Continuation for Nonlinear Elliptic Partial Differential Equations Discretized by the Multiquadric Method

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.