A superconvergent multiscale scheme for the Helmholtz equation in rapidly oscillating heterogeneous media

Shan Jiang; Weipeng Miao; Huayi Wei; Nianyu Yi; Shan Jiang; Weipeng Miao; Huayi Wei; Nianyu Yi

doi:10.3934/nhm.2026024

Networks and Heterogeneous Media

2026, Volume 21, Issue 2: 531-550. doi: 10.3934/nhm.2026024

Previous Article Next Article

Research article

A superconvergent multiscale scheme for the Helmholtz equation in rapidly oscillating heterogeneous media

1.
School of Mathematics and Statistics, Nantong University, Nantong 226019, China
2.
School of Mathematics and Computational Science, Xiangtan University, Xiangtan 411105, China

Received: 04 November 2025 Revised: 10 March 2026 Accepted: 20 March 2026 Published: 31 March 2026

We present a superconvergent and highly efficient multiscale finite element scheme for solving the Helmholtz equation in heterogeneous media. The proposed scheme is specifically designed to address the challenges posed by the multiscale nature of the Helmholtz equation, which often leads to high computational costs and numerical instabilities in traditional methods. By constructing tailored multiscale basis functions, the proposed approach effectively captures both the local fine-scale features and global macroscale behaviors of the multiscale solution. The fundamental mechanism underlying the multiscale scheme is systematically expounded. Through rigorous error analyses, we derive a second-order superconvergence result in the $ H^1 $ norm. Additionally, numerical experiments are conducted to validate the performance of the scheme. Both theoretical findings and numerical results confirm that the proposed multiscale scheme not only achieves higher solution accuracy but also exhibits superconvergence rates and significantly reduces computational costs.
- Helmholtz equation,
- multiscale finite element method,
- multiscale decomposition,
- $ H^1 $ norm,
- superconvergence
Citation: Shan Jiang, Weipeng Miao, Huayi Wei, Nianyu Yi. A superconvergent multiscale scheme for the Helmholtz equation in rapidly oscillating heterogeneous media[J]. Networks and Heterogeneous Media, 2026, 21(2): 531-550. doi: 10.3934/nhm.2026024

Related Papers:

Abstract

We present a superconvergent and highly efficient multiscale finite element scheme for solving the Helmholtz equation in heterogeneous media. The proposed scheme is specifically designed to address the challenges posed by the multiscale nature of the Helmholtz equation, which often leads to high computational costs and numerical instabilities in traditional methods. By constructing tailored multiscale basis functions, the proposed approach effectively captures both the local fine-scale features and global macroscale behaviors of the multiscale solution. The fundamental mechanism underlying the multiscale scheme is systematically expounded. Through rigorous error analyses, we derive a second-order superconvergence result in the $ H^1 $ norm. Additionally, numerical experiments are conducted to validate the performance of the scheme. Both theoretical findings and numerical results confirm that the proposed multiscale scheme not only achieves higher solution accuracy but also exhibits superconvergence rates and significantly reduces computational costs.

References

[1]	F. Ihlenburg, I. Babuska, Finite element solution of the Helmholtz equation with high wave number. Part Ⅰ: The $h$-version of the FEM, Comput. Math. Appl., 30 (1995), 9–37. https://doi.org/10.1016/0898-1221(95)00144-N doi: 10.1016/0898-1221(95)00144-N
[2]	A. A. Oberai, P. M. Pinsky, A multiscale finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Eng., 154 (1998), 281–297. https://doi.org/10.1016/S0045-7825(97)00130-8 doi: 10.1016/S0045-7825(97)00130-8
[3]	I. Babuska, S. A. Sauter, Is the pollution effect of the FEM avoidable for the Helmholtz equation considering high wave numbers? SIAM J. Numer. Anal., 34 (1997), 2392–2423. https://doi.org/10.1137/S0036142994269186 doi: 10.1137/S0036142994269186
[4]	D. Gallistl, D. Peterseim, Stable multiscale Petrov-Galerkin finite element method for high frequency acoustic scattering, Comput. Methods Appl. Mech. Eng., 295 (2015), 1–17. https://doi.org/10.1016/j.cma.2015.06.017 doi: 10.1016/j.cma.2015.06.017
[5]	S. Jiang, M. Presho, Y. Q. Huang, An adapted Petrov-Galerkin multiscale finite element for singularly perturbed reaction-diffusion problems, Int. J. Comput. Math., 93 (2016), 1200–1211. https://doi.org/10.1080/00207160.2015.1041935 doi: 10.1080/00207160.2015.1041935
[6]	T. Chaumont-Frelet, On high order methods for the heterogeneous Helmholtz equation, Comput. Math. Appl., 72 (2016), 2203–2225. https://doi.org/10.1016/j.camwa.2016.08.026 doi: 10.1016/j.camwa.2016.08.026
[7]	T. Chaumont-Frelet, F. Valentin, A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains, SIAM J. Numer. Anal., 58 (2020), 1029–1067. https://doi.org/10.1137/19M1255616 doi: 10.1137/19M1255616
[8]	M. Ohlberger, B. Verfurth, A new heterogeneous multiscale method for the Helmholtz equation with high contrast, Multiscale Model. Simul., 16 (2018), 385–411. https://doi.org/10.1137/16m1108820 doi: 10.1137/16m1108820
[9]	E. T. Chung, Y. Efendiev, W. T. Leung, Constraint energy minimizing generalized multiscale finite element method, Comput. Methods Appl. Mech. Eng., 339 (2018), 298–319. https://doi.org/10.1016/j.cma.2018.04.010 doi: 10.1016/j.cma.2018.04.010
[10]	S. B. Fu, K. Gao, R. L. Gibson Jr., E. T. Chung, An efficient high-order multiscale finite element method for frequency-domain elastic wave modeling, Comput. Geosci., 23 (2019), 997–1010. https://doi.org/10.1007/s10596-019-09865-0 doi: 10.1007/s10596-019-09865-0
[11]	S. B. Fu, G. L. Li, R. Craster, S. Guenneau, Wavelet-based edge multiscale finite element method for Helmholtz problems in perforated domains, Multiscale Model. Simul., 19 (2021), 1684–1709. https://doi.org/10.1137/19M1267180 doi: 10.1137/19M1267180
[12]	I. Singer, E. Turkel, Sixth-order accurate finite difference schemes for the Helmholtz equation, J. Comput. Acoust., 14 (2006), 339–351. https://doi.org/10.1142/S0218396X06003050 doi: 10.1142/S0218396X06003050
[13]	T. T. Wu, R. M. Xu, An optimal compact sixth-order finite difference scheme for the Helmholtz equation, Comput. Math. Appl., 75 (2018), 2520–2537. https://doi.org/10.1016/j.camwa.2017.12.023 doi: 10.1016/j.camwa.2017.12.023
[14]	N. Kumar, R. K. Dubey, A new development of sixth order accurate compact scheme for the Helmholtz equation, J. Appl. Math. Comput., 62 (2020), 637–662. https://doi.org/10.1007/s12190-019-01301-x doi: 10.1007/s12190-019-01301-x
[15]	W. Jiang, X. H. Chen, B. N. Lv, S. S. Jiang, An accurate and efficient multiscale finite-difference frequency-domain method for the scalar Helmholtz equation, Geophysics, 87 (2022), T43–T60. https://doi.org/10.1190/GEO2021-0217.1 doi: 10.1190/GEO2021-0217.1
[16]	U. Kalachikova, M. Vasilyeva, I. Harris, E. T. Chung, Generalized multiscale finite element method for scattering problem in heterogeneous media, J. Comput. Appl. Math., 424 (2023), 114977. https://doi.org/10.1016/j.cam.2022.114977 doi: 10.1016/j.cam.2022.114977
[17]	P. Freese, M. Hauck, D. Peterseim, Super-localized orthogonal decomposition for high-frequency Helmholtz problems, SIAM J. Sci. Comput., 46 (2024), A2377–A2397. https://doi.org/10.1137/21M1465950 doi: 10.1137/21M1465950
[18]	S. Jiang, A. Protasov, M. L. Sun, Balanced truncation based on generalized multiscale finite element method for the parameter-dependent elliptic problem, Adv. Appl. Math. Mech., 10 (2018), 1527–1548. https://doi.org/10.4208/aamm.OA-2018-0073 doi: 10.4208/aamm.OA-2018-0073
[19]	S. Jiang, Y. Cheng, Y. Cheng, Y. Q. Huang, Generalized multiscale finite element method and balanced truncation for parameter-dependent parabolic problems, Mathematics, 11 (2023), 4965. https://doi.org/10.3390/math11244965 doi: 10.3390/math11244965
[20]	Y. F. Chen. T. Y. Hou, Y. X. Wang, Exponential convergence for multiscale linear elliptic PDEs via adaptive edge basis functions, Multiscale Model. Simul., 19 (2021), 980–1010. https://doi.org/10.1137/20M1352922 doi: 10.1137/20M1352922
[21]	Y. F. Chen. T. Y. Hou, Y. X. Wang, Exponentially convergent multiscale methods for 2D high frequency heterogeneous Helmholtz equations, Multiscale Model. Simul., 21 (2023), 849–883. https://doi.org/10.1137/22M1507802 doi: 10.1137/22M1507802
[22]	Y. F. Chen. T. Y. Hou, Y. X. Wang, Exponentially convergent multiscale finite element method, Commun. Appl. Math. Comput., 6 (2024), 862–878. https://doi.org/10.1007/s42967-023-00260-2 doi: 10.1007/s42967-023-00260-2
[23]	S. Jiang, X. Ding, M. L. Sun, Parameter-uniform superconvergence of multiscale computation for singular perturbation exhibiting twin boundary layers, J. Appl. Anal. Comput., 13 (2023), 3330–3351. https://doi.org/10.11948/20230020 doi: 10.11948/20230020
[24]	Y. Cheng, S. Jiang, M. Stynes, Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem, Math. Comput., 92 (2023), 2065–2095. https://doi.org/10.1090/mcom/3844 doi: 10.1090/mcom/3844
[25]	Y. Cheng, X. S. Wang, M. Stynes, Optimal balanced-norm error estimate of the LDG method for reaction-diffusion problems I: The one-dimensional case, J. Sci. Comput., 100 (2024), 50. https://doi.org/10.1007/s10915-024-02602-5 doi: 10.1007/s10915-024-02602-5
[26]	C. P. Ma, C. Alber, R. Scheichl, Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems, SIAM J. Numer. Anal., 61 (2023), 1546–1584. https://doi.org/10.1137/21M1466748 doi: 10.1137/21M1466748
[27]	E. Giammatteo, A. Heinlein, M. Schlottbom, An extension of the approximate component mode synthesis method to the heterogeneous Helmholtz equation, IMA J. Numer. Anal., 45 (2025), 2844–2879. https://doi.org/10.1093/imanum/drae076 doi: 10.1093/imanum/drae076
[28]	S. Saini, P. Das, S. Kumar, Parameter uniform higher order numerical treatment for singularly perturbed Robin type parabolic reaction diffusion multiple scale problems with large delay in time, Appl. Numer. Math., 196 (2024), 1–21. https://doi.org/10.1016/j.apnum.2023.10.003 doi: 10.1016/j.apnum.2023.10.003
[29]	D. Sarkar, S. Kumar, P. Das, H. Ramos, Higher-order convergence analysis for interior and boundary layers in a semi-linear reaction-diffusion system networked by a $k$-star graph with non-smooth source terms, Networks Heterogen. Media, 19 (2024), 1085–1115. https://doi.org/10.3934/nhm.2024048 doi: 10.3934/nhm.2024048
[30]	Y. L. Li, H. J. Wu, Higher-order FEM and CIP-FEM for Helmholtz equation with high wave number and perfectly matched layer truncation, J. Sci. Comput., 104 (2025), 47. https://doi.org/10.1007/s10915-025-02959-1 doi: 10.1007/s10915-025-02959-1
[31]	M. Bernkopf, T. Chaumont-Frelet, J. M. Melenk, Wavenumber-explicit stability and convergence analysis of $hp$ finite element discretizations of Helmholtz problems in piecewise smooth media, Math. Comput., 94 (2025), 73–122. https://doi.org/10.1090/mcom/3958 doi: 10.1090/mcom/3958

Reader Comments

Your name:*

Email:*
© 2026 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)