Continuation for Nonlinear Elliptic Partial Differential Equations Discretized by the Multiquadric Method
A. I. Fedoseyev, M. J. Friedman, E. J. Kansa
TL;DR
This work develops a meshless Multiquadric (MQ) radial basis function discretization for parametrized nonlinear elliptic PDEs and couples it with numerical continuation to detect bifurcations in 1D and 2D problems. By reformulating the MQ discretization to yield a small, dense system and a nodal-value representation, the authors enable use of standard continuation tools (AUTO, CONTENT) to trace solution branches and locate bifurcations. Across 1D Gelfand–Bratu and Brusselator problems and 2D Bratu and Brusselator problems, MQ achieves high accuracy with far fewer unknowns than traditional finite difference methods, though conditioning of the MQ operator remains a challenge. The paper also demonstrates that adaptive node placement near boundaries and adaptive shape parameters can substantially improve bifurcation-detection accuracy, pointing to practical pathways for robust MQ-based continuation in nonlinear PDEs.
Abstract
The Multiquadric Radial Basis Function (MQ) Method is a meshless collocation method with global basis functions. It is known to have exponentional convergence for interpolation problems. We descretize nonlinear elliptic PDEs by the MQ method. This results in modest size systems of nonlinear algebraic equations which can be efficiently continued by standard continuation software such as AUTO and CONTENT. Examples are given of detection of bifurcations in 1D and 2D PDEs. These examples show high accuracy with small number of unknowns, as compared with known results from the literature.