IDEAS home Printed from https://ideas.repec.org/a/eee/apmaco/v397y2021ics0096300320308560.html
   My bibliography  Save this article

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

Author

Listed:
  • 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
    as

    Download full text from publisher

    File URL: http://www.sciencedirect.com/science/article/pii/S0096300320308560
    Download Restriction: Full text for ScienceDirect subscribers only
    File URL: https://libkey.io/10.1016/j.amc.2020.125903?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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).
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Qiang Wang & Pyeoungkee Kim & Wenzhen Qu, 2022. "A Hybrid Localized Meshless Method for the Solution of Transient Groundwater Flow in Two Dimensions," Mathematics, MDPI, vol. 10(3), pages 1-14, February.
    2. Tingting Sun & Peng Wang & Guanjun Zhang & Yingbin Chai, 2022. "A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids," Mathematics, MDPI, vol. 10(16), pages 1-20, August.
    3. 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.
    4. 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).
    5. 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.
    6. Jue Qu & Hongjun Xue & Yancheng Li & Yingbin Chai, 2022. "An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems," Mathematics, MDPI, vol. 10(9), pages 1-12, April.
    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).

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Li, Yancheng & Liu, Cong & Li, Wei & Chai, Yingbin, 2023. "Numerical investigation of the element-free Galerkin method (EFGM) with appropriate temporal discretization techniques for transient wave propagation problems," Applied Mathematics and Computation, Elsevier, vol. 442(C).
    2. 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).
    3. Jue Qu & Hongjun Xue & Yancheng Li & Yingbin Chai, 2022. "An Enriched Finite Element Method with Appropriate Interpolation Cover Functions for Transient Wave Propagation Dynamic Problems," Mathematics, MDPI, vol. 10(9), pages 1-12, April.
    4. Sina Dang & Gang Wang & Yingbin Chai, 2023. "A Novel “Finite Element-Meshfree” Triangular Element Based on Partition of Unity for Acoustic Propagation Problems," Mathematics, MDPI, vol. 11(11), pages 1-21, May.
    5. Yancheng Li & Sina Dang & Wei Li & Yingbin Chai, 2022. "Free and Forced Vibration Analysis of Two-Dimensional Linear Elastic Solids Using the Finite Element Methods Enriched by Interpolation Cover Functions," Mathematics, MDPI, vol. 10(3), pages 1-21, January.
    6. Qu, Wenzhen & Sun, Linlin & Li, Po-Wei, 2021. "Bending analysis of simply supported and clamped thin elastic plates by using a modified version of the LMFS," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 185(C), pages 347-357.
    7. Gui, Qiang & Li, Wei & Chai, Yingbin, 2023. "The enriched quadrilateral overlapping finite elements for time-harmonic acoustics," Applied Mathematics and Computation, Elsevier, vol. 451(C).
    8. Xunbai Du & Sina Dang & Yuzheng Yang & Yingbin Chai, 2022. "The Finite Element Method with High-Order Enrichment Functions for Elastodynamic Analysis," Mathematics, MDPI, vol. 10(23), pages 1-27, December.
    9. Cong Liu & Shaosong Min & Yandong Pang & Yingbin Chai, 2023. "The Meshfree Radial Point Interpolation Method (RPIM) for Wave Propagation Dynamics in Non-Homogeneous Media," Mathematics, MDPI, vol. 11(3), pages 1-27, January.
    10. Yingbin Chai & Kangye Huang & Shangpan Wang & Zhichao Xiang & Guanjun Zhang, 2023. "The Extrinsic Enriched Finite Element Method with Appropriate Enrichment Functions for the Helmholtz Equation," Mathematics, MDPI, vol. 11(7), pages 1-25, March.
    11. Chai, Yingbin & Li, Wei & Liu, Zuyuan, 2022. "Analysis of transient wave propagation dynamics using the enriched finite element method with interpolation cover functions," Applied Mathematics and Computation, Elsevier, vol. 412(C).
    12. Tingting Sun & Peng Wang & Guanjun Zhang & Yingbin Chai, 2022. "A Modified Radial Point Interpolation Method (M-RPIM) for Free Vibration Analysis of Two-Dimensional Solids," Mathematics, MDPI, vol. 10(16), pages 1-20, August.
    13. Nikan, O. & Avazzadeh, Z., 2021. "A localisation technique based on radial basis function partition of unity for solving Sobolev equation arising in fluid dynamics," Applied Mathematics and Computation, Elsevier, vol. 401(C).
    14. Xi, Qiang & Fu, Zhuojia & Wu, Wenjie & Wang, Hui & Wang, Yong, 2021. "A novel localized collocation solver based on Trefftz basis for potential-based inverse electromyography," Applied Mathematics and Computation, Elsevier, vol. 390(C).

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:eee:apmaco:v:397:y:2021:i:c:s0096300320308560. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Catherine Liu (email available below). General contact details of provider: https://www.journals.elsevier.com/applied-mathematics-and-computation .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.