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Manuscript Title: FESSDE, a program for the finite-element solution of the coupled- channel Schrodinger equation using high-order accuracy approximations.
Authors: A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin
Program title: FESSDE
Catalogue identifier: ACVU_v1_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 85(1995)65
Programming language: Fortran.
Computer: IBM RS/6000 Model 320H.
Operating system: AIX 3.2.2, ULTRIX 4.2A, DEC OSF/1 V1.2, OPEN VMS.
RAM: 4600K words
Word size: 64
Peripherals: disc.
Keywords: General purpose, Finite element method, High-order accuracy Approximations, Sturm-liouville problem, Schrodinger equation, Eigensolutions, Ordinary Differential equations, Atomic, Molecular, Chemical physics.
Classification: 4.3.

Nature of problem:
Coupled second-order differential equations of the form
    d       dY(x)                                                        
  - -- [P(x)-----] + [U(x) - lambdaR(x)]Y(x) = 0, x contained in [a,b],  
    dx       dx                                                          
 with boundary conditions                                                
              Y(a) = 0 or  -----|   = 0,                                 
                            dx  |x=a                                     
              Y(b) = 0 or  -----|   = 0,                                 
                            dx  |x=b 
are solved. Here lambda is an eigenvalue, Y(x) is an eigenvector, P(x), U(x) and R(x) are symmetrical matrices, P(x) is a diagonal matrix, elements of which are the differentiable functions on a given interval [a,b], and R(x) is a positive matrix throughout the interior of interval [a,b]. Such systems of coupled differential equations usually arise in atomic, molecular and chemical physics calculations after separating the scattering (radial) coordinate from the rest of variables in the multi- dimensional Schrodinger equation. The purpose of this paper is to present the finite element method procedure based on the use of high- order accuracy approximations for high-precision calculation of the approximate eigensolutions for systems of coupled ordinary differential equations.

Solution method:
The coupled differential equations are solved by the finite element method using high-order accuracy approximations [1]. The generalized algebraic eigenvalue problem A Y = lambda B Y arising after the replacement of the differential problem by the finite-element approximation of high order of accuracy is solved by the subspace iteration method [2] using the SSPACE program [2].

The computer memory requirements depend on: a) the number of equations to be solved, b) the order of shape functions and the number of finite elements chosen; and c) the number of eigensolutions required. Restrictions due to dimension sizes may be easily alleviated by altering PARAMETER statements (see Long Write-Up and listing for details). The user must also supply subroutines which evaluate the differential equation coefficient matrices P(x), U(x) and R(x) at a given x.

Running time:
The running time depends critically upon: a) the number of coupled differential equations; b) the number of required eigensolutions; c) the order and number of finite elements on interval [a,b]. The test run which accompanies this paper took 51 s on the DECstation 3000 Model 800.

[1] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev and I.V. Puzynin, Comput. Phys. Commun. (see preceding paper).
[2] K.-J. Bathe, Finite Element Procedures in Engineering Analysis (Prentice-Hall, Englewood Cliffs, New York, 1982).