An overlapping domain decomposition Schwarz method applied to the method of fundamental solution

• Published in: Computational & applied mathematics 2021-12, Vol.40 (8), Article 295
• Main Authors: Shanazari, Kamal, Mohammadi, Nasrin
• Format: Article
• Language: English
• Subjects: Applications of Mathematics; Applied physics; Computational mathematics; Computational Mathematics and Numerical Analysis; Domain decomposition methods; Elliptic functions; Gauss-Seidel method; Ill-conditioned problems (mathematics); Interpolation; Iterative methods; Mathematical analysis; Mathematical Applications in Computer Science; Mathematical Applications in the Physical Sciences; Mathematics; Mathematics and Statistics; Partial differential equations; Radial basis function; Robustness (mathematics)
• ISSN: 2238-3603; 1807-0302
• DOI: 10.1007/s40314-021-01669-2

In this paper, we propose an overlapping domain decomposition method (DDM) to improve the method of fundamental solution (MFS) for the elliptic partial differential equations. The MFS often solves the Poisson-type equations by the use of a particular solution which is obtained by the radial basis functions (RBFs) interpolation. Although the MFS is a boundary-type method, the interior points are essentially needed in approximating the particular solution. Consequently, when dealing with large-scale problems, the condition number of the RBF interpolation matrix considerably increases. To treat the ill-conditioning, we apply an overlapping DDM to localize the globally supported RBFs. In addition, we present a robust formulation of the Schwarz method based on MFS and the particular solution matrix. Furthermore, we show that the iteration involved in the Schwarz method is equivalent to the iterative Gauss–Seidel method. Some numerical examples will be given to demonstrate the performance of the suggested method.