American Institute of Mathematical Sciences

Advanced
August  2020, 25(8): 2923-2948. doi: 10.3934/dcdsb.2020046

Discontinuous Galerkin method for the Helmholtz transmission problem in two-level homogeneous media

Qingjie Hu 1, , Zhihao Ge 2, and  Yinnian He 1,,

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.

Keywords: Helmholtz transmission problem, discontinuous Galerkin method, transmission condition, stability, error estimates.
Mathematics Subject Classification: Primary: 65N12, 65N15, 65N30; Secondary: 78A40.
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, 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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 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.  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. 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.  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. 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.  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. 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.  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. 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.  Google Scholar

[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.  Google Scholar

[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.  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. 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.  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. 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.  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 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.  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. 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.  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. 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.  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. 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.  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 Options

Download full-size image

Download as PowerPoint slide

Figure 2.  A sample mesh $ \mathcal{T}_{\frac{1}{8}} $ of $ \Omega_1 $ (left) and $ \Omega_2 $ (right)
Figure Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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 Options

Download full-size image

Download as PowerPoint slide

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
Figure Options

Download full-size image

Download as PowerPoint slide

[1]

Michele Di Cristo. Stability estimates in the inverse transmission scattering problem. Inverse Problems & Imaging, 2009, 3 (4) : 551-565. doi: 10.3934/ipi.2009.3.551
[2]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[3]

Jun Zhang, Xinyue Fan. An efficient spectral method for the Helmholtz transmission eigenvalues in polar geometries. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4799-4813. doi: 10.3934/dcdsb.2019031
[4]

Chaoxu Pei, Mark Sussman, M. Yousuff Hussaini. A space-time discontinuous Galerkin spectral element method for the Stefan problem. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3595-3622. doi: 10.3934/dcdsb.2017216
[5]

Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko, Puntip Toghaw. Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures. Communications on Pure & Applied Analysis, 2007, 6 (3) : 719-740. doi: 10.3934/cpaa.2007.6.719
[6]

Na An, Chaobao Huang, Xijun Yu. Error analysis of discontinuous Galerkin method for the time fractional KdV equation with weak singularity solution. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 321-334. doi: 10.3934/dcdsb.2019185
[7]

Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230
[8]

David Colton, Lassi Päivärinta, John Sylvester. The interior transmission problem. Inverse Problems & Imaging, 2007, 1 (1) : 13-28. doi: 10.3934/ipi.2007.1.13
[9]

Emanuela R. S. Coelho, Valéria N. Domingos Cavalcanti, Vinicius A. Peralta. Exponential stability for a transmission problem of a nonlinear viscoelastic wave equation. Communications on Pure & Applied Analysis, 2021, 20 (5) : 1987-2020. doi: 10.3934/cpaa.2021055
[10]

Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340
[11]

Jiayu Han. Nonconforming elements of class $L^2$ for Helmholtz transmission eigenvalue problems. Discrete & Continuous Dynamical Systems - B, 2018, 23 (8) : 3195-3212. doi: 10.3934/dcdsb.2018281
[12]

Armin Lechleiter. The factorization method is independent of transmission eigenvalues. Inverse Problems & Imaging, 2009, 3 (1) : 123-138. doi: 10.3934/ipi.2009.3.123
[13]

Yoshifumi Aimoto, Takayasu Matsuo, Yuto Miyatake. A local discontinuous Galerkin method based on variational structure. Discrete & Continuous Dynamical Systems - S, 2015, 8 (5) : 817-832. doi: 10.3934/dcdss.2015.8.817
[14]

Tiexiang Li, Tsung-Ming Huang, Wen-Wei Lin, Jenn-Nan Wang. On the transmission eigenvalue problem for the acoustic equation with a negative index of refraction and a practical numerical reconstruction method. Inverse Problems & Imaging, 2018, 12 (4) : 1033-1054. doi: 10.3934/ipi.2018043
[15]

Jing-Jing Xiang, Juan Wang, Li-Ming Cai. Global stability of the dengue disease transmission models. Discrete & Continuous Dynamical Systems - B, 2015, 20 (7) : 2217-2232. doi: 10.3934/dcdsb.2015.20.2217
[16]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[17]

David Colton, Yuk-J. Leung. On a transmission eigenvalue problem for a spherically stratified coated dielectric. Inverse Problems & Imaging, 2016, 10 (2) : 369-378. doi: 10.3934/ipi.2016004
[18]

Alexandre Thorel. A biharmonic transmission problem in Lp-spaces. Communications on Pure & Applied Analysis, 2021, 20 (9) : 3193-3213. doi: 10.3934/cpaa.2021102
[19]

Natalia P. Bondarenko, Vjacheslav A. Yurko. A new approach to the inverse discrete transmission eigenvalue problem. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021073
[20]

Selim Esedoḡlu, Fadil Santosa. Error estimates for a bar code reconstruction method. Discrete & Continuous Dynamical Systems - B, 2012, 17 (6) : 1889-1902. doi: 10.3934/dcdsb.2012.17.1889

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (278)
  • HTML views (313)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy