FAST ENERGY-NORM CONVERGENCE AND HIGH EFFICIENCY IN A MULTISCALE FINITE ELEMENT COMPUTATION FOR THE HELMHOLTZ EQUATION

Meiling Sun; Yao Cheng; Peng Zhu; Shan Jiang

doi:10.11948/20250329

2027 Volume 17 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2027 > 17(2): 767-787

Next Article Previous Article

Meiling Sun, Yao Cheng, Peng Zhu, Shan Jiang. FAST ENERGY-NORM CONVERGENCE AND HIGH EFFICIENCY IN A MULTISCALE FINITE ELEMENT COMPUTATION FOR THE HELMHOLTZ EQUATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 767-787. doi: 10.11948/20250329

Citation:

Meiling Sun, Yao Cheng, Peng Zhu, Shan Jiang. FAST ENERGY-NORM CONVERGENCE AND HIGH EFFICIENCY IN A MULTISCALE FINITE ELEMENT COMPUTATION FOR THE HELMHOLTZ EQUATION[J]. Journal of Applied Analysis & Computation, 2027, 17(2): 767-787. doi: 10.11948/20250329

FAST ENERGY-NORM CONVERGENCE AND HIGH EFFICIENCY IN A MULTISCALE FINITE ELEMENT COMPUTATION FOR THE HELMHOLTZ EQUATION

1.
School of Mathematics and Statistics, Nantong University, Nantong 226019, China
2.
Department of Mathematics, Nantong Vocational University, Nantong 226007, China
3.
School of Mathematical Sciences, Suzhou University of Science and Technology, Suzhou 215009, China
4.
College of Data Science, Jiaxing University, Jiaxing 314001, China

Author Bio: Email: sunmeiling81@163.com(M. L. Sun); Email: ycheng@usts.edu.cn(Y. Cheng); Email: pzh_zjxu@163.com(P. Zhu)
Corresponding author: Email: jiangshan@ntu.edu.cn(S. Jiang)
Fund Project: The authors were supported by Key Project of Research Program on High-Quality Public Course Teaching Reform of Jiangsu Universities (2026GZJX015) to Meiling Sun, NSFC (11801396), Natural Science Foundation of Jiangsu Province (BK20170374) to Yao Cheng, Natural Science Foundation of Zhejiang Province (LY23A010005) to Peng Zhu, NSFC (11771224), China Scholarship Council (CSC) to Shan Jiang

Abstract

Abstract

This paper develops a fast-convergent and highly efficient multiscale finite element method for the Helmholtz equation. Through the construction of tailored multiscale basis functions, the proposed method is capable of capturing both local fine-scale characteristics and global macroscale behaviors of multiscale solutions. We further systematically elaborate on the associated data structure, in which a global mapping matrix intrinsically encodes rich microscopic information inside each macroscopic element. Rigorous error analysis is then carried out, combining approximation and interpolation error estimates to establish a second-order convergence result in the energy norm, which furnishes a solid theoretical foundation for the accuracy of the proposed scheme. Representative numerical experiments are presented to verify the performance of the method with regard to computational efficiency and solution stability.
- Multiscale scheme /
- data structure /
- error analysis /
- energy norm /
- fast convergence
MSC: 35J25, 65N12, 65N30

FullText(HTML)

References

[1]	M. M. Abedi, D. Pardo and T. Alkhalifah, Gabor-enhanced physics-informed neural networks for fast simulations of acoustic wavefields, Neural Networks, 2026, 193, Paper No. 107978. Google Scholar
[2]	M. M. Abedi, D. Pardo and T. Alkhalifah, Least-squares-embedded optimization for accelerated convergence of PINNs in high-frequency acoustic wavefield simulations, Comput. Geosci., 2026, 212, Paper No. 106162. Google Scholar
[3]	J. Baiges and R. Codina, A variational multiscale method with subscales on the element boundaries for the Helmholtz equation, Internat. J. Numer. Methods Engrg., 2012, 93, 664–684. Google Scholar
[4]	H. T. Cao and H. J. Wu, IPCDGM and multiscale IPDPGM for the Helmholtz problem with large wave number, J. Comput. Appl. Math., 2020, 369, Paper No. 112590. Google Scholar
[5]	T. Chaumont-Frelet and F. Valentin, A multiscale hybrid-mixed method for the Helmholtz equation in heterogeneous domains, SIAM J. Numer. Anal., 2020, 58, 1029–1067. doi: 10.1137/19M1255616 CrossRef Google Scholar
[6]	Y. F. Chen, T. Y. Hou and Y. X. Wang, Exponentially convergent multiscale methods for 2D high frequency heterogeneous Helmholtz equations, Multiscale Model. Simul., 2023, 21(3), 849–883. doi: 10.1137/22M1507802 CrossRef Google Scholar
[7]	Y. Cheng, S. Jiang and M. Stynes, Supercloseness of the local discontinuous Galerkin method for a singularly perturbed convection-diffusion problem, Math. Comput., 2023, 92(343), 2065–2095. doi: 10.1090/mcom/3844 CrossRef Google Scholar
[8]	P. Ciarlet Jr. and C. Stohrer, Finite-element heterogeneous multiscale method for the Helmholtz equation, C. R. Acad. Sci. Paris, Ser. I, 2014, 352, 755–760. doi: 10.1016/j.crma.2014.07.006 CrossRef Google Scholar
[9]	S. B. Fu, K. Gao, R. L. Gibson Jr. and E. T. Chung, An efficient high-order multiscale finite element method for frequency-domain elastic wave modeling, Computat. Geoscien., 2019, 23, 997–1010. doi: 10.1007/s10596-019-09865-0 CrossRef Google Scholar
[10]	M. Hauck and D. Peterseim, Multi-resolution localized orthogonal decomposition for Helmholtz problems, Multiscale Model. Simul., 2022, 20, 657–684. doi: 10.1137/21M1414607 CrossRef Google Scholar
[11]	F. Ihlenburg and I. Babuska, Finite element solution of the Helmholtz equation with high wave number Part Ⅰ: The $ h $-version of the FEM, Comput. Math. Appl., 1995, 30(9), 9–37. doi: 10.1016/0898-1221(95)00144-N CrossRef $ h $-version of the FEM" target="_blank">Google Scholar
[12]	S. Jiang, Y. Cheng, Y. Cheng and Y. Q. Huang, Generalized multiscale finite element method and balanced truncation for parameter-dependent parabolic problems, Math., 2023, 11, Paper No. 4965. Google Scholar
[13]	S. Jiang, A. Protasov and M. L. Sun, Balanced truncation based on generalized multiscale finite element method for the parameter-dependent elliptic problem, Advan. Appl. Math. Mech., 2018, 10(6), 1527–1548. doi: 10.4208/aamm.OA-2018-0073 CrossRef Google Scholar
[14]	W. Jiang, X. H. Chen, S. S. Jiang and J. Zhang, Multiscale finite-difference method for frequency-domain acoustic wave modeling, Geophys., 2021, 86, 193–210. Google Scholar
[15]	U. Kalachikova, M. Vasilyeva, I. Harris and E. T. Chung, Generalized multiscale finite element method for scattering problem in heterogeneous media, J. Comput. Appl. Math., 2023, 424, Paper No. 114977. Google Scholar
[16]	B. Khairullin and S. Rykovanov, Structure-preserving Helmholtz PINNs for absorbing media with transparent boundaries, Communi. Nonlinear Sci. Numer. Simul., 2026, 159, Paper No. 109926. Google Scholar
[17]	N. Kumar and R. K. Dubey, A new development of sixth order accurate compact scheme for the Helmholtz equation, J. Appl. Math. Comput., 2020, 62, 637–662. doi: 10.1007/s12190-019-01301-x CrossRef Google Scholar
[18]	B. Lerer, I. Ben-Yair and E. Treister, Multigrid-augmented deep learning preconditioners for the Helmholtz equation using compact implicit layers, SIAM J. Sci. Comput., 2024, 46, S123–S144. doi: 10.1137/23M1583302 CrossRef Google Scholar
[19]	G. L. Li, On the convergence rates of GMsFEMs for heterogeneous elliptic problems without oversampling techniques, Multiscale Model. Simul., 2019, 17(2), 593–619. doi: 10.1137/18M1172715 CrossRef Google Scholar
[20]	Y. H. Liu and Y. Cheng, Local discontinuous Galerkin method for a singularly perturbed fourth-order problem of reaction-diffusion type, J. Comput. Appl. Math., 2024, 440, Paper No. 115641. Google Scholar
[21]	C. P. Ma, C. Alber and R. Scheichl, Wavenumber explicit convergence of a multiscale generalized finite element method for heterogeneous Helmholtz problems, SIAM J. Numer. Anal., 2023, 61, 1546–1584. doi: 10.1137/21M1466748 CrossRef Google Scholar
[22]	A. A. Oberai and P. M. Pinsky, A multiscale finite element method for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg., 1998, 154, 281–297. doi: 10.1016/S0045-7825(97)00130-8 CrossRef Google Scholar
[23]	M. Ohlberger and B. Verfurth, A new heterogeneous multiscale method for the Helmholtz equation with high contrast, Multiscale Model. Simul., 2018, 16, 385–411. doi: 10.1137/16M1108820 CrossRef Google Scholar
[24]	D. Peterseim, Eliminating the pollution effect in Helmholtz problems by local subscale correction, Math. Comput., 2017, 86, 1005–1036. Google Scholar
[25]	A. T. Tufa, T. T. Dufera, M. D. Firdi and A. T. Deresse, A physics-informed neural networks for solving 2D frequency-domain electromagnetic waves in the differential form of Maxwell's equations, Advan. Math. Phys., 2026, 2026(1), Paper No. 5520631. Google Scholar
[26]	X. S. Wang and Y. Cheng, An improved pointwise error estimate of the LDG method for 1-d singularly perturbed reaction-diffusion problem, Appl. Numer. Math., 2024, 196, 199–217. doi: 10.1016/j.apnum.2023.10.001 CrossRef Google Scholar
[27]	X. S. Wang, S. Jiang and Y. Cheng, Pointwise error estimate of the LDG method for 2D singularly perturbed reaction-diffusion problem, Numer. Algorithms, 2025, 100(1), 189–225. doi: 10.1007/s11075-024-01958-x CrossRef Google Scholar
[28]	Q. Zhuang, C. Z. Yao, Z. Q. Zhang and G. E. Karniadakis, Two-scale neural networks for partial differential equations with small parameters, Communi. Comput. Phys., 2025, 38(3), 603–629. doi: 10.4208/cicp.OA-2024-0040 CrossRef Google Scholar