Generalized Multiquadric Based Direct Global Differential Quadrature Method for Accurate and Stable Solution of Elliptic Problems

This paper investigates the use of the generalized multiquadric (GMQ) radial basis function within the Direct Global Differential Quadrature (DGDQ) method for solving two-dimensional elliptic PDEs. The GMQ kernel, ϕβ(r)=(c2+r2)β, extends the classical MQ and allows systematic variation of the exponent β and shape parameter c. Numerical experiments on the Poisson equation demonstrate that accuracy and conditioning are highly sensitive to these parameters. Very small shape parameters cause ill-conditioning, while moderate values yield stable and accurate approximations. Negative values produce robust though moderately accurate results, whereas larger positive values achieve near-spectral accuracy at the cost of increased sensitivity to conditioning. The novelty of this work lies in extending DGDQ to the full GMQ family, providing a unified polynomial augmentation strategy, and mapping the accuracy–stability trade-off. These findings offer practical guidance for parameter selection and motivate future work on stabilization techniques.