Iranian Journal of Mathematical Chemistry

Iranian Journal of Mathematical Chemistry

An Innovative Numerical Approach for Solving Two-Dimensional Helmholtz Problem

Document Type : Research Paper

Authors
Soheila Heidari
Mahmoud Behroozifar
Department of Mathematics‎, ‎Babol Noshirvani University of Technology‎, ‎Babol‎, ‎Iran
10.22052/ijmc.2026.258327.2112
Abstract
‎This paper proposes a new spectral approach for the two-dimensional Helmholtz equation with Dirichlet boundary conditions‎. ‎We construct a closed-form two-dimensional generalized interpolation function that exactly satisfies arbitrary (non-homogeneous) boundary data‎, ‎reducing the original problem to an equivalent one with homogeneous boundary conditions‎. ‎To approximate the homogenized problem‎, ‎we introduce a two-variable family of homogenizing Gegenbauer polynomials (HGPs) that vanish identically on the domain boundary‎, ‎and show that the resulting basis is semi-orthogonal with a banded (pentadiagonal) Gram matrix‎, ‎preserving the computational efficiency of spectral methods‎. ‎A Galerkin discretization then reduces the problem to a system of algebraic equations‎. ‎We derive an explicit upper bound on the approximation error and prove a spectral (exponential) convergence rate for analytic solutions‎. ‎Eight benchmark problems‎ - ‎including Helmholtz and Poisson equations with separable and non-separable exact solutions‎, ‎constant and variable coefficients‎, ‎and a classical torsion-problem benchmark‎ - ‎are used to assess accuracy‎, ‎stability under boundary and source-term perturbations‎, ‎and CPU efficiency‎. ‎The proposed method achieves low RMSE values with low polynomial degree $(M \leq 4)$ and modest CPU time‎, ‎and compares favorably with existing finite element‎, ‎finite difference‎, ‎and spectral element approaches reported in the literature‎.
Keywords
Helmholtz problem
Homogenizing Gegenbauer polynomials
Generalized interpolation
Spectral method
Non-homogeneous boundary conditions
Subjects
[1] L. L. Thompson, A review of finite-element methods for time-harmonic acoustics, J. Acoust. Soc. Am. 119 (2006) 1315–1330, https://doi.org/10.1121/1.2164987.
[2] A. De La Bourdonnaye, Some formulations coupling finite element and integral equation methods for Helmholtz equation and electromagnetism, Numer. Math. 69 (1995) 257–268, https://doi.org/10.1007/s002110050091.
[3] J.-H. Li, X.-Y. Hu, S.-H. Zeng, J.-G. Lu, G.-P. Huo, B. Han and R.-H. Peng, Threedimensional forward calculation for loop source transient electromagnetic method based on electric field Helmholtz equation, Chin. J. Geophys. 56 (2013) 4256–4267.
[4] B. R. McLeod, A. A. Pilla and M. W. Sampsel, Electromagnetic fields induced by Helmholtz aiding coils inside saline-filled boundaries, Bioelectromagnetics 4 (1983) 357–370, https://doi.org/10.1002/bem.2250040407.
[5] D. M. Pozar, Microwave Engineering, Fourth ed., John Wiley & Sons, Inc, 2012.
[6] N. H. Tuan, V. A. Khoa, M. N. Minh and T. Tran, Reconstruction of the electric field of the Helmholtz equation in three dimensions, J. Comput. Appl. Math. 309 (2017) 56–78.
[7] I. Baydoun, D. Baresch, R. Pierrat and A. Derode, Radiative transfer of acoustic waves in continuous complex media: Beyond the Helmholtz equation, Phys. Rev. E 94 (2016) #053005.
[8] W. Tobocman, Calculation of acoustic wave scattering by means of the Helmholtz integral equation. II, J. Acoust. Soc. Am. 76 (1984) 1549–1554, https://doi.org/10.1121/1.391465.
[9] S.-A. Yang, Acoustic scattering by a hard or soft body across a wide frequency range by the Helmholtz integral equation method, J. Acoust. Soc. Am. 102 (1997) 2511-2520, https://doi.org/10.1121/1.420305.
[10] D. J. Griffiths and D. F. Schroeter, Introduction to Quantum Mechanics, Cambridge University Press, 2018.
[11] P. Amore, Solving the Helmholtz equation for membranes of arbitrary shape: numerical results, J. Phys. A: Math. Theor. 41 (2008) #265206, https://doi.org/ 10.1088/1751-8113/41/26/265206.
[12] S. Yu. Reutskiy, Vibration analysis of arbitrarily shaped membranes, Comput. Model. Eng. Sci. 51 (2009) 115–142, https://doi.org/10.3970/cmes.2009.051.115.
[13] B. Borden, Mathematical problems in radar inverse scattering, Inverse Problems. 18 (2001) R1–R28, https://doi.org/10.1088/0266-5611/18/1/201.
[14] W. S. Hall and X. Q. Mao, A boundary element investigation of irregular frequencies in electromagnetic scattering, Eng. Anal. Bound. Elem. 16 (1995) 245-252, https://doi.org/10.1016/0955-7997(95)00068-2.
[15] J. D. Anderson Jr., Fundamentals of Aerodynamics, McGraw-Hill, New York, 1991.
[16] S. A. Khan, Wavelength-dependent modifications in Helmholtz Optics, Int. J. Theor. Phys. 44 (2005) 95–125, https://doi.org/10.1007/s10773-005-1488-0.
[17] E. Hecht, Optik, Walter de Gruyter GmbH & Co KG, 2023.
[18] L. F. Canino, J. J. Ottusch, M. A. Stalzer, J. L. Visher and S. M. Wandzura, Numerical solution of the Helmholtz equation in 2D and 3D using a high-order Nyström discretization, J. Comput. Phys. 146 (1998) 627–663.
[19] G. Bao, G. W. Wei and S. Zhao, Numerical solution of the Helmholtz equation with high wavenumbers, Internat. J. Numer. Methods Engrg. 59 (2004) 389-408, https://doi.org/10.1002/nme.883.
[20] M. M. Gupta, J. Kouatchou and J. Zhang, Comparison of second-and fourth-order discretizations for multigrid Poisson solvers, J. Comput. Phys. 132 (1997) 226-232, https://doi.org/10.1006/jcph.1996.5466.
[21] D. Veerababu and P. K. Ghosh, Solving 2-D Helmholtz equation in the rectangular, circular, and elliptical domains using neural networks, J. Sound Vib. 607 (2025)#119022, https://doi.org/10.1016/j.jsv.2025.119022.
[22] A. Kaya and M. A. Freitag, Conditioning analysis for discrete Helmholtz problems, Comput. Math. Appl. 118 (2022) 171–182, https://doi.org/10.1016/j.camwa.2022.05.016.
[23] O. Z. Mehdizadeh and M. Paraschivoiu, Investigation of a two-dimensional spectral element method for Helmholtz’s equation, J. Comput. Phys. 189 (2003) 111–129, https://doi.org/10.1016/S0021-9991(03)00204-3.
[24] M. Dehghan, M. Nourian and M. B. Menhaj, Numerical solution of Helmholtz equation by the modified Hopfield finite difference techniques, Numer. Methods Partial Differential Equations 25 (2009) 637–656, https://doi.org/10.1002/num.20366.
[25] R. K. Mohanty, Sixth order compact multi-phase block-AGE iteration methods for computing 2D Helmholtz equation, MethodsX. 12 (2024) #102633, https://doi.org/10.1016/j.mex.2024.102633.
[26] D. A. Juraev, P. Agarwal, E. E. Elsayed and N. Targyn, Applications of the Helmholtz equation, Adv. Eng. Days 8 (2023) 28–30.
[27] A. Szabo and N. S. Ostlund, Modern Quantum Chemistry: Introduction to Advanced Electronic Structure Theory, Courier Corporation, 1996.
[28] J. M. Thijssen, Computational Physics, 2nd ed., Cambridge University Press, Cambridge, 2007.
[29] G. Szegö, Orthogonal Polynomials, American Mathematical Society Colloquium Publications, Vol. 23, American Mathematical Society, Providence, RI, 1939.
[30] J. P. Boyd, Chebyshev and Fourier Spectral Methods, 2nd ed., Dover Publications, Mineola, NY, 2000.
[31] J. Shen, T. Tang and L.-L.Wang, Spectral Methods: Algorithms, Analysis and Applications, Springer, Berlin, 2011.
[32] C. Canuto, M. Y. Hussaini, A. Quarteroni and T. A. Zang, Spectral Methods: Fundamentals in Single Domains, Springer, Berlin, 2006.
[33] E. Kreyszig, Introductory Functional Analysis with Applications, John Wiley & Sons, 1991.
[34] E. Isaacson and H. B. Keller, Analysis of Numerical Methods, Courier Corporation, 2012.
[35] J. Stoer, R. Bulirsch, R. Bartels, W. Gautschi and C. Witzgall, Introduction to Numerical Analysis, Springer, New York, 1980.
[36] S. M. El-Sayed and D. Kaya, Comparing numerical methods for Helmholtz equation model problem, Appl. Math. Comput. 150 (2004) 763–773, https://doi.org/10.1016/S0096-3003 (03)00305-9.
[37] I. Singer and E. Turkel, High-order finite difference methods for the Helmholtz equation, Comput. Methods Appl. Mech. Engrg. 163 (1998) 343–358, https://doi.org/10.1016/S0045-7825(98)00023-1.
[38] H. C. Elman, D. J. Silvester and A. J. Wathen, Finite Elements and Fast Iterative Solvers: with Applications in Incompressible Fluid Dynamics, 2nd ed., Oxford University Press, Oxford, 2014.
[39] S. J. Kahlaf and A. A. Mhassin, Numerical solution of a two-dimensional Helmholtz equation with Dirichlet boundary conditions, J. Interdiscip. Math. 24 (2021) 971–982, https://doi.org/10.1080/09720502.2021.1885815.
[40] R. Haberman, Applied Partial Differential Equations with Fourier Series and Boundary Value Problems, 5th ed., Pearson, Boston, 2012.
[41] B. Mebrate and P. R. Koya, Numerical solution of a two dimensional poisson equation with dirichlet boundary conditions, Am. J. Appl. Math. 3 (2015) 297–304, https://doi.org/10.11648/j.ajam.20150306.19.
[42] S. Timoshenko and J. N. Goodier, Theory of Elasticity, 3rd ed., McGraw-Hill, New York, 1970.
Iranian Journal of Mathematical Chemistry
Volume 17, Issue 4
Autumn 2026
Pages 415-437