Finite Element Method of High-Order Accuracy for Solving Two Dimensional Elliptic Boundary-Value Problems of Two and Three Identical Atoms in a Line

We considered models of three identical atoms in a line with molecular pair interactions and diatomic molecule scattered by an atom or tunneling through potential barriers. The models are formulated as 2D elliptic boundary-value problems (BVPs) in the Jacobi and polar coordinates. The BVP in Jacobi coordinates solved by finite element method of high-order accuracy for discrete spectrums of models under consideration. To solve the scattering problems the BVP in polar coordinates are reduced by means of Kantorovich method to a system of second-order ordinary differential equations with respect to the radial variable using the expansion of the desired solutions in the set of angular basis functions that depend on the radial variable as a parameter. The efficiency of the elaborated method, algorithms and programs is demonstrated by benchmark calculations of the resonance scattering, metastable and bound states of the considered models and also by a comparison of results for bound states of the three atomic system in the framework of direct solving the BVP by FEM and Kantorovich reduction.