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Volume 18, Issue 1
Meshfree Methods for the Helmholtz Equation with Variable Wave Speed

Zhiyong Liu & Qiuyan Xu

DOI: 10.4208/aamm.OA-2024-0117

Adv. Appl. Math. Mech., 18 (2026), pp. 296-321.

Published online: 2025-10

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  • Abstract

In this paper, we develop one-level and multilevel meshfree radial basis functions (RBF) collocation methods for solving the Helmholtz equation with variable wave speed on a bounded connected Lipschitz domain. The approximate solution is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. We prove the convergence of one-level and multilevel collocation method for the modelling problem in Sobolev spaces. The convergence rates depend on the regularity of the solution, the smoothness of the computational domain, the bounds of frequency and wave speed, the approximation of scaled kernel-based spaces, the increasing rules of scattered data, and the selection of scaling parameters.

  • Keywords

Helmholtz equation, radial basis functions, meshfree methods, collocation, variable wave speed.

  • AMS Subject Headings

65N12, 65N15, 65N35

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-18-296, author = {Liu , Zhiyong and Xu , Qiuyan}, title = {Meshfree Methods for the Helmholtz Equation with Variable Wave Speed}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2025}, volume = {18}, number = {1}, pages = {296--321}, abstract = {

In this paper, we develop one-level and multilevel meshfree radial basis functions (RBF) collocation methods for solving the Helmholtz equation with variable wave speed on a bounded connected Lipschitz domain. The approximate solution is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. We prove the convergence of one-level and multilevel collocation method for the modelling problem in Sobolev spaces. The convergence rates depend on the regularity of the solution, the smoothness of the computational domain, the bounds of frequency and wave speed, the approximation of scaled kernel-based spaces, the increasing rules of scattered data, and the selection of scaling parameters.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2024-0117}, url = {http://global-sci.org/intro/article_detail/aamm/24528.html} }
TY - JOUR T1 - Meshfree Methods for the Helmholtz Equation with Variable Wave Speed AU - Liu , Zhiyong AU - Xu , Qiuyan JO - Advances in Applied Mathematics and Mechanics VL - 1 SP - 296 EP - 321 PY - 2025 DA - 2025/10 SN - 18 DO - http://doi.org/10.4208/aamm.OA-2024-0117 UR - https://global-sci.org/intro/article_detail/aamm/24528.html KW - Helmholtz equation, radial basis functions, meshfree methods, collocation, variable wave speed. AB -

In this paper, we develop one-level and multilevel meshfree radial basis functions (RBF) collocation methods for solving the Helmholtz equation with variable wave speed on a bounded connected Lipschitz domain. The approximate solution is constructed by employing successive refinement scattered data sets and scaled compactly supported radial basis functions with varying support radii. We prove the convergence of one-level and multilevel collocation method for the modelling problem in Sobolev spaces. The convergence rates depend on the regularity of the solution, the smoothness of the computational domain, the bounds of frequency and wave speed, the approximation of scaled kernel-based spaces, the increasing rules of scattered data, and the selection of scaling parameters.

Liu , Zhiyong and Xu , Qiuyan. (2025). Meshfree Methods for the Helmholtz Equation with Variable Wave Speed. Advances in Applied Mathematics and Mechanics. 18 (1). 296-321. doi:10.4208/aamm.OA-2024-0117
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