A well-conditioned and efficient implementation of dual reciprocity method for poisson equation

Khan S. Riaz Khan M. Alqahtani A.M. Shah H.H. Issakhov A. Shah Q. Ei-Shorbagy M.A.
2021 American Institute of Mathematical Sciences

AIMS Mathematics
2021 #6 Issue 11 12560 - 12582 pp.

One of the attractive and practical techniques to transform the domain integrals to equivalent boundary integrals is the dual reciprocity method (DRM). The success of DRM relies on the proper treatment of the non-homogeneous term in the governing differential equation. For this purpose, radial basis functions (RBFs) interpolations are performed to approximate the non-homogeneous term accurately. Moreover, when the interpolation points are large, the global RBFs produced dense and ill-conditioned interpolation matrix, which poses severe stability and computational issues. Fortunately, there exist interpolation functions with local support known as compactly supported radial basis functions (CSRBFs). These functions produce a sparse and well-conditioned interpolation matrix, especially for large-scale problems. Therefore, this paper aims to apply DRM based on multiquadrics (MQ) RBFs and CSRBFs for evaluation of the Poisson equation, especially for large-scale problems. Furthermore, the convergence analysis of DRM with MQ and CSRBFs is performed, along with error estimate and stability analysis. Several experiments are performed to ensure the well-conditioned, efficient, and accurate behavior of the CSRBFs compared to the MQ-RBFs, especially for large-scale interpolation points.

Compactly supported radial basis functions , Condition number , Dual reciprocity method , Multiquadrics , Poisson equation , Stability analysis

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School of Mathematics and Statistics, Central South University, Hunan, Changsha, 410083, China
LSEC and ICMSEC, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, University of Chinese Academy of Sciences, Beijing, 100190, China
Department of Mathematical Sciences, College of Science Princess Nourah bint Abdulrahman University, Saudi Arabia
Department of Mathematical Sciences, Balochistan University of Information Technology Engineering and Management Sciences, Quetta, Pakistan
Department of Mathematical and Computer Modeling, Al-Farabi Kazakh National University, Almaty, 050040, Kazakhstan
Department of Mathematics and Cybernetics, Kazakh British Technical University, Almaty, 050000, Kazakhstan
Department of Basic Sciences, University of Engineering and Technology, Peshawar, Pakistan
Department of Mathematics, College of Science and Humanities in Al-Kharj, Prince Sattam bin Abdulaziz University, Al-Kharj, 11942, Saudi Arabia
Department of Basic Engineering Science, Faculty of Engineering, Menoufia University, Shebin El-Kom, 32511, Egypt

School of Mathematics and Statistics
LSEC and ICMSEC
Department of Mathematical Sciences
Department of Mathematical Sciences
Department of Mathematical and Computer Modeling
Department of Mathematics and Cybernetics
Department of Basic Sciences
Department of Mathematics
Department of Basic Engineering Science

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