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Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains

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  • Wang, Fajie
  • Zhao, Qinghai
  • Chen, Zengtao
  • Fan, Chia-Ming

Abstract

In this paper, a novel collocation method is presented for the efficient and accurate evaluation of the two-dimensional elliptic partial differential equation. In the new method, the physical domain is discretized into a series of overlapping small (local) subdomains, and in each of the subdomain, a localized Chebyshev collocation method is applied in which the unknown functions at every node can be computed by using a linear combination of unknowns at its near-by nodes. The Chebyshev polynomials employed here can provide the spectral accuracy of new approach. The concept of the local subdomain is introduced to derive a sparse system, which ensures the feasibility for large-scale simulation. This paper aims at proposing a new method to solve general partial differential equations accurately and efficiently. Several numerical examples including Poisson equation, Helmholtz-type equation and transient heat conduction equation are provided to demonstrate the validity and applicability of the proposed method. Numerical experiments indicate that the localized Chebyshev collocation method is very promising for the efficient and accurate solution of large-scale problems.

Suggested Citation

  • Wang, Fajie & Zhao, Qinghai & Chen, Zengtao & Fan, Chia-Ming, 2021. "Localized Chebyshev collocation method for solving elliptic partial differential equations in arbitrary 2D domains," Applied Mathematics and Computation, Elsevier, vol. 397(C).
    • Handle: RePEc:eee:apmaco:v:397:y:2021:i:c:s0096300320308560
    DOI: 10.1016/j.amc.2020.125903
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    References listed on IDEAS

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    1. You, Xiangyu & Li, Wei & Chai, Yingbin, 2020. "A truly meshfree method for solving acoustic problems using local weak form and radial basis functions," Applied Mathematics and Computation, Elsevier, vol. 365(C).
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    Cited by:

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    3. Gábor Turzó & Ildikó-Renáta Száva & Sándor Dancsó & Ioan Száva & Sorin Vlase & Violeta Munteanu & Teofil Gălățanu & Zsolt Asztalos, 2022. "A New Approach in Heat Transfer Analysis: Reduced-Scale Straight Bars with Massive and Square-Tubular Cross-Sections," Mathematics, MDPI, vol. 10(19), pages 1-24, October.
    4. Zhang, Shangyuan & Nie, Yufeng, 2023. "Localized Chebyshev and MLS collocation methods for solving 2D steady state nonlocal diffusion and peridynamic equations," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 206(C), pages 264-285.
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    6. Li, Yang & Liu, Dejun & Yin, Zhexu & Chen, Yun & Meng, Jin, 2023. "Adaptive selection strategy of shape parameters for LRBF for solving partial differential equations," Applied Mathematics and Computation, Elsevier, vol. 440(C).
    7. Liang Zhang & Qinghai Zhao & Jianliang Chen, 2022. "Reliability-Based Topology Optimization of Thermo-Elastic Structures with Stress Constraint," Mathematics, MDPI, vol. 10(7), pages 1-22, March.
    8. Xingxing Yue & Buwen Jiang & Xiaoxuan Xue & Chao Yang, 2022. "A Simple, Accurate and Semi-Analytical Meshless Method for Solving Laplace and Helmholtz Equations in Complex Two-Dimensional Geometries," Mathematics, MDPI, vol. 10(5), pages 1-9, March.
    9. Sun, Linlin & Fu, Zhuojia & Chen, Zhikang, 2023. "A localized collocation solver based on fundamental solutions for 3D time harmonic elastic wave propagation analysis," Applied Mathematics and Computation, Elsevier, vol. 439(C).

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