Discrete weighted least-squares method for the Poisson and biharmonic problems on domains with smooth boundary

In this article a discrete weighted least-squares method for the numerical solution of elliptic partial differential equations exhibiting smooth solution is presented. It is shown how to create well-conditioned matrices of the resulting system of linear equations using algebraic polynomials, carefully selected matching points and weight factors. Two simple algorithms generating suitable matching points, the Chebyshev matching points for standard two-dimensional domains and the approximate Fekete points of Sommariva and Vianello for general domains, are described. The efficiency of the presented method is demonstrated by solving the Poisson and biharmonic problems with the homogeneous Dirichlet boundary conditions defined on circular and annular domains using basis functions in the form satisfying and in the form not satisfying the prescribed boundary conditions.