117.  doi: 10.1090/S0025-5718-07-01985-0.  Google Scholar

[22]

X. Feng and H. Wu, Discontinuous Galerkin methods for the Helmholtz equation with large wave number, SIAM J. Numer. Anal., 47 (2009), 2872-2896.  doi: 10.1137/080737538.  Google Scholar

[23]

X. Feng and Y. Xing, Absolutely stable local discontinuous Galerkin methods for the Helmholtz equation with large wave number, Math. Comp., 82 (2013), 1269-1296.  doi: 10.1090/S0025-5718-2012-02652-4.  Google Scholar

[24]

H. GengT. Yin and L. Xu, A priori error estimates of the DtN-FEM for the transmission problem in acoustics, J. Comput. Appl. Math., 313 (2017), 1-17.  doi: 10.1016/j.cam.2016.09.004.  Google Scholar

[25]

U. Hetmaniuk, Stability estimates for a class of Helmholtz problems, Commun. Math. Sci., 5 (2007), 665-678.  doi: 10.4310/CMS.2007.v5.n3.a8.  Google Scholar

[26]

R. Hiptmair and P. Meury, Stabilized FEM-BEM coupling for Helmholtz transmission problems, SIAM J. Numer. Anal., 44 (2006), 2107-2130.  doi: 10.1137/050639958.  Google Scholar

[27]

G. C. Hsiao and L. Xu, A system of boundary integral equations for the transmission problem in acoustics, Appl. Numer. Math., 61 (2011), 1017-1029.  doi: 10.1016/j.apnum.2011.05.003.  Google Scholar

[28]

F. Ihlenburg and I. Babuška, Finite element solution of the Helmholtz equation with high wave number. Ⅰ. The $h$-version of the FEM, Comput. Math. Appl., 30 (1995), 9-37.  doi: 10.1016/0898-1221(95)00144-N.  Google Scholar

[29]

F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, 132, Springer-Verlag, New York, 1998. doi: 10.1007/b98828.  Google Scholar

[30]

G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.  doi: 10.1088/0266-5611/13/4/006.  Google Scholar

[31]

C. Johnson and J.-C. Nédélec, On the coupling of boundary integral and finite element methods, Math. Comp., 35 (1980), 1063-1079.  doi: 10.1090/S0025-5718-1980-0583487-9.  Google Scholar

[32]

R. Kittappa and R. E. Kleinman, Acoustic scattering by penetrable homogeneous objects, J. Mathematical Phys., 16 (1975), 421-432.  doi: 10.1063/1.522517.  Google Scholar

[33]

R. Kress and G. F. Roach, Transmission problems for the Helmholtz equation, J. Mathematical Phys., 19 (1978), 1433-1437.  doi: 10.1063/1.523808.  Google Scholar

[34]

A. Moiola and E. A. Spence, Acoustic transmission problems: Wavenumber-explicit bounds and resonance-free regions, Math. Models Methods Appl. Sci., 29 (2019), 317-354.  doi: 10.1142/S0218202519500106.  Google Scholar

[35]

L. MuJ. WangX. Ye and S. Zhao, A numerical study on the weak Galerkin method for the Helmholtz equation, Commun. Comput. Phys., 15 (2014), 1461-1479.  doi: 10.4208/cicp.251112.211013a.  Google Scholar

[36]

J. Shen and L.-L. Wang, Analysis of a spectral-Galerkin approximation to the Helmholtz equation in exterior domains, SIAM J. Numer. Anal., 45 (2007), 1954-1978.  doi: 10.1137/060665737.  Google Scholar

[37]

T. Warburton and J. S. Hesthaven, On the constants in $hp$-finite element trace inverse inequalities, Comput. Methods Appl. Mech. Engrg., 192 (2003), 2765-2773.  doi: 10.1016/S0045-7825(03)00294-9.  Google Scholar

[38]

M. F. Wheeler, An elliptic collocation-finite element method with interior penalties, SIAM J. Numer. Anal., 15 (1978), 152-161.  doi: 10.1137/0715010.  Google Scholar

[39]

H. Wu, Pre-asymptotic error analysis of CIP-FEM and FEM for the Helmholtz equation with high wave number. Part Ⅰ: Linear version, IMA J. Numer. Anal., 34 (2014), 1266-1288.  doi: 10.1093/imanum/drt033.  Google Scholar

Figure 1.  The sketch of the two homogeneous media $\Omega_1$ and $\Omega_2$ with wave numbers $k_1$ and $k_2$, respectively
Figure 2.  A sample mesh $ \mathcal{T}_{\frac{1}{8}} $ of $ \Omega_1 $ (left) and $ \Omega_2 $ (right)
Figure 3.  $ |u_{h, 1}|_{1, h} $ (left) and $ |u_{h, 2}|_{1, h} $ (right) for $ h = 1/64 $, $ h = 1/128 $, $ h = 1/256 $
Figure 4.  Relative error of the DG solutions with the penalty parameters $ \gamma_{1, e} = 0.03+0.06i, \beta_{1, e} = 1 $ and each of the following $ \gamma_{0, e}: $ $ \gamma_{0, e} = 0.01 $ (solid line with $ \diamond $), $ \gamma_{0, e} = 0.1 $ (solid line), $ \gamma_{0, e} = 1 $ (dashdot line), $ \gamma_{0, e} = 10 $ (dashed line) in the $ H^1 $-seminorm for $ k_1 = 50, k_2 = 40 $ (left) and $ k_1 = 100, k_2 = 90 $ (right)
Figure 5.  Relative error of the DG solutions with the penalty parameters $ \gamma_{0, e} = 100, \beta_{1, e} = 1 $ and each of the following $ \gamma_{1, e}: $ $ \gamma_{1, e} = 0.01 $ (solid line with $ \diamond $), $ \gamma_{1, e} = 0.1 $ (solid line), $ \gamma_{1, e} = 1 $ (dashdot line), $ \gamma_{1, e} = 10 $ (dashed line) in the $ H^1 $-seminorm for $ k_1 = 50, k_2 = 40 $ (left) and $ k_1 = 100, k_2 = 90 $ (right)
Figure 6.  Relative error of the DG solutions with the penalty parameters $ \gamma_{1, e} = 0.03+0.06i, \gamma_{0, e} = 100 $ and each of the following $ \beta_{1, e}: $ $ \beta_{1, e} = 0.01 $ (solid line with $ \diamond $), $ \beta_{1, e} = 0.1 $ (solid line), $ \beta_{1, e} = 1 $ (dashdot line), $ \beta_{1, e} = 10 $ (dashed line) in the $ H^1 $-seminorm for $ k_1 = 50, k_2 = 40 $ (left) and $ k_1 = 100, k_2 = 90 $ (right)
Figure 7.  Relative error of the DG solution for $ k_1 = 50, k_2 = 45 $; $ k_1 = 100, k_2 = 95 $ and $ k_1 = 150, k_2 = 145 $
Figure 8.  Surface plots of exact solution $ u_1 $ (left) and the DG solution $ u_{h, 1} $ (right) for $ k_1 = 100, k_2 = 90 $ and $ h = 1/64 $
Figure 9.  Surface plots of exact solution $ u_2 $ (left) and the DG solution $ u_{h, 2} $ (right) for $ k_1 = 100, k_2 = 90 $ and $ h = 1/64 $
Figure 10.  Surface plots of exact solution $ u_1 $ (left) and the DG solution $ u_{h, 1} $ (right) for $ k_1 = 100, k_2 = 50 $ and $ h = 1/64 $
Figure 11.  Surface plots of exact solution $ u_2 $ (left) and the DG solution $ u_{h, 2} $ (right) for $ k_1 = 100, k_2 = 50 $ and $ h = 1/64 $
Figure 12.  The traces of the DG solution using piecewise linear polynomials in the $ xz- $plane for $ k_1 = 100, k_2 = 90 $ with mesh size $ h = 1/64 $ (left) and $ h = 1/128 $ (right). The purple lines give the trace of the exact solution in $ xz $-plane
Figure 13.  The traces of the DG solution using piecewise quadratic polynomials in the $ xz- $plane for $ k_1 = 100, k_2 = 90 $ with mesh size $ h = 1/64 $ (left) and $ h = 1/128 $ (right). The purple lines give the trace of the exact solution in $ xz $-plane
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