Abstract
New h-, p-, and hp-versions of the least-squares collocation method (LSCM) are proposed and implemented for solving the Dirichlet problem for a Poisson equation. Examples are considered of solving problems with singularities such as large gradients, fast growing rate of solution derivatives with increasing order of differentiation, discontinuities of second-order derivatives at angular points of the domain boundary, and solution oscillations with various frequencies for an infinite discontinuity point for derivatives of any order. The new versions of the method are based on a special selection of collocation points in the roots of Chebyshev polynomials of the first kind. The basis functions are defined as products of Chebyshev polynomials. The behavior of numerical solutions on a sequence of grids with an increasing degree of the approximating polynomial is analyzed by using exact analytical solutions. Formulas for the extension operation of transition from a coarse grid to a finer one on a multi-grid complex in the Fedorenko method are obtained.
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