# American Institute of Mathematical Sciences

August  2020, 25(8): 2923-2948. 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  August 2020 Early access  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 and Continuous Dynamical Systems - B, 2020, 25 (8) : 2923-2948. doi: 10.3934/dcdsb.2020046
##### References:
 [1] M. Ainsworth, P. 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. [2] G. B. Alvarez, A. F. D. Loula, E. 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. [3] T. S. Angell, R. 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. [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. [5] D. N. Arnold, F. Brezzi, B. 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. [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. [7] I. Babuška, F. Ihlenburg, E. 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. [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. [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. [10] A. Ben Abda, F. Ben Hassen, J. 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. [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. [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. [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. [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. [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. [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. [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. [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. [19] V. Dolejší, M. 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. [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. [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. [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. [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. [24] H. Geng, T. 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. [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. [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. [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. [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. [29] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, 132, Springer-Verlag, New York, 1998. doi: 10.1007/b98828. [30] G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.  doi: 10.1088/0266-5611/13/4/006. [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. [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. [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. [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. [35] L. Mu, J. Wang, X. 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. [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. [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. [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. [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.

show all references

##### References:
 [1] M. Ainsworth, P. 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. [2] G. B. Alvarez, A. F. D. Loula, E. 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. [3] T. S. Angell, R. 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. [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. [5] D. N. Arnold, F. Brezzi, B. 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. [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. [7] I. Babuška, F. Ihlenburg, E. 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. [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. [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. [10] A. Ben Abda, F. Ben Hassen, J. 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. [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. [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. [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. [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. [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. [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. [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. [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. [19] V. Dolejší, M. 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. [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. [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. [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. [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. [24] H. Geng, T. 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. [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. [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. [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. [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. [29] F. Ihlenburg, Finite Element Analysis of Acoustic Scattering, Applied Mathematical Sciences, 132, Springer-Verlag, New York, 1998. doi: 10.1007/b98828. [30] G. Inglese, An inverse problem in corrosion detection, Inverse Problems, 13 (1997), 977-994.  doi: 10.1088/0266-5611/13/4/006. [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. [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. [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. [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. [35] L. Mu, J. Wang, X. 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. [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. [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. [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. [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.
The sketch of the two homogeneous media $\Omega_1$ and $\Omega_2$ with wave numbers $k_1$ and $k_2$, respectively
A sample mesh $\mathcal{T}_{\frac{1}{8}}$ of $\Omega_1$ (left) and $\Omega_2$ (right)
$|u_{h, 1}|_{1, h}$ (left) and $|u_{h, 2}|_{1, h}$ (right) for $h = 1/64$, $h = 1/128$, $h = 1/256$
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)
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)
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)
Relative error of the DG solution for $k_1 = 50, k_2 = 45$; $k_1 = 100, k_2 = 95$ and $k_1 = 150, k_2 = 145$
Surface plots of exact solution $u_1$ (left) and the DG solution $u_{h, 1}$ (right) for $k_1 = 100, k_2 = 90$ and $h = 1/64$
Surface plots of exact solution $u_2$ (left) and the DG solution $u_{h, 2}$ (right) for $k_1 = 100, k_2 = 90$ and $h = 1/64$
Surface plots of exact solution $u_1$ (left) and the DG solution $u_{h, 1}$ (right) for $k_1 = 100, k_2 = 50$ and $h = 1/64$
Surface plots of exact solution $u_2$ (left) and the DG solution $u_{h, 2}$ (right) for $k_1 = 100, k_2 = 50$ and $h = 1/64$
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
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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