The collocation solution of Poisson problems based on approximate Fekete points

The aim of this work is to show how the collocation method may be used for the approximate solution of Poisson problems on planar domains with a smooth boundary in a stable and efficient way. The most important aspect of this work consists in the use of approximate Fekete points recently developed by Sommariva and Vianello. Numerical experiments concerning the collocation solution of Poisson problems defined on the unit disc and an eccentric annulus with the homogeneous Dirichlet boundary conditions are presented. Two sets of trial functions, consisting of algebraic polynomials, satisfying and not satisfying the prescribed boundary conditions are considered. As the presented results show, these easily computable collocation points, giving well-conditioned collocation matrices, open new horizons for the collocation solution of elliptic partial differential equations considered on planar and higher-dimensional domains.