Kernel-Based Meshless Collocation Methods for Solving Coupled Bulk–Surface Partial Differential Equations

• Published in: Journal of scientific computing Vol. 81; no. 1; pp. 375 - 391
• Main Authors: Chen, Meng, Ling, Leevan
• Format: Journal Article
• Language: English
• Published: New York Springer US 01.10.2019; Springer Nature B.V
• Subjects: Algorithms; Boundary conditions; Collocation methods; Computational Mathematics and Numerical Analysis; Estimates; Finite element analysis; Fourier transforms; Least squares; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Meshless methods; Partial differential equations; Robustness (mathematics); Smoothness; Theoretical; 65N35; 41A63; Error estimate; Coupled bulk–surface PDEs; Smoothness orders of global and restricted kernels; 65D15; Meshless collocation methods
• ISSN: 0885-7474; 1573-7691
• DOI: 10.1007/s10915-019-01020-2

A meshless kernel-based method is developed to solve coupled second-order elliptic PDEs in bulk domains and surfaces, subject to Robin boundary conditions. It combines a least-squares kernel collocation method with a surface-type intrinsic approach. Therefore, we can use each pair for discrete point sets, RBF kernels (globally and restrictedly), trial spaces, and some essential assumptions, for the search of least-squares solutions in bulks and on surfaces respectively. We first give error estimates for domain-type Robin-boundary problems. Based on this and existing results for surface PDEs, we discuss the theoretical requirements for the employed Sobolev kernels. Then, we select the orders of smoothness for the kernels in bulks and on surfaces. Lastly, several numerical experiments are demonstrated to test the robustness of the coupled method for accuracy and convergence rates under different settings.