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Manuscript Title: POTHEA: A program for computing eigenvalues and eigenfunctions and their first derivatives with respect to the parameter of the parametric self-adjoined 2D elliptic partial differential equation
Authors: A.A. Gusev, O. Chuluunbaatar, S.I. Vinitsky, A.G. Abrashkevich
Program title: POTHEA
Catalogue identifier: AESX_v1_0
Distribution format: tar.gz
Journal reference: Comput. Phys. Commun. 185(2014)2636
Programming language: FORTRAN 77.
Computer: Personal computer.
Operating system: Unix/Linux Windows.
RAM: depends on
a) the number of differential equations;
b) the number and order of finite elements; and
c) the number of eigensolutions required
Keywords: Eigenvalue and multichannel scattering problems, Kantorovich method, Finite element method, Multichannel adiabatic approximation, Ordinary differential equations, High-order accuracy approximations.
PACS: 02.30.Hq, 02.60.Jh, 02.60.Lj, 03.65.Nk, 31.15.Ja, 31.15.Pf, 34.50.-s, 34.80.Bm.
Classification: 2.1, 2.4.

External routines: SSPACE[1], GAULEG [2]

Nature of problem:
Solutions of boundary value problems (BVPs) for the elliptic partial differential equations (PDEs) of the Schrödinger type find wide application in molecular, atomic and nuclear physics, for example, in three-dimensional tunneling of a diatomic molecule incident upon a potential barrier, fission model of collision of heavy ions, fragmentation of light nuclei, a hydrogen atom in magnetic field, photoionization of helium like atoms, one photon ionization of atoms, electron-impact ionization of molecular Hydrogen and photodissociation of molecules in strong laser field [3,4]. In the coupled-channel adiabatic approach (CCAA) [4], known in mathematical physics as Kantorovich method (KM), desirable solution of the original boundary value problem (BVP) is expanded over surface eigenfunctions in fast variables (for example, angular variables) of an auxiliary BVP for an appropriate PDE dependent on a slow variable (for example, radial variable) as a parameter. Averaging of the original BVP over the surface eigenfunctions leads to 1D BVP for a system of the coupled second-order ordinary differential equations (SOODEs) containing the potential matrix elements and first-derivative coupling terms that are integrals of the products of the surface eigenfunctions and/or the first derivatives of the surface eigenfunctions with respect to a parameter [4]. The purpose of this paper is to present the finite element method procedure based on the use of high-order accuracy approximations for calculating eigenvalues, surface eigenfunctions and their first derivatives with respect to a parameter of the parametric BVP for self-adjoined 2D PDE with the Dirichlet and/or Neumann type boundary conditions on a finite 2D region which arise at the reduction of the 3D BVP to 1D BVP for a system of coupled SOODEs in the framework of CCAA. The program developed calculates potential matrix elements that are integrals of the products of the surface eigenfunctions and/or the first derivatives of the surface eigenfunctions with respect to a parameter. These eigenvalues and potential matrix elements can be used for solving the bound state and multi-channel scattering problems for a system of the coupled SOODEs with the help of the KANTBP programs [5].

Solution method:
We seek the desirable solution of the parametric 2D BVP in the form of expansion in the basis functions of the auxiliary Sturm-Liouville problem with respect to one of the fast variables. They are chosen in analytical form or calculated by the ODPEVP program [6]. The coefficients of the expansion are vector-eigenfunctions of the parametric homogeneous 1D BVP for a system of the coupled SOODEs obtained by averaging the original 2D problem over the basis functions. First derivatives with respect to the parameter of these vector-eigenfunctions and eigenvalues are solutions of the parametric inhomogeneous 1D BVP, obtained by taking a derivative of the parametric homogeneous 1D BVP with respect to the parameter [7]. Then, we solve the reduced parametric homogeneous and inhomogeneous 1D BVPs by the finite element method using high-order accuracy approximations [6]. The generalized algebraic eigenvalue problem A F = E B F with respect to a pair of unknown (E, F) arising after the replacement of the differential problem by the finite-element approximation is solved by the subspace iteration method using the SSPACE program [1]. First derivatives of the vector-eigenfunctions and eigenvalues with respect to the parameter are obtained by solving the inhomogeneous algebraic equations in accordance with the algorithm used in [6]. Finally, we evaluate the desirable matrix elements using the calculated eigenvalues, vector-eigenfunctions and their derivatives, which can be applied to generate the coupled system equations in the slow variable in the CCAA.
Benchmark calculations of eigenvalues and eigenfunctions of the ground and first excited states of a Helium atom in the framework of a coupled-channel hyperspherical adiabatic approach are presented. Additionally a convergence of the eigenvalues versus both the number of the parametric vector-eigenfunctions and the number of their components is studied. As a test desk, the program is applied to the calculation of the eigensolutions and their first derivatives with respect to the parameter of the parametric 2D BVP including evaluation of matrix elements, which are used in the benchmark calculations.

Restrictions:
The computer memory requirements depend on:
a) the number of differential equations;
b) the number and order of finite elements; and
c) the number of eigensolutions required.
Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up). The user must also supply subroutine POTCLC for evaluating potential matrix elements. The user must also supply additional three DOUBLE PRECISION functions (see Long Write-Up for details).

Running time:
The running time depends critically upon:
a) the number of differential equations;
b) the number and order of finite elements; and
c) the number of eigensolutions required.
The test run which accompanies this paper took 15 s with calculation of matrix potentials on computer Intel Core i5 CPU 3.33 GHz, 4 GB RAM, Windows 7. This test run requires 10MB of disk storage.

References:
[1] K.J. Bathe, Finite Element Procedures in Engineering Analysis, Englewood Cliffs, Prentice Hall, New York, 1982.
[2] W.H. Press, S.A. Teukolsky, W.T. Vetterling and B.P. Flannery, Numerical Recipes: The Art of Scientific Computing, Cambridge University Press, Cambridge, 1986.
[3] M. Shapiro and P. Brumer, Quantum Control of Molecular Processes, Weley VCH, Verlag GmbH and Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany, 2012.
[4] U. Fano, Rep. Progr. Phys. 46 (1983) 97-165; U. Fano and A.R.P. Rau, Atomic Collisions and Spectra, Academic Press, Florida, 1986.
[5] O. Chuluunbaatar, A.A. Gusev, A.G. Abrashkevich, A. Amaya-Tapia, M.S. Kaschiev, S.Y. Larsen and S.I. Vinitsky, Comput. Phys. Commun. 177(2007) 649-675.
[6] O. Chuluunbaatar, A.A. Gusev, S.I. Vinitsky and A.G. Abrashkevich, Comput. Phys. Commun. 180(2009)1358-1375.
[7] A.G. Abrashkevich, M.S. Kaschiev and S.I. Vinitsky, J. Comp. Phys. 163 (2000) 328-348.