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Manuscript Title: FESSDE 2.2: A new version of 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 2.2
Catalogue identifier: ACVU_v2_0
Distribution format: gz
Journal reference: Comput. Phys. Commun. 115(1998)90
Programming language: Fortran.
Computer: DEC 3000 ALPHA AXP 800.
Operating system: UNIX4.0, AIX3.2.5, SunOs4.1.2, HP/UX9.01, Irix4.05.
RAM: 4.4M words
Word size: 64
Peripherals: disc.
Keywords: Finite element method, Sturm-liouville problem, High-order accuracy, Approximations, Schrodinger equation, Eigensolutions, Ordinary, Differential equations, Atomic, Molecular, Chemical physics, General purpose.
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 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 multidimensional 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 [2]. The generalized algebraic eigenvalue problem A Y = lambda B Y arising from the replacement of the differential problem by the finite-element approximation of high order of accuracy is solved by the subspace iteration method [3].

Summary of revisions:

  1. The code has been modified and optimized to compile under the Fortran 90 compiler.
  2. Output has been cleaned up and reduced.
  3. A new parameter NSITV has been added to the list of input program parameters. It specifies the number of simultaneous iteration vectors to be used.
  4. The SSPACE routine [3] used in the previous version of the FESSDE [4] for the solution of the generalized eigenvalue problem has been replaced with the widely available standard routine F02FJF from the NAG Program Library [1]. This routine is designed for solution of generalized eigenvalue problems for large symmetric banded matrices. The procedure chooses a vector subspace of the full solution space and iterates upon successive solutions in the subspace. Subroutines FACTRS, DOT, IMAGE, MONIT, PROD and SOLVE have been added to the program. The function of each routine is briefly described below:
    • Subroutine F02FJF [1] finds eigenvalues and eigenvectors of a real sparse generalized symmetric eigenvalue problem using the subspace iteration method [3].
    • Subroutine FACTRS calculates L(D)L**T factorization of the stiffness matrix.
    • DOUBLE PRECISION function DOT computes the generalized dot product w**TBz for given n element vectors z and w and mass matrix B.
    • Subroutine IMAGE solves the positive-definite system of equations Aw = Bz for w, where z is a given n element vector, and A and B are the stiffness and mass matrices, respectively.
    • Subroutine MONIT can be used by user to monitor the progress of the F02FJF program.
    • Subroutine PROD evaluates product of the two vectors stored in compact form.
    • Subroutine SOLVE solves a system of linear algebraic equations AX = B for X using L(D)L**T factorization.

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 of [4] 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 the program took 3.22 min of the DECstation 3000 Model 800.

[1] NAG Fortran Library Manual, Mark 15 (The Numerical Algorithms Group Limited, Oxford, 1991).
[2] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 40.
[3] K.J. Bathe, Finite Element Procedures in Engineering Analysis (Prentice-Hall, Englewood Cliffs, NJ, 1982).
[4] A.G. Abrashkevich, D.G. Abrashkevich, M.S. Kaschiev, I.V. Puzynin, Comput. Phys. Commun. 85 (1995) 65.