NESC0806: 2DEPEP, Partial Differencial Equation Solution and Eigenvalues for Potential and Diffusion Problems

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NESC0806 2DEPEP (Abstract last modified 23-DEC-1980)

2DEPEP, Partial Differencial Equation Solution and Eigenvalues for Potential and Diffusion Problems

Abstract Table of Contents:

1. NAME OR DESIGNATION OF PROGRAM - 2DEPEP

2. COMPUTER FOR WHICH PROGRAM IS DESIGNED AND OTHER MACHINE VERSION PACKAGES AVAILABLE -

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Explanation of the status codes.

Program-name Package-ID Status 2DEPEP NESC0806/01 Tested

Machines used: Package-ID Orig.Computer Test Computer NESC0806/01 CDC 7600 CDC 7600

3. DESCRIPTION OF PROBLEM OR FUNCTION - 2DEPEP  solves   the  partial
differential equation system:
   C1(X,Y,U,V,T)*DU/DT=D(OXX)/DX+D(OXY)/DY+F1(X,Y,U,V,T)
   C2(X,Y,U,V,T)*DV/DT=D(OYX)/DX+D(OYY)/DY+F2(X,Y,U,V,T)   in    a
general two-dimensional region, R, with
   U=FB1(S)
   V=FB2(S) for S on BR1, and
   OXX*NX+OXY*NY=GB1(S,U,V,T)
   OYX*NX+OYY*NY=GB2(S,U,V,T) for S on BR2,  where BR1 and BR2 are
distinct parts of the boundary.  (NX,NY)= unit outward normal.
   U=U0(X,Y)
   V=V0(X,Y) for T=T0, and
   OXX=OXX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OXY=OXY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OYX=OYX(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T)
   OYY=OYY(X,Y,DU/DX,DU/DY,DV/DX,DV/DY,T).
   The Jacobian  matrices of  derivatives of  OXX,OXY,OYX,OYY with
respect to  DU/DX,DU/DY,DV/DX,DV/DY and of  F1,F2 with  respect to
U,V and of GB1,GB2 with respect to U,V must be symmetric.
   The related elliptic and eigenvalue problems are also solved by
2DEPEP and single equations can  be handled efficiently.  Examples
of applications  of  the  program are  elasticity,  one-  or  two-
component diffusion,  heat conduction, minimal  surface, potential
problems and  the time-independent and  time-dependent Schrodinger
equations.

4. METHOD OF SOLUTION - The  user supplies  an initial  triangulation
which  defines the  region R.   This triangulation  is refined  as
controlled  by a  user-supplied function.   For singular  elliptic
problems, optimal order  convergence is possible if  this function
approximates the function  D3MAX(UV), which is the  maximum of the
third derivatives of U and V.
   The problem is discretized by  Galerkin's method, using a trial
function space of piecewise quadratic  polynomials with respect to
the triangulation.   An approximate solution is  calculated, using
either the implicit  or the Crank-Nicolson  method for  each time-
step.  The non-linear  equations which must  be solved  each time-
step are solved  by Newton's method.  One  iteration is sufficient
since the solution on the previous  time-step is used for starting
values.  In  each application  of Newton's  method the  symmetric,
banded   Jacobian  matrix   is  inverted   directly  by   Gaussian
elimination, with  the matrix  stored out-of-core  when necessary,
according  to the  frontal  method.   For linear  time-independent
problems the Jacobian is the same  each time-step, so the Cholesky
factorization done  in the  first step  is used  on all  following
steps.   The  isoparametric  method  is   used  to  handle  curved
boundaries.
   When C1=C2=0 and  all functions are independent of  T, that is,
when the problem is a steady-state  (elliptic) problem, each time-
step  corresponds  to  one iteration  of  Newton's  method.   (One
iteration  is  sufficient  for linear  problems.)   An  eigenvalue
problem can be converted to a related parabolic problem where each
time-step  is equivalent  to  an iteration  of  the inverse  power
method for finding  the smallest eigenvalue and  the corresponding
eigenfunction.  A single  equation can be handled with  no loss of
efficiency in storage or execution time.

5. RESTRICTIONS ON THE COMPLEXITY OF THE PROBLEM - At    most     two
simultaneous partial  differential equations  can be  solved.  The
input data set is limited to 200  cards.  This can be increased by
changing  the value  of  the variable  MXCARD  and increasing  the
dimensions of the arrays L, INDX, and LNAM in the preprocessor.

6. TYPICAL RUNNING TIME - Execution time is problem dependent. NESC
executed the sample problems in 30 seconds on a CDC7600.

7. UNUSUAL FEATURES OF THE PROGRAM - Local mesh refinement capability
makes 2DEPEP ideal for singular problems, such as elastic crack
problems.

8. RELATED AND AUXILIARY PROGRAMS -

9. STATUS
NESC0806/01: 23-DEC-1980 Tested at NEADB

10. REFERENCES  -
NESC0806/01:
- Granville Sewell:
  2DEPEP, 2-D Elliptic, Parabolic and Eigenvalue Problems, User's
  Manual
  Purdue University (October 1978).

11. MACHINE REQUIREMENTS - 120K (octal)  words of memory  are required
together with an  auxiliary storage device, such as  disk or tape,
for temporary use in large problems  (unit 2), and another storage
device for  temporary use to  store the preprocessor  output (unit
4).

12. PROGRAMMING LANGUAGE(S) USED -
NESC0806/01: FORTRAN-IV

13. OPERATING SYSTEM UNDER WHICH PROGRAM IS EXECUTED - SCOPE.

14. OTHER PROGRAMMING OR OPERATING INFORMATION OR RESTRICTIONS - The
program is written to process input data for one problem only per
execution.

15. NAME AND ESTABLISHMENT OF AUTHOR -
                 E. G. Sewell
                 Computer Science Department
                 Purdue University
                 West Lafayette, Indiana  47907

17. CATEGORIES -

- P. General Mathematical and Computing System Routines

Keywords: BANDED MATRIX, EIGENVALUES, FINITE ELEMENT METHOD, ITERATIVE METHODS, PARTIAL DIFFERENTIAL EQUATIONS, TWO-DIMENSIONAL