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

• Main Authors: Suliman Khan, M. Riaz Khan, Aisha M. Alqahtani, Hasrat Hussain Shah, Alibek Issakhov, Qayyum Shah, M. A. EI-Shorbagy
• Format: Article
• Language: English
• Published: AIMS Press 2021-08-01
• Series: AIMS Mathematics
• Subjects: dual reciprocity method; multiquadrics; compactly supported radial basis functions; stability analysis; condition number; poisson equation
• Online Access: https://aimspress.com/article/doi/10.3934/math.2021724?viewType=HTML

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.