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VECFEM3 Reference Manual: veme02
Type: FORTRAN routine
veme02 - solves a non-linear functional equation by mixed
- CALL VEME02(
- T, LU, U, EEST, LIVEM, IVEM, LLVEM, LVEM, LRVEM, RVEM,
LNEK, NEK, LRPARM, RPARM, LIPARM, IPARM, LDNOD, DNOD,
LRDPRM, RDPARM, LIDPRM, IDPARM, LNODN, NODNUM, LNOD, NOD,
LNOPRM, NOPARM, LBIG, RBIG, IBIG, MASKL, MASKF, USERB,
USRFU, USERF, VEM50X, VEM63X)
- LU, LIVEM, LLVEM, LRVEM, LNEK, LRPARM, LIPARM, LDNOD,
LRDPRM, LIDPRM, LNODN, LNOPRM, LBIG
- IVEM(LIVEM), NEK(LNEK), IPARM(LIPARM), DNOD(LDNOD),
IDPARM(LIDPRM), NODNUM(LNODN), IBIG(*)
- DOUBLE PRECISION
- T, U(LU), EEST(LU), RVEM(LRVEM), RPARM(LRPARM), RDPARM(LRDPRM),
NOD(LDNOD), NOPARM(LNOPRM), RBIG(LBIG)
- LVEM(LLVEM), MASKL(NK,NK,NGROUP), MASKF(NK,NGROUP)
- USERB, USRFU, USERF, VEM50X, VEM63X
veme02 is a subroutine for the numerical solution of a non-linear
steady functional equation on a space of smooth functions H. The
problem for the solution U in H has to be given in the form :
 functional equation
for all V in H with V(COMPV)|D(COMPV)=0 for all COMPV=1,..,NK:
 Dirichlet conditions
for all COMPU=1,...,NK : U(COMPU)|D(COMPU)=B(COMPU)
where F is a linear form depending on the searched solution U,
see userf. B must not depend on U, see userb.
Using the finite element method the linear form is discretized.
The resulting system of non-linear equations is solved by the
Newton-Raphson method. An error indicator estimates the quality
of the computed solution. In every Newton-Raphson step and for
the calculation of the error indicator the program package lsolpp is used. The computed solution has
to be postprocessed by the routines vemu03,
vemu04 and vemu05
and can be handed over to a postprocessor program, see vemide or vempat.
- T double precision, scalar, input, local
- Real number (e.g. the current time).
- LU integer, scalar, input, local
- Length of solution vector U, LU
- U double precision, array: U(LU),
- The solution vector at the global nodes. U(i)
is the value at the global node i+PTRMBK(MYPROC),
see vemdis. If STARTU
is set, U gives an initial guess for the
- EEST double precision, array: EEST(LU),
- The vector of the error indicator at the global nodes. EEST(i)
is the value at the global node i+PTRMBK(MYPROC),
see vemdis. For a mesh
adaptation the error indicator can be interpolated to the
centre point of elements (see vemu03)
or to the geometrical nodes (see vemu05).
If ERREST is not set, EEST is
- LIVEM integer, scalar, input, local
- Length of the integer information vector, LIVEM>=
NDNOD+ NK+ LM.
- IVEM integer, array: IVEM(LIVEM),
- Integer information vector.
- (1)=MESH, input, local
- Start address of the mesh information in IVEM,
- (2)=ERR, output, global
- Error number.
||program terminated without error.
||program stops, since prescribed CPU time
||MAXIT is reached.
||the Newton-Raphson iteration does not
||the Newton-Raphson correction is too
||error in lsolpp.
||LBIG is too small.
arrays or solution array is too small.
- (3)=STEP, input/output, global
- Restart indicator.
veme02 offers the possibility to split the
computation of the numerical solution into
several parts. This may be useful for very
difficult problems with a high computational
amount per Newton-Raphson step. The iteration
stops if SECOND is greater than JOBEND=
RVEM(1), where SECOND is a real
function in the VECFEM
library which gives the running CPU time in
seconds. SECOND is checked after every call of lsolpp. So if you want to
stop the iteration after RUNTIME CPU seconds, you
have to set JOBEND= RVEM(1)=
SECOND()+RUNTIME before the call of veme02. If
the current value of SECOND is greater than JOBEND
on any processor, veme02 stops and sets STEP=
IVEM(3)=2. Then you have to save the
mesh arrays NEK, RPARM, IPARM,
DNOD, RDPARM, IDPARM,
NODNUM, NOD, NOPARM,
the solution array U and the
information arrays IVEM, RVEM,
LVEM for every individual process (e.g.
Before you restart veme02 you have to read the
mesh arrays, solution and information arrays (keep
the TIDS in IVEM!) (e.g.
and set STEP=2 to indicate a restart. vemdis need not be
||the first call of veme02.
||the restart of veme02.
- (5)=NIVEM, output, local
- Used length of IVEM.
- (6)=NRVEM, output, local
- Used length of RVEM.
- (7)=NLVEM, output, local
- Used length of LVEM.
- (10), input, local
- Unit for paging.
- (11), input, local
- Unit for paging.
- (12), input, local
- Unit for paging. If it is necessary, veme02
writes parts (pages) of RBIG/IBIG
to external data sets. For that purpose veme02
needs three temporary files on units IVEM(10),
IVEM(11) and IVEM(12). They
are only used if the storage for RBIG/IBIG
is too small. The needed lengths of these files
can not be computed in advance because they
depend on the given mesh and the functional
equation. Every process has to get its own data
set, otherwise the results will be chaotic. If
the data sets are allocated on different discs,
the i/o-time will be reduced, since the I/O is
done in parallel.
- (40)=LOUT, input, local
- Unit number of the standard output file, normally
- (41)=OUTCNT, input, local
- Output control flag.
||only error messages are printed.
||a protocol and every OUTCNT-th
iteration step in lsolpp are
- (45)=MSPACK, input, global
- Maximal number of stripes to pack the global
matrix, normally 100. The packing of the global
matrix is divided into several steps (called
stripes). Before the packing starts, the needed
number of stripes is estimated. If this number is
greater than MSPACK, the computation
will be stopped. You have to give more storage
for RBIG/IBIG or to
increase MSPACK. In general a problem
needing more than 100 stripes is too large for
the given storage, or else the mesh is numbered
- (46)=PCLASS, input, local
- Packing limit, normally 0. The global matrix is
stored in packed form into RBIG/IBIG.
The needed storage can be controlled by PCLASS.
need minimal CPU time. The needed storage is large
||compromise of needed storage and CPU time
||The storage for the global matrix will be
minimal. lsolpp will need much
- (51)=ORDER, input, global
- Order of the integration formulas for the
computation of the element matrices (0<ORDER<19).
ORDER gives the maximal degree of the
polynomials, which will be integrated exactly.
You should select ORDER greater than
the square of the order of the used proposal
- (60)=MAXIT, input, global
- Maximal number of Newton-Raphson steps. If MAXIT=0,
no limit for the number of iterations is set.
- (70)=MS, input, global
- Method selection in lsolpp.
- (71)=MSPREC, input, global
- Normalization method in lsolpp.
- (72)=ITMAX, input, global
- Permitted maximal number of matrix-vector
multiplications per Newton-Raphson step in lsolpp.
- (73)=MUNIT, input, global
- If MUNIT is greater than 20 and less
than 100 the global matrix of the last LINSOL
call is written to unit MUNIT, see lsolpp.
- (200)=NPROC, input, global
- Number of processes, see combgn.
- (201)=MYPROC, input, local
- Logical process id number, see combgn.
- (202)=NMSG, input/output, global
- Message counter. The difference of the input and
the output values gives the number of
communications during the veme02 call.
- (204)=TIDS(1), input, global
- Begin of the list TIDS that defines
the mapping of the logical process ids to the
physical process ids. See combgn.
- (MESH), input, local
- Start of mesh information, see mesh.
- LRVEM integer, scalar, input, local
- Length of the real information vector, LRVEM>=40+
- RVEM double precision, array: RVEM(LRVEM),
- Real information vector.
- (1)=JOBEND, output, local
- If the real function SECOND in the VECFEM
library is greater than JOBEND on any
processor, veme02 stops the calculation. You can save the
mesh arrays, the solution vector and the information
array for a restart (see STEP). If JOBEND
is equal to zero, no limit for the run time is set.
- (2)=EPS, output, local
- Smallest positive number with 1.+EPS>1.
- (3)=EPSEST, input, global
- Desired accuracy for the solution of the linear systems
to compute the error indicator, e.g. 1.D-2.
- (10)=TOL, input, global
- Desired accuracy for the relative error of the solution,
- LLVEM integer, scalar, input, local
- Length of the logical information vector, LLVEM>=20+2*NK+
- LVEM logical, array: LVEM(LLVEM),
- Logical information vector.
- (1)=LSYM, input, global
- LSYM=true indicates a symmetrical Frechet
derivative of F, see usrfu.
- (4)=SMLLCR, input, global
- If SMLLCR=true, a Newton-Raphson correction
which is too small will be accepted, in the other case
the solution processes is stopped, normally false. If the
problem is very ill-posed a small Newton-Raphson
correction is not an indicator for a good solution, so SMLLCR=true
should only be set, if you are sure that your problem is
well-posed or you solve a test problem.
- (5)=STARTU, input, global
- STARTU=true indicates that U
specified an initial guess for the Newton-Raphson
iteration. Use vemu08 or vemu06 to set an initial guess.
- (6)=ERRSTP, input, global
- If ERRSTP=true, the estimation of the
discretization error is considered in the stopping
criterion of the Newton-Raphson iteration, normally true.
The iteration ends if the estimated discretization is
reached, so that no computing time is wasted to reach a
too strict TOL.
- (7)=ERREST, input, global
- If ERREST=true, the error indicator is
computed, else EEST is undefined, normally true.
- (8)=USESNI, input, global
- If USESNI=true, the simplified Newton-Raphson
method may be used, normally true. The use of the
simplified Newton-Raphson method will reduce the CPU time,
since the mounting of a new global matrix is saved, but
it might be risky in the case of ill-posed problems.
- (9)=NOSMTH, input, global
- NOSMTH=true indicates that lsolpp returns the smoothed
solution, normally false. For some applications the use
of the non smoothed solution can improve the convergence
of the Newton-Raphson iteration.
- (10)=LMVM, input, global
- If LMVM=true, the Newton-Raphson iteration is
continued even if lsolpp
reaches the maximal number of matrix-vector
multiplications, normally true.
- (11)=NORMMA, input, global
- If NORMMA=true, the Newton-Raphson iteration
is controlled by the maximum error over all components,
else they are controlled by the error of each individual
component, normally false.
- LNEK integer, scalar, input, local
- Length of the element array.
- NEK integer, array: NEK(LNEK),
- Array of the elements, see mesh.
- LRPARM integer, scalar, input, local
- Length of the real parameter array.
- RPARM double precision, array: RPARM(LRPARM),
- Real parameter array, see mesh.
- LIPARM integer, scalar, input, local
- Length of the integer parameter array.
- IPARM integer, array: IPARM(LIPARM),
- Integer parameter array, see mesh.
- LDNOD integer, scalar, input, local
- Length of the array of the Dirichlet nodes.
- DNOD integer, array: DNOD(LDNOD),
- Array of the Dirichlet nodes, see mesh.
- LRDPRM integer, scalar, input, local
- Length of the real Dirichlet parameter array.
- RDPARM double precision, array: RDPARM(LRDPRM),
- Array of the real Dirichlet parameters, see mesh.
- LIDPRM integer, scalar, input, local
- Length of the integer Dirichlet parameter array.
- IDPARM integer, array: IDPARM(LIDPRM),
- Array of the integer Dirichlet parameters, see mesh.
- LNODN integer, scalar, input, local
- Length of the array of the id numbers of the geometrical
- NODNUM integer, array: NODNUM(LNODN),
- Array of the id numbers of the geometrical nodes, see mesh.
- LNOD integer, scalar, input, local
- Length of the array of the coordinates of the geometrical
- NOD double precision, array: NOD(LNOD),
- Array of the coordinates of the geometrical nodes, see mesh.
- LNOPRM integer, scalar, input, local
- Length of the array of the node parameters.
- NOPARM double precision, array: NOPARM(LNOPRM),
- Array of the node parameters, see mesh.
- LBIG integer, scalar, input, local
- Length of the real work array. It is impossible to
compute the needed length for LBIG before the
first veme02 run. It depends on the given mesh and the
functional equation. The needed length of LBIG
is controlled by the parameters MSPACK and PCLASS.
It should be as large as possible.
- RBIG double precision, array: RBIG(LBIG),
work array, local
- Real work array.
- IBIG integer, array: IBIG(*), work
- Integer work array, RBIG and IBIG
have to be defined by the EQUIVALENCE statement.
- MASKL logical, array: MASKL(NK,NK,NGROUP),
- If MASKL(COMPV,COMPU,G)=true, the coefficient
of the COMPV-th component of the test function in the
linear form F depends on the COMPU-th component of the
solution at the elements in the group G, see usrfu.
- MASKF logical, array: MASKF(NK,NGROUP),
- If MASKF(COMPV,G)=true, the COMPV-th component
of the test function gives a contribution to the linear
form F over the elements in group G, see userf. If you select the masks in
an optimal manner, the computation will use the CPU and
storage resources optimally. If you are uncertain about
the correct settings, you can set the masks equal to true
or call vemfre to check your
- USERB external, local
- Name of the subroutine in which the Dirichlet conditions
are described, see userb.
- USRFU external, local
- Name of the subroutine in which the Frechet derivative of
the linear form F with respect to U is described, see usrfu.
- USERF external, local
- Name of the subroutine in which the coefficients of the
linear forms F are described, see userf.
- VEM50X external, global
- Name of the subroutine in which the element matrices are
computed. The general case is VEM500. Special
versions for structural analysis are not yet available.
- VEM63X external, global
- Name of the subroutine in which the storage of VEM50X
is specified. The general case is VEM630.
The functional equation - is discretized by the finite
element method. The solution of the resulting system of linear
equations is an approximation of the solution of -.
In every element the solution is replaced by a polynomial
which interpolates the solution at the global nodes of the
element. The parametric representation is the polynomial
interpolation of the assigned geometrical nodes.
The discretization produces a system of non-linear equations.
It is solved by the Newton-Raphson method. The iteration stops
and the solution is accepted if the space discretization error is
reached or if it converges nearly quadratically and the
correction of the next iteration will be smaller than TOL.
It may be necessary to use underrelaxation. For good convergence
the simplified Newton-Raphson method can be used.
Systems of linear equations have to be solved, which are in
general very large and extremely sparse. The linear equations
solver lsolpp in veme02 uses iterative
The discretization error is estimated by inserting other test
functions into the functional equation as they are used for the
computation of the solution. Halving the mesh size and the order
generates the functions. To compute the error function EEST
the global matrix of the last Newton-Raphson step is used.
The accuracy is determined by the error tolerance TOL,
which controls the error of the solution of the discretized
functional equation. Additionally the accuracy of the numerical
solution depends on the mesh width and the order of the proposal
functions, which is controlled by the error estimation, the order
of the geometrical approximation and the used order of
integration formulas, which is not considered in the estimation.
- The existence and uniqueness of the solution of the
problem have to be ensured.
- lsolpp uses iterative methods.
In some cases problems with the convergence can be
avoided if the coefficients of the linear form are
ordered in a way that the derivatives of the COMPV-th
component of the solution occur in the coefficients of
the derivatives of the COMPV-th component from the test
functions and/or the COMPV-th component of the solution
occurs in the coefficients of the COMPV-th component from
the test functions.
- The Newton-Raphson iteration or the lsolpp
iteration may be divergent. In general this occurs if the
coefficients for the derivatives of the test functions
get a dominant term of the solution relative to terms of
the derivative of the solution or the coefficients for
the test functions get a dominant term of the derivative
of the solution relative to terms of the solution. If it
is possible, scale the coefficients in the linear form so
that they are in the same order of magnitude. In case of
divergence you should check the Frechet derivatives and
the mesh. Sometimes an increase of the integration order
will produce convergence.
- If the error estimation is active, the order of the used
proposal functions for a component must be greater than
one, if the derivative of this component of the test
function or the solution is used in the functional
equation. In the other case, order one is sufficient.
[FAQ], [THEOMAN], [DATAMAN], [DATAMAN2], [LINSOL], [P_MPI].
userb, userf, usrfu, vembuild,
vemgen(later), vemopt(later), vemu06, vemu08.
Program by L. Grosz, C. Roll, P. Sternecker, 1989-1996.
Copyrights by Universitaet Karlsruhe 1989-1996. Copyrights by
Lutz Grosz 1996. All rights reserved. More details see VECFEM.
By L. Grosz, Auckland, 4 June, 2000.