Solving the Poisson equation with singularities by the least-squares collocation method

New h-, p- and hp-versions of the least-squares collocation method are proposed and implemented for solving the Dirichlet problem for the Poisson equation. The paper considers some examples of solving problems with singularities such as large gradients, high growth rate of solution derivatives with increasing the order of differentiation, discontinuity of the second-order derivatives at the angular points of the domain boundary, and the oscillating solution with different frequencies in the presence of an infinite discontinuity for derivatives of any order. The new versions of the method are based on a special selection of collocation points in the roots of the Chebyshev polynomials of the first kind. Basis functions are defined as a product of the Chebyshev polynomials. The behavior of the numerical solution on a sequence of grids and with an increase in the degree of the approximating polynomial has been analyzed using exact analytical solutions. The formulas for the continuation operation necessary for the transition from a coarse mesh to a finer one on a multi-grid complex in the Fedorenko method have been obtained.