Newton iteration with multiquadrics for the solution of nonlinear PDEs

Newton iteration is a standard tool for the numerical solution of nonlinear partial differential equations. We show how globally supported multiquadric radial basis functions can be used for this task. One of the insights gained is that the use of coarse meshes during the initial iterations along with a multiquadric parameter which is adjusted with the meshsize increases the efficiency and stability of the resulting algorithm. Some experiments with Nash iteration are also included.