Finite-element solution of the coupled-channel Schrödinger equation using high-order accuracy approximations

• Published in: Computer physics communications Vol. 85; no. 1; pp. 40 - 64
• Main Authors: Abrashkevich, A.G., Abrashkevich, D.G., Kaschiev, M.S., Puzynin, I.V.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 1995; Elsevier Science
• Subjects: Computational techniques; Exact sciences and technology; Finite-element and galerkin methods; Mathematical methods in physics; Physics; Second order equation; Neumann problem; Sturm Liouville problem; Boundary conditions; Convergence; Coupled channel approximation; Finite element method; Linear equation; Numerical solution; Differential equations; Dirichlet problem; Schroedinger equation; Bound states
• ISSN: 0010-4655; 1879-2944
• DOI: 10.1016/0010-4655(94)00106-C

The finite element method (FEM) is applied to solve the bound state (Sturm-Liouville) problem for systems of ordinary linear second-order differential equations. The convergence, accuracy and the range of applicability of the high-order FEM approximations (up to tenth order) are studied systematically on the basis of numerical experiments for a wide set of quantum-mechanical problems. The analytical and tabular forms of giving the coefficients of differential equations are considered. The Dirichlet and Neumann boundary conditions are discussed. It is shown that the use of the FEM high-order accuracy approximations considerably increases the accuracy of the FE solutions with substantial reduction of the requirements on the computational resources. The results of the FEM calculations for various quantum-mechanical problems dealing with different types of potentials used in atomic and molecular calculations (including the hydrogen atom in a homogeneous magnetic field) are shown to be well converged and highly accurate.