A polynomial-augmented RBF collocation method using ghost points for 2D elliptic problems with nonlocal boundary conditions

• Published in: Computational particle mechanics
• Main Author: Oruç, Ömer
• Format: Journal Article
• Language: English
• Published: 12.06.2025
• ISSN: 2196-4378; 2196-4386
• DOI: 10.1007/s40571-025-00979-0

Two-dimensional (2D) elliptic partial differential equations (PDEs) with nonlocal boundary conditions are solved numerically by using polynomial-augmented radial basis function (RBF) collocation method using ghost points. Unlike traditional RBF collocation methods which select centers on problem domain we utilize ghost points as centers that may be placed outside of problem domain. This approach is used first time in the context of RBF collocation methods for the elliptic PDEs with nonlocal boundary conditions and significantly improves accuracy of the method when compared with traditional RBF collocation methods. Moreover, adding polynomial basis to RBF along with using ghost points decreases sensitivity of the proposed method to changes in shape parameter of the RBF. We perform some numerical simulations using the proposed method. In cases where the exact solution is available, the accuracy of the proposed method is examined by comparing the obtained numerical solutions with the exact solutions. In order to measure the reliability of the proposed method in cases where the exact solution is not available, the numerical solutions obtained via the proposed method are compared with the solutions obtained by the finite element method. Numerical simulations and comparisons confirm accuracy and reliability of the proposed method.