UDC 517.958:530.145.6 The Algorithms of the Numerical Solution to the Parametric Two-Dimensional Boundary-Value Problem and Calculation Derivative of Solution with Respect to the Parameter and Matrix Elements by the Finite-Element Method A. A. Gusev Laboratory of Information Technologies Joint Institute for Nuclear Research 6, Joliot-Curie str., Dubna, Moscow region, 141980, Russia The effective and stable algorithms for numerical solution with the given accuracy of the parametrical two-dimensional (2D) boundary value problem (BVP)are presented. <...> This BVP formulated for self-adjoined elliptic differential equations with the Dirichlet and/or Neumann type boundary conditions on a finite region of two variables. <...> The original problem is reduced to the parametric homogeneous 1D BVP for a set of ordinary second order differential equations (ODEs). <...> This reduction is implemented by using expansion of the required solution over an appropriate set of orthogonal eigenfunctions of an auxiliary Sturm-Liouville problem by one of the variables. <...> It is obtained by taking a derivative of the reduced problem with respect to the parameter. <...> These problems are solved by the finite-element method with automatical shift of the spectrum. <...> The presented algorithm implemented in Fortran 77 as the POTHEA program calculates with a given accuracy a set ∼ 50 of eigenvalues (potential curves), eigenfunctions and their first derivatives with respect to the parameter, and matrix elements that are integrals of the products of eigenfunctions and/or the derivatives of the eigenfunctions with respect to the parameter. <...> The calculated potential curves and matrix elements can be used for forming the variable coefficients matrixes of a system of ODEs which arises in the reduction of the 3D BVP (d = 3) in the framework of a coupled-channel adiabatic approach or the Kantorovich method. <...> The efficiency and stability of the algorithm are demonstrated by numerical analysis of eigensolutions 2D BVP and evaluated matrix elements which apply to solve the 3D BVP for the Schr¨ odinger equation in hyperspherical coordinates describing a Helium atom with zero angular momentum with help of KANTBP program. <...> Key words and phrases: parametrical two-dimensional boundary value problem, elliptical equation in partial derivatives, finite element method, Kantorovich method, hyperspherical coordinates, Helium atom. 1. <...> Introduction as well as physics of semiconductor nanostructures are described by boundary value problems (BVPs) for the multidimensional equation of Schr¨ Mathematical models of few-body systems in molecular, atomic and nuclear physics <...>