Parametric bases for elliptic boundary value problem

We consider the calculation schemes in the framework of Kantorovich method that consist in the reduction of a 3D elliptic boundary-value problem (BVP) to a set of second-order ordinary differential equations (ODEs) using the parametric basis functions. These functions are solution of the 2D parametric BVP. The coefficients in the ODEs are the parametric eigenvalues and the potential matrix elements expressed by the integrals of the eigenfunctions multiplied by their first derivatives with respect to the parameter. We calculate the parametric basis functions numerically in the general case using the high-accuracy finite element method. The efficiency of the proposed calculation schemes and algorithms is demonstrated by the example of the BVP describing the bound states of helium atom.