# [petsc-users] Krylov Method Takes Too Long to Solve

Barry Smith bsmith at mcs.anl.gov
Fri Apr 22 23:26:27 CDT 2016

```  Preconditioning for this type of discretization

>>>> The finite element formulation is the displacement/pressure mixed formulation

requires some thought. You cannot simply blindly apply some simple minded preconditioner like ICC or ILU, you will need to do some literature searches on preconditioner techniques for these discretitization. Based on the analogy of mixed methods for fluid problems I think the likely building block for efficient preconditioners will be so-called "block methods", what we call in PETSc PCFIELDSPLIT. Loosely speaking these "block methods" allow you to "eliminate" the displacement variables and iterate on the Schur complement, which is on the pressure variables. Likely you can use algebraic multigrid in the "elimination" of the displacement variables very effectively. Getting this right is a nontrivial project and requires you to understand your problem; you will never get good results without understanding it. So read the manual pages for PCFIELDSPLIT and then PCGAMG and google scholar for "preconditioners for finite element displacement/pressure mixed formulation".  Note that some people writing about the displacement/pressure mixed formulation will emphasis is good performance against "locking" but fail to mention that it is non-trivial to solve the resulting linear systems. Often used more sophisticated discretizations resolves one set of numerical problems but introduces another set (like difficult to solve linear systems) that still must be solved to make the methods practical; there is no free lunch.

Barry

> On Apr 22, 2016, at 10:50 PM, Jie Cheng <chengj5 at rpi.edu> wrote:
>
> Hi Jed and Barry
>
> Thanks for your help. After I reconfigured PETSc without debugging, it did become much faster. But still not fast enough. The problem remains: what could be the factor that slows the solver down? In this example, only 4662 nodes and 2920 elements are used. Typically I use 3D hexahedral element, the number of degrees of freedom in the mixed formulation I use is 3*4662+2920 = 16906. At this point, my code runs only in serial. I am planning to parallelize it and then work with much finer mesh in the future. If as Barry said, the number of degrees of freedom is too small for iterative solvers, then which direct solver do you recommend to use?
>
> Here is the log when I ran without debugging. It is not fast enough because this is only 1 step, and there are 99 more steps.
>
> Thank you.
> Jie
>
> ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------
>
> ./main on a arch-darwin-c-opt named jiecheng.local with 1 processor, by Jie Fri Apr 22 22:36:38 2016
> Using Petsc Release Version 3.6.3, Dec, 03, 2015
>
>                          Max       Max/Min        Avg      Total
> Time (sec):           1.520e+02      1.00000   1.520e+02
> Objects:              6.800e+01      1.00000   6.800e+01
> Flops:                2.590e+11      1.00000   2.590e+11  2.590e+11
> Flops/sec:            1.704e+09      1.00000   1.704e+09  1.704e+09
> MPI Messages:         0.000e+00      0.00000   0.000e+00  0.000e+00
> MPI Message Lengths:  0.000e+00      0.00000   0.000e+00  0.000e+00
> MPI Reductions:       0.000e+00      0.00000
>
> Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
>                             e.g., VecAXPY() for real vectors of length N --> 2N flops
>                             and VecAXPY() for complex vectors of length N --> 8N flops
>
> Summary of Stages:   ----- Time ------  ----- Flops -----  --- Messages ---  -- Message Lengths --  -- Reductions --
>                         Avg     %Total     Avg     %Total   counts   %Total     Avg         %Total   counts   %Total
>  0:      Main Stage: 1.5195e+02 100.0%  2.5896e+11 100.0%  0.000e+00   0.0%  0.000e+00        0.0%  0.000e+00   0.0%
>
> ------------------------------------------------------------------------------------------------------------------------
> See the 'Profiling' chapter of the users' manual for details on interpreting output.
> Phase summary info:
>    Count: number of times phase was executed
>    Time and Flops: Max - maximum over all processors
>                    Ratio - ratio of maximum to minimum over all processors
>    Mess: number of messages sent
>    Avg. len: average message length (bytes)
>    Reduct: number of global reductions
>    Global: entire computation
>    Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
>       %T - percent time in this phase         %F - percent flops in this phase
>       %M - percent messages in this phase     %L - percent message lengths in this phase
>       %R - percent reductions in this phase
>    Total Mflop/s: 10e-6 * (sum of flops over all processors)/(max time over all processors)
> ------------------------------------------------------------------------------------------------------------------------
> Event                Count      Time (sec)     Flops                             --- Global ---  --- Stage ---   Total
>                    Max Ratio  Max     Ratio   Max  Ratio  Mess   Avg len Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
> ------------------------------------------------------------------------------------------------------------------------
>
> --- Event Stage 0: Main Stage
>
> VecMDot            49906 1.0 8.4904e+00 1.0 2.61e+10 1.0 0.0e+00 0.0e+00 0.0e+00  6 10  0  0  0   6 10  0  0  0  3074
> VecNorm            51599 1.0 1.0644e+00 1.0 1.74e+09 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0  1639
> VecScale           51587 1.0 4.5893e-01 1.0 8.72e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1900
> VecCopy             1682 1.0 2.9097e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> VecSet              1781 1.0 1.9067e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> VecAXPY             3324 1.0 5.8797e-02 1.0 1.12e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1911
> VecAYPX               12 1.0 3.0708e-04 1.0 4.06e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1321
> VecMAXPY           51581 1.0 7.2805e+00 1.0 2.78e+10 1.0 0.0e+00 0.0e+00 0.0e+00  5 11  0  0  0   5 11  0  0  0  3817
> VecAssemblyBegin      18 1.0 4.0531e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> VecAssemblyEnd        18 1.0 0.0000e+00 0.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> VecNormalize       51581 1.0 1.5588e+00 1.0 2.62e+09 1.0 0.0e+00 0.0e+00 0.0e+00  1  1  0  0  0   1  1  0  0  0  1678
> MatMult            51555 1.0 6.2301e+01 1.0 1.01e+11 1.0 0.0e+00 0.0e+00 0.0e+00 41 39  0  0  0  41 39  0  0  0  1622
> MatSolve           51561 1.0 5.8966e+01 1.0 1.01e+11 1.0 0.0e+00 0.0e+00 0.0e+00 39 39  0  0  0  39 39  0  0  0  1714
> MatLUFactorNum         6 1.0 1.4240e-01 1.0 2.23e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1569
> MatILUFactorSym        1 1.0 1.0796e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> MatAssemblyBegin       6 1.0 0.0000e+00 0.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> MatAssemblyEnd         6 1.0 6.2988e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> MatGetRowIJ            1 1.0 5.9605e-06 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> MatGetOrdering         1 1.0 5.3310e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> MatZeroEntries         6 1.0 5.8441e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> KSPGMRESOrthog     49906 1.0 1.5348e+01 1.0 5.22e+10 1.0 0.0e+00 0.0e+00 0.0e+00 10 20  0  0  0  10 20  0  0  0  3401
> KSPSetUp               7 1.0 1.5509e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
> KSPSolve               6 1.0 1.3919e+02 1.0 2.59e+11 1.0 0.0e+00 0.0e+00 0.0e+00 92100  0  0  0  92100  0  0  0  1861
> PCSetUp                6 1.0 1.5490e-01 1.0 2.23e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0  1442
> PCApply               20 1.0 1.3901e+02 1.0 2.59e+11 1.0 0.0e+00 0.0e+00 0.0e+00 91100  0  0  0  91100  0  0  0  1861
> ------------------------------------------------------------------------------------------------------------------------
>
> Memory usage is given in bytes:
>
> Object Type          Creations   Destructions     Memory  Descendants' Mem.
> Reports information only for process 0.
>
> --- Event Stage 0: Main Stage
>
>               Vector    56             56      7659456     0
>               Matrix     2              2     24271340     0
>        Krylov Solver     2              2        36720     0
>       Preconditioner     2              2         1832     0
>            Index Set     5              5       237080     0
>               Viewer     1              0            0     0
> ========================================================================================================================
> Average time to get PetscTime(): 2.14577e-07
> #PETSc Option Table entries:
> -log_summary
> #End of PETSc Option Table entries
> Compiled without FORTRAN kernels
> Compiled with full precision matrices (default)
> sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
> Configure options: --with-debugging=no
>> On Apr 22, 2016, at 6:30 PM, Jed Brown <jed at jedbrown.org> wrote:
>>
>> Allow me to call your attention to some fine print:
>>
>>        ##########################################################
>>        #                                                        #
>>        #                          WARNING!!!                    #
>>        #                                                        #
>>        #   This code was compiled with a debugging option,      #
>>        #   To get timing results run ./configure                #
>>        #   using --with-debugging=no, the performance will      #
>>        #   be generally two or three times faster.              #
>>        #                                                        #
>>        ##########################################################
>>
>>
>> This is standard mailing list etiquette.  It is important to preserve
>> threading for people who find this discussion later and so that we do
>> not waste our time re-answering the same questions that have already
>> been answered in private side-conversations.  You'll likely get an
>> answer faster that way too.
>>
>> Jie Cheng <chengj5 at rpi.edu> writes:
>>
>>> Hello Jed
>>>
>>> Thanks for your reply. Here is the log: (the first a couple of lines are printed by my own code during a time step, “iteration” means the number of Newton’s iteration, “GMRES_iteration” denotes the number of GMRES iterations it takes in a particular Newton’s iteration, err1 and err2 are my criteria of convergence)
>>>
>>>
>>> Number of nonzeros allocated: 988540
>>> Step =    1, LoadFactor =   1.0000e-02
>>> REASON: 2
>>> Iteration =    1     GMRES_Iteration =    2     Err1 =   1.0000e+00,     Err2 =   1.8375e-11
>>> REASON: 2
>>> Iteration =    2     GMRES_Iteration =    4     Err1 =   4.1151e-02,     Err2 =   5.9467e-11
>>> REASON: 2
>>> Iteration =    3     GMRES_Iteration =    2     Err1 =   1.0265e-02,     Err2 =   2.1268e-12
>>> REASON: 2
>>> Iteration =    4     GMRES_Iteration =    2     Err1 =   8.9824e-04,     Err2 =   1.2622e-14
>>> REASON: 2
>>> Iteration =    5     GMRES_Iteration =    2     Err1 =   1.5741e-04,     Err2 =   8.1248e-16
>>> REASON: 2
>>> Iteration =    6     GMRES_Iteration =    2     Err1 =   7.1515e-06,     Err2 =   1.6605e-16
>>> ************************************************************************************************************************
>>> ***             WIDEN YOUR WINDOW TO 120 CHARACTERS.  Use 'enscript -r -fCourier9' to print this document            ***
>>> ************************************************************************************************************************
>>>
>>> ---------------------------------------------- PETSc Performance Summary: ----------------------------------------------
>>>
>>> ./main on a arch-darwin-c-debug named jiecheng.local with 1 processor, by Jie Fri Apr 22 18:06:33 2016
>>> Using Petsc Release Version 3.6.3, Dec, 03, 2015
>>>
>>>                         Max       Max/Min        Avg      Total
>>> Time (sec):           4.119e+02      1.00000   4.119e+02
>>> Objects:              6.800e+01      1.00000   6.800e+01
>>> Flops:                2.590e+11      1.00000   2.590e+11  2.590e+11
>>> Flops/sec:            6.287e+08      1.00000   6.287e+08  6.287e+08
>>> Memory:               3.308e+07      1.00000              3.308e+07
>>> MPI Messages:         0.000e+00      0.00000   0.000e+00  0.000e+00
>>> MPI Message Lengths:  0.000e+00      0.00000   0.000e+00  0.000e+00
>>> MPI Reductions:       0.000e+00      0.00000
>>>
>>> Flop counting convention: 1 flop = 1 real number operation of type (multiply/divide/add/subtract)
>>>                            e.g., VecAXPY() for real vectors of length N --> 2N flops
>>>                            and VecAXPY() for complex vectors of length N --> 8N flops
>>>
>>> Summary of Stages:   ----- Time ------  ----- Flops -----  --- Messages ---  -- Message Lengths --  -- Reductions --
>>>                        Avg     %Total     Avg     %Total   counts   %Total     Avg         %Total   counts   %Total
>>> 0:      Main Stage: 4.1191e+02 100.0%  2.5896e+11 100.0%  0.000e+00   0.0%  0.000e+00        0.0%  0.000e+00   0.0%
>>>
>>> ------------------------------------------------------------------------------------------------------------------------
>>> See the 'Profiling' chapter of the users' manual for details on interpreting output.
>>> Phase summary info:
>>>   Count: number of times phase was executed
>>>   Time and Flops: Max - maximum over all processors
>>>                   Ratio - ratio of maximum to minimum over all processors
>>>   Mess: number of messages sent
>>>   Avg. len: average message length (bytes)
>>>   Reduct: number of global reductions
>>>   Global: entire computation
>>>   Stage: stages of a computation. Set stages with PetscLogStagePush() and PetscLogStagePop().
>>>      %T - percent time in this phase         %F - percent flops in this phase
>>>      %M - percent messages in this phase     %L - percent message lengths in this phase
>>>      %R - percent reductions in this phase
>>>   Total Mflop/s: 10e-6 * (sum of flops over all processors)/(max time over all processors)
>>> ------------------------------------------------------------------------------------------------------------------------
>>>
>>>
>>>      ##########################################################
>>>      #                                                        #
>>>      #                          WARNING!!!                    #
>>>      #                                                        #
>>>      #   This code was compiled with a debugging option,      #
>>>      #   To get timing results run ./configure                #
>>>      #   using --with-debugging=no, the performance will      #
>>>      #   be generally two or three times faster.              #
>>>      #                                                        #
>>>      ##########################################################
>>>
>>>
>>> Event                Count      Time (sec)     Flops                             --- Global ---  --- Stage ---   Total
>>>                   Max Ratio  Max     Ratio   Max  Ratio  Mess   Avg len Reduct  %T %F %M %L %R  %T %F %M %L %R Mflop/s
>>> ------------------------------------------------------------------------------------------------------------------------
>>>
>>> --- Event Stage 0: Main Stage
>>>
>>> VecMDot            49906 1.0 2.1200e+01 1.0 2.61e+10 1.0 0.0e+00 0.0e+00 0.0e+00  5 10  0  0  0   5 10  0  0  0  1231
>>> VecNorm            51599 1.0 1.5052e+00 1.0 1.74e+09 1.0 0.0e+00 0.0e+00 0.0e+00  0  1  0  0  0   0  1  0  0  0  1159
>>> VecScale           51587 1.0 5.6397e+00 1.0 8.72e+08 1.0 0.0e+00 0.0e+00 0.0e+00  1  0  0  0  0   1  0  0  0  0   155
>>> VecCopy             1682 1.0 5.0184e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> VecSet              1781 1.0 3.5445e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> VecAXPY             3324 1.0 4.0115e-01 1.0 1.12e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   280
>>> VecAYPX               12 1.0 1.0931e-03 1.0 4.06e+05 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   371
>>> VecMAXPY           51581 1.0 2.4876e+01 1.0 2.78e+10 1.0 0.0e+00 0.0e+00 0.0e+00  6 11  0  0  0   6 11  0  0  0  1117
>>> VecAssemblyBegin      18 1.0 1.6809e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> VecAssemblyEnd        18 1.0 1.2112e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> VecNormalize       51581 1.0 8.1458e+00 1.0 2.62e+09 1.0 0.0e+00 0.0e+00 0.0e+00  2  1  0  0  0   2  1  0  0  0   321
>>> MatMult            51555 1.0 1.5226e+02 1.0 1.01e+11 1.0 0.0e+00 0.0e+00 0.0e+00 37 39  0  0  0  37 39  0  0  0   664
>>> MatSolve           51561 1.0 1.7415e+02 1.0 1.01e+11 1.0 0.0e+00 0.0e+00 0.0e+00 42 39  0  0  0  42 39  0  0  0   580
>>> MatLUFactorNum         6 1.0 6.3280e-01 1.0 2.23e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   353
>>> MatILUFactorSym        1 1.0 2.1535e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> MatAssemblyBegin       6 1.0 3.2902e-05 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> MatAssemblyEnd         6 1.0 1.7512e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> MatGetRowIJ            1 1.0 6.8307e-04 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> MatGetOrdering         1 1.0 1.3239e-02 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> MatZeroEntries         6 1.0 9.4421e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> KSPGMRESOrthog     49906 1.0 4.6091e+01 1.0 5.22e+10 1.0 0.0e+00 0.0e+00 0.0e+00 11 20  0  0  0  11 20  0  0  0  1133
>>> KSPSetUp               7 1.0 3.3112e-03 1.0 0.00e+00 0.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0     0
>>> KSPSolve               6 1.0 3.8880e+02 1.0 2.59e+11 1.0 0.0e+00 0.0e+00 0.0e+00 94100  0  0  0  94100  0  0  0   666
>>> PCSetUp                6 1.0 6.8447e-01 1.0 2.23e+08 1.0 0.0e+00 0.0e+00 0.0e+00  0  0  0  0  0   0  0  0  0  0   326
>>> PCApply               20 1.0 3.8806e+02 1.0 2.59e+11 1.0 0.0e+00 0.0e+00 0.0e+00 94100  0  0  0  94100  0  0  0   667
>>> ------------------------------------------------------------------------------------------------------------------------
>>>
>>> Memory usage is given in bytes:
>>>
>>> Object Type          Creations   Destructions     Memory  Descendants' Mem.
>>> Reports information only for process 0.
>>>
>>> --- Event Stage 0: Main Stage
>>>
>>>              Vector    56             56      7659456     0
>>>              Matrix     2              2     24271340     0
>>>       Krylov Solver     2              2        36720     0
>>>      Preconditioner     2              2         1832     0
>>>           Index Set     5              5       237080     0
>>>              Viewer     1              0            0     0
>>> ========================================================================================================================
>>> Average time to get PetscTime(): 9.53674e-08
>>> #PETSc Option Table entries:
>>> -log_summary
>>> #End of PETSc Option Table entries
>>> Compiled without FORTRAN kernels
>>> Compiled with full precision matrices (default)
>>> sizeof(short) 2 sizeof(int) 4 sizeof(long) 8 sizeof(void*) 8 sizeof(PetscScalar) 8 sizeof(PetscInt) 4
>>> Configure options: --with-mpi-dir=/usr/local
>>> -----------------------------------------
>>> Libraries compiled on Tue Mar 22 14:05:36 2016 on calcium-68.dynamic2.rpi.edu
>>> Machine characteristics: Darwin-15.4.0-x86_64-i386-64bit
>>> Using PETSc directory: /usr/local/petsc/petsc-3.6.3
>>> Using PETSc arch: arch-darwin-c-debug
>>> -----------------------------------------
>>>
>>> Using C compiler: /usr/local/bin/mpicc  -fPIC -Wall -Wwrite-strings -Wno-strict-aliasing -Wno-unknown-pragmas -g3 -O0  \${COPTFLAGS} \${CFLAGS}
>>> Using Fortran compiler: /usr/local/bin/mpif90  -fPIC -Wall -Wno-unused-variable -ffree-line-length-0 -Wno-unused-dummy-argument -g -O0   \${FOPTFLAGS} \${FFLAGS}
>>> -----------------------------------------
>>>
>>> Using include paths: -I/usr/local/petsc/petsc-3.6.3/arch-darwin-c-debug/include -I/usr/local/petsc/petsc-3.6.3/include -I/usr/local/petsc/petsc-3.6.3/include -I/usr/local/petsc/petsc-3.6.3/arch-darwin-c-debug/include -I/usr/local/include
>>> -----------------------------------------
>>>
>>> Using libraries: -Wl,-rpath,/usr/local/petsc/petsc-3.6.3/arch-darwin-c-debug/lib -L/usr/local/petsc/petsc-3.6.3/arch-darwin-c-debug/lib -lpetsc -llapack -lblas -Wl,-rpath,/usr/local/lib -L/usr/local/lib -Wl,-rpath,/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang/7.3.0/lib/darwin -L/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/lib/clang/7.3.0/lib/darwin -lmpi_usempif08 -lmpi_usempi_ignore_tkr -lmpi_mpifh -lgfortran -Wl,-rpath,/usr/local/gfortran/lib/gcc/x86_64-apple-darwin14/4.9.2 -L/usr/local/gfortran/lib/gcc/x86_64-apple-darwin14/4.9.2 -Wl,-rpath,/usr/local/gfortran/lib -L/usr/local/gfortran/lib -lgfortran -lgcc_ext.10.5 -lquadmath -lm -lclang_rt.osx -lmpi_cxx -lc++ -Wl,-rpath,/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/../lib/clang/7.3.0/lib/darwin -L/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/../lib/clang/7.3.0/lib/darwin -lclang_rt.osx -Wl,-rpath,/usr/local/lib -L/usr/local/lib -ldl -lmpi -lSystem -Wl,-rpath,/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/../lib/clang/7.3.0/lib/darwin -L/Applications/Xcode.app/Contents/Developer/Toolchains/XcodeDefault.xctoolchain/usr/bin/../lib/clang/7.3.0/lib/darwin -lclang_rt.osx -ldl
>>> -----------------------------------------
>>>
>>>> On Apr 22, 2016, at 5:12 PM, Jed Brown <jed at jedbrown.org> wrote:
>>>>
>>>>
>>>> Jie Cheng <chengj5 at rpi.edu> writes:
>>>>
>>>>> Hi
>>>>>
>>>>> I’m implementing finite element method on nonlinear solid mechanics. The main portion of my code that involves PETSc is that in each step, the tangent stiffness matrix A is formed and the increment of the nodal degrees of freedom is solved. Standard Newton’s iteration. The problem is: when I use Krylov methods to solve the linear system, the KSPsolve process takes too long, although only 2 or 3 iterations are needed.
>>>>>
>>>>> The finite element formulation is the displacement/pressure mixed formulation, which I believe is symmetric and positive-definite. However if I pick conjugate gradient method with ICC preconditioned, PETSc gives me a -8 converged reason, which indicates a non-positive-definite matrix. After a couple of trials and errors, the only pair that works is GMRES plus PCKSP. But as I said, the KSPsolve function takes too much time.
>>>>>
>>>>> A typical problem I’m trying has 16906 rows and 16906 cols. The message printed out by ksp_view is as following:
>>>>>
>>>>> KSP Object: 1 MPI processes
>>>>> type: gmres
>>>>>   GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
>>>>>   GMRES: happy breakdown tolerance 1e-30
>>>>> maximum iterations=10000, initial guess is zero
>>>>> tolerances:  relative=1e-05, absolute=1e-50, divergence=10000
>>>>> left preconditioning
>>>>> using PRECONDITIONED norm type for convergence test
>>>>> PC Object: 1 MPI processes
>>>>> type: ksp
>>>>> KSP and PC on KSP preconditioner follow
>>>>> ---------------------------------
>>>>>   KSP Object:    (ksp_)     1 MPI processes
>>>>>     type: gmres
>>>>>       GMRES: restart=30, using Classical (unmodified) Gram-Schmidt Orthogonalization with no iterative refinement
>>>>>       GMRES: happy breakdown tolerance 1e-30
>>>>>     maximum iterations=10000, initial guess is zero
>>>>>     tolerances:  relative=1e-05, absolute=1e-50, divergence=10000
>>>>>     left preconditioning
>>>>>     using PRECONDITIONED norm type for convergence test
>>>>>   PC Object:    (ksp_)     1 MPI processes
>>>>>     type: ilu
>>>>>       ILU: out-of-place factorization
>>>>>       0 levels of fill
>>>>>       tolerance for zero pivot 2.22045e-14
>>>>>       matrix ordering: natural
>>>>>       factor fill ratio given 1, needed 1
>>>>>         Factored matrix follows:
>>>>>           Mat Object:             1 MPI processes
>>>>>             type: seqaij
>>>>>             rows=16906, cols=16906
>>>>>             package used to perform factorization: petsc
>>>>>             total: nonzeros=988540, allocated nonzeros=988540
>>>>>             total number of mallocs used during MatSetValues calls =0
>>>>>               using I-node routines: found 7582 nodes, limit used is 5
>>>>>     linear system matrix = precond matrix:
>>>>>     Mat Object:       1 MPI processes
>>>>>       type: seqaij
>>>>>       rows=16906, cols=16906
>>>>>       total: nonzeros=988540, allocated nonzeros=988540
>>>>>       total number of mallocs used during MatSetValues calls =0
>>>>>         using I-node routines: found 7582 nodes, limit used is 5
>>>>> ---------------------------------
>>>>> linear system matrix = precond matrix:
>>>>> Mat Object:   1 MPI processes
>>>>>   type: seqaij
>>>>>   rows=16906, cols=16906
>>>>>   total: nonzeros=988540, allocated nonzeros=988540
>>>>>   total number of mallocs used during MatSetValues calls =0
>>>>>     using I-node routines: found 7582 nodes, limit used is 5
>>>>>
>>>>> Could anyone give me any suggestion please?
>>>>>
>>>>> Thanks
>>>>> Jie Cheng
>

```