[petsc-users] 2D Poisson on nonuniform meshes

Tue Sep 1 06:32:38 CDT 2015

On Tue, Sep 1, 2015 at 6:24 AM, Filippo Leonardi <
filippo.leonardi at sam.math.ethz.ch> wrote:

> On Tuesday 01 September 2015 05:13:54 you wrote:
>
> > On Tue, Sep 1, 2015 at 4:38 AM, Filippo Leonardi <
>
> >
>
> > filippo.leonardi at sam.math.ethz.ch> wrote:
>
> > > On Tuesday 01 September 2015 04:15:50 you wrote:
>
> > > > On Tue, Sep 1, 2015 at 4:12 AM, Filippo Leonardi <
>
> > > >
>
> > > > filippo.leonardi at sam.math.ethz.ch> wrote:
>
> > > > > Thanks Matt,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > I use the "natural scaling" of the Laplacian Lapl * x = b. The
> grid is
>
> > > > >
>
> > > > > *still* uniform.
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > I give you a couple of results:
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1 ) This is e.g. gmres + gamg + 4 levels. As you can see it
> converges
>
> > > > >
>
> > > > > after 2 iterations which is totally strange. The solution is
>
> > > > > incorrect.
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2 ) This is mg + galerkin matrices + levels. Result is incorrect
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 3 ) Default: works fine
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 4) LU: works fine
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > I remove the constant nullspace as usual.
>
> > > >
>
> > > > If you have a constant null space, how does 4) work? It should fail
> with
>
> > >
>
> > > a
>
> > >
>
> > > > zero pivot since
>
> > > >
>
> > > > the matrix is singular.
>
> > >
>
> > > I do not care about the constant so any shift in the LU decomposition
> is
>
> > > fine and as far as I understood PETSc is robust in this respect.
>
> >
>
> > 1) The KSP view does not say it is shifting. Are you using the latest
>
> > release?
>
>
>
> yes, 3.6. Does PETSc warn for that even if I set the nullspace? I can also
> check MUMPS or something else.
>

I am not sure what you think PETSc does here. If shifting were enabled, it
would add some
diagonal matrix to the input matrix and continue the factorization. This
would mean that the
factors were not, in fact, a factorization of the input matrix, and you
would not get the exact
solution in one iterate.

> >
>
> > 2) If it shifted, it would not solve in a single iterate.
>
>
>
> even with preonly?
>

You would have a large residual. Do you?

> >
>
> > 3) Your GAMG results imply that something is wrong with the coarse solve.
>
> > This is exactly what would happen if
>
> > that problem was not solved accurately (its off by > 10 orders of
>
> > magnitude).
>
>
>
> yes, but GAMG builds is own coarse solvers so either the problem is
> already in the definition of A and b (likely) or it is a bug in gamg.
>

Yes. GAMG uses the constants to build the basis, on the assumption that
they are in the (near) nullspace of the
operator with no boundary conditions. Since this is far off, I think this
must not be true for your A.

> >
>
> > It sounds like your operator is not singular, and its not the Laplacian
>
> > since it does not look like the Neumann version
>
> > has constants as a null space.
>
>
>
> I'm using periodic boundaries, and constants are in kern(A)
>

Did you check?

   Matt

> Thanks a lot for your time.
>
> >
>
> > Thanks,
>
> >
>
> > Matt
>
> >
>
> > > > Matt
>
> > > >
>
> > > > > I use the usual setcomputeoperator business and compute rhs.
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Instead, if I scale to Lapl * dx * dy * x = b * dx * dy (things
> work
>
> > >
>
> > > fine
>
> > >
>
> > > > > also with MG). This is fine with uniform grid but I do not know
> how to
>
> > >
>
> > > do
>
> > >
>
> > > > > that properly on nonuniform.
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Let me know if you need code pieces or other details.
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1)
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0 KSP Residual norm 1.793027353791e-05
>
> > > > >
>
> > > > > 0 KSP preconditioned resid norm 1.793027353791e-05 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 7.912994695029e-03 ||r(i)||/||b|| 1.000000000000e+00
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1 KSP Residual norm 2.662212708666e-20
>
> > > > >
>
> > > > > 1 KSP preconditioned resid norm 2.662212708666e-20 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 3.120963188535e-02 ||r(i)||/||b|| 3.944098674167e+00
>
> > > > >
>
> > > > > Linear solve converged due to CONVERGED_RTOL iterations 1
>
> > > > >
>
> > > > > 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: gamg
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > MG: type is MULTIPLICATIVE, levels=4 cycles=v
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Cycles per PCApply=1
>
> > > > >
>
> > > > > Using Galerkin computed coarse grid matrices
>
> > > > >
>
> > > > > GAMG specific options
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Threshold for dropping small values from graph 0
>
> > > > >
>
> > > > > AGG specific options
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Symmetric graph false
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Coarse grid solver -- level -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_coarse_) 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=1, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_coarse_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: bjacobi
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > block Jacobi: number of blocks = 1
>
> > > > >
>
> > > > > Local solve is same for all blocks, in the following KSP and PC
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > objects:
>
> > > > >
>
> > > > > KSP Object: (mg_coarse_sub_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: preonly
>
> > > > >
>
> > > > > maximum iterations=1, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_coarse_sub_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: lu
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > LU: out-of-place factorization
>
> > > > >
>
> > > > > tolerance for zero pivot 2.22045e-14
>
> > > > >
>
> > > > > using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
>
> > > > >
>
> > > > > matrix ordering: nd
>
> > > > >
>
> > > > > factor fill ratio given 5, needed 1.10953
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Factored matrix follows:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=33, cols=33
>
> > > > >
>
> > > > > package used to perform factorization: petsc
>
> > > > >
>
> > > > > total: nonzeros=1013, allocated nonzeros=1013
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > using I-node routines: found 21 nodes, limit used is 5
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=33, cols=33
>
> > > > >
>
> > > > > total: nonzeros=913, allocated nonzeros=913
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=33, cols=33
>
> > > > >
>
> > > > > total: nonzeros=913, allocated nonzeros=913
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 1
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_1_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_1_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_1_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=401, cols=401
>
> > > > >
>
> > > > > total: nonzeros=9777, allocated nonzeros=9777
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 2
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_2_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_2_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_2_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=2296, cols=2296
>
> > > > >
>
> > > > > total: nonzeros=25038, allocated nonzeros=25038
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 3
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_3_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_3_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_3_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2)
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0 KSP Residual norm 1.010294060491e-05
>
> > > > >
>
> > > > > 0 KSP preconditioned resid norm 1.010294060491e-05 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.086379257868e-03 ||r(i)||/||b|| 1.000000000000e+00
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1 KSP Residual norm 8.803303876758e-20
>
> > > > >
>
> > > > > 1 KSP preconditioned resid norm 8.803303876758e-20 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2.101751936637e-03 ||r(i)||/||b|| 1.934639235254e+00
>
> > > > >
>
> > > > > Linear solve converged due to CONVERGED_RTOL iterations 1
>
> > > > >
>
> > > > > 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: mg
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > MG: type is MULTIPLICATIVE, levels=4 cycles=v
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Cycles per PCApply=1
>
> > > > >
>
> > > > > Using Galerkin computed coarse grid matrices
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Coarse grid solver -- level -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_coarse_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: preonly
>
> > > > >
>
> > > > > maximum iterations=1, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_coarse_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: lu
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > LU: out-of-place factorization
>
> > > > >
>
> > > > > tolerance for zero pivot 2.22045e-14
>
> > > > >
>
> > > > > using diagonal shift on blocks to prevent zero pivot [INBLOCKS]
>
> > > > >
>
> > > > > matrix ordering: nd
>
> > > > >
>
> > > > > factor fill ratio given 5, needed 4.83507
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Factored matrix follows:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=256, cols=256
>
> > > > >
>
> > > > > package used to perform factorization: petsc
>
> > > > >
>
> > > > > total: nonzeros=11140, allocated nonzeros=11140
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=256, cols=256
>
> > > > >
>
> > > > > total: nonzeros=2304, allocated nonzeros=2304
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 1
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_1_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_1_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_1_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=1024, cols=1024
>
> > > > >
>
> > > > > total: nonzeros=9216, allocated nonzeros=9216
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 2
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_2_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_2_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_2_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=4096, cols=4096
>
> > > > >
>
> > > > > total: nonzeros=36864, allocated nonzeros=36864
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > Down solver (pre-smoother) on level 3
> -------------------------------
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_3_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: chebyshev
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Chebyshev: eigenvalue estimates: min = -1.79769e+307, max = -inf
>
> > > > >
>
> > > > > Chebyshev: eigenvalues estimated using gmres with translations [0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0.1; 0 1.1]
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > KSP Object: (mg_levels_3_esteig_) 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=10, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > maximum iterations=2
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using nonzero initial guess
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: (mg_levels_3_) 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: sor
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > SOR: type = local_symmetric, iterations = 1, local iterations = 1,
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > omega = 1
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Up solver (post-smoother) same as down solver (pre-smoother)
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 3)
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0 KSP Residual norm 2.990548680394e-08
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 0 KSP preconditioned resid norm 2.990548680394e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2.966915112865e-04 ||r(i)||/||b|| 1.000000000000e+00
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1 KSP Residual norm 2.368703753809e-08
>
> > > > >
>
> > > > > 1 KSP preconditioned resid norm 2.368703753809e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 3.109667150576e-04 ||r(i)||/||b|| 1.048114634993e+00
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2 KSP Residual norm 2.236515515253e-08
>
> > > > >
>
> > > > > 2 KSP preconditioned resid norm 2.236515515253e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2.655771471955e-04 ||r(i)||/||b|| 8.951289035670e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 3 KSP Residual norm 1.957630940223e-08
>
> > > > >
>
> > > > > 3 KSP preconditioned resid norm 1.957630940223e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2.158250768646e-04 ||r(i)||/||b|| 7.274393390250e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 4 KSP Residual norm 1.771416739221e-08
>
> > > > >
>
> > > > > 4 KSP preconditioned resid norm 1.771416739221e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 2.075204223802e-04 ||r(i)||/||b|| 6.994484657830e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 5 KSP Residual norm 1.617056152936e-08
>
> > > > >
>
> > > > > 5 KSP preconditioned resid norm 1.617056152936e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.908090487285e-04 ||r(i)||/||b|| 6.431227098516e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 6 KSP Residual norm 1.467170521607e-08
>
> > > > >
>
> > > > > 6 KSP preconditioned resid norm 1.467170521607e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.683698119419e-04 ||r(i)||/||b|| 5.674911668750e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 7 KSP Residual norm 1.321526248809e-08
>
> > > > >
>
> > > > > 7 KSP preconditioned resid norm 1.321526248809e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.473617884651e-04 ||r(i)||/||b|| 4.966835344432e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 8 KSP Residual norm 1.174828665876e-08
>
> > > > >
>
> > > > > 8 KSP preconditioned resid norm 1.174828665876e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.458902362707e-04 ||r(i)||/||b|| 4.917236615167e-01
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 9 KSP Residual norm 1.012151625603e-08
>
> > > > >
>
> > > > > 9 KSP preconditioned resid norm 1.012151625603e-08 true resid norm
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 1.303866699420e-04 ||r(i)||/||b|| 4.394688253015e-01
>
> > > > >
>
> > > > > 10 KSP Residual norm 7.876439970777e-09
>
> > > > >
>
> > > > > 10 KSP preconditioned resid norm 7.876439970777e-09 true resid norm
>
> > > > >
>
> > > > > 1.162414266035e-04 ||r(i)||/||b|| 3.917922224990e-01
>
> > > > >
>
> > > > > 11 KSP Residual norm 4.981821461221e-09
>
> > > > >
>
> > > > > 11 KSP preconditioned resid norm 4.981821461221e-09 true resid norm
>
> > > > >
>
> > > > > 9.075981784793e-05 ||r(i)||/||b|| 3.059063518683e-01
>
> > > > >
>
> > > > > 12 KSP Residual norm 2.426217812448e-09
>
> > > > >
>
> > > > > 12 KSP preconditioned resid norm 2.426217812448e-09 true resid norm
>
> > > > >
>
> > > > > 5.145334699067e-05 ||r(i)||/||b|| 1.734237247556e-01
>
> > > > >
>
> > > > > 13 KSP Residual norm 1.049060114422e-09
>
> > > > >
>
> > > > > 13 KSP preconditioned resid norm 1.049060114422e-09 true resid norm
>
> > > > >
>
> > > > > 2.107527819651e-05 ||r(i)||/||b|| 7.103431475044e-02
>
> > > > >
>
> > > > > 14 KSP Residual norm 5.887222551677e-10
>
> > > > >
>
> > > > > 14 KSP preconditioned resid norm 5.887222551677e-10 true resid norm
>
> > > > >
>
> > > > > 9.047187295472e-06 ||r(i)||/||b|| 3.049358323817e-02
>
> > > > >
>
> > > > > 15 KSP Residual norm 4.536407095834e-10
>
> > > > >
>
> > > > > 15 KSP preconditioned resid norm 4.536407095834e-10 true resid norm
>
> > > > >
>
> > > > > 6.032647097712e-06 ||r(i)||/||b|| 2.033306268708e-02
>
> > > > >
>
> > > > > 16 KSP Residual norm 3.564018350025e-10
>
> > > > >
>
> > > > > 16 KSP preconditioned resid norm 3.564018350025e-10 true resid norm
>
> > > > >
>
> > > > > 5.116056274149e-06 ||r(i)||/||b|| 1.724368941991e-02
>
> > > > >
>
> > > > > 17 KSP Residual norm 2.744968239957e-10
>
> > > > >
>
> > > > > 17 KSP preconditioned resid norm 2.744968239957e-10 true resid norm
>
> > > > >
>
> > > > > 3.648248676660e-06 ||r(i)||/||b|| 1.229643767306e-02
>
> > > > >
>
> > > > > 18 KSP Residual norm 2.281847233955e-10
>
> > > > >
>
> > > > > 18 KSP preconditioned resid norm 2.281847233955e-10 true resid norm
>
> > > > >
>
> > > > > 2.900148270786e-06 ||r(i)||/||b|| 9.774962074955e-03
>
> > > > >
>
> > > > > 19 KSP Residual norm 1.881670741813e-10
>
> > > > >
>
> > > > > 19 KSP preconditioned resid norm 1.881670741813e-10 true resid norm
>
> > > > >
>
> > > > > 2.473850917169e-06 ||r(i)||/||b|| 8.338125032436e-03
>
> > > > >
>
> > > > > 20 KSP Residual norm 1.581922522755e-10
>
> > > > >
>
> > > > > 20 KSP preconditioned resid norm 1.581922522755e-10 true resid norm
>
> > > > >
>
> > > > > 1.965254213304e-06 ||r(i)||/||b|| 6.623897680059e-03
>
> > > > >
>
> > > > > 21 KSP Residual norm 1.349952347592e-10
>
> > > > >
>
> > > > > 21 KSP preconditioned resid norm 1.349952347592e-10 true resid norm
>
> > > > >
>
> > > > > 1.709953736235e-06 ||r(i)||/||b|| 5.763406336838e-03
>
> > > > >
>
> > > > > 22 KSP Residual norm 1.148754892680e-10
>
> > > > >
>
> > > > > 22 KSP preconditioned resid norm 1.148754892680e-10 true resid norm
>
> > > > >
>
> > > > > 1.452723184020e-06 ||r(i)||/||b|| 4.896409667135e-03
>
> > > > >
>
> > > > > 23 KSP Residual norm 9.889008860663e-11
>
> > > > >
>
> > > > > 23 KSP preconditioned resid norm 9.889008860663e-11 true resid norm
>
> > > > >
>
> > > > > 1.259317356950e-06 ||r(i)||/||b|| 4.244534504846e-03
>
> > > > >
>
> > > > > 24 KSP Residual norm 8.468070834423e-11
>
> > > > >
>
> > > > > 24 KSP preconditioned resid norm 8.468070834423e-11 true resid norm
>
> > > > >
>
> > > > > 1.097968496396e-06 ||r(i)||/||b|| 3.700707484468e-03
>
> > > > >
>
> > > > > 25 KSP Residual norm 7.340678871498e-11
>
> > > > >
>
> > > > > 25 KSP preconditioned resid norm 7.340678871498e-11 true resid norm
>
> > > > >
>
> > > > > 9.388293102378e-07 ||r(i)||/||b|| 3.164328181036e-03
>
> > > > >
>
> > > > > 26 KSP Residual norm 6.452497596580e-11
>
> > > > >
>
> > > > > 26 KSP preconditioned resid norm 6.452497596580e-11 true resid norm
>
> > > > >
>
> > > > > 7.907509627807e-07 ||r(i)||/||b|| 2.665229481464e-03
>
> > > > >
>
> > > > > 27 KSP Residual norm 5.817349095331e-11
>
> > > > >
>
> > > > > 27 KSP preconditioned resid norm 5.817349095331e-11 true resid norm
>
> > > > >
>
> > > > > 6.712549222809e-07 ||r(i)||/||b|| 2.262467568992e-03
>
> > > > >
>
> > > > > 28 KSP Residual norm 5.382715584394e-11
>
> > > > >
>
> > > > > 28 KSP preconditioned resid norm 5.382715584394e-11 true resid norm
>
> > > > >
>
> > > > > 5.883671521302e-07 ||r(i)||/||b|| 1.983093987351e-03
>
> > > > >
>
> > > > > 29 KSP Residual norm 5.072666199584e-11
>
> > > > >
>
> > > > > 29 KSP preconditioned resid norm 5.072666199584e-11 true resid norm
>
> > > > >
>
> > > > > 5.417512216577e-07 ||r(i)||/||b|| 1.825974795533e-03
>
> > > > >
>
> > > > > 30 KSP Residual norm 4.806650672230e-11
>
> > > > >
>
> > > > > 30 KSP preconditioned resid norm 4.806650672230e-11 true resid norm
>
> > > > >
>
> > > > > 5.151866485899e-07 ||r(i)||/||b|| 1.736438789084e-03
>
> > > > >
>
> > > > > 31 KSP Residual norm 4.675001898732e-11
>
> > > > >
>
> > > > > 31 KSP preconditioned resid norm 4.675001898732e-11 true resid norm
>
> > > > >
>
> > > > > 4.825947510614e-07 ||r(i)||/||b|| 1.626587659919e-03
>
> > > > >
>
> > > > > 32 KSP Residual norm 4.530536181592e-11
>
> > > > >
>
> > > > > 32 KSP preconditioned resid norm 4.530536181592e-11 true resid norm
>
> > > > >
>
> > > > > 4.648650267512e-07 ||r(i)||/||b|| 1.566829548764e-03
>
> > > > >
>
> > > > > 33 KSP Residual norm 4.357157235509e-11
>
> > > > >
>
> > > > > 33 KSP preconditioned resid norm 4.357157235509e-11 true resid norm
>
> > > > >
>
> > > > > 4.567209908707e-07 ||r(i)||/||b|| 1.539380041209e-03
>
> > > > >
>
> > > > > 34 KSP Residual norm 4.135715715343e-11
>
> > > > >
>
> > > > > 34 KSP preconditioned resid norm 4.135715715343e-11 true resid norm
>
> > > > >
>
> > > > > 4.469703503011e-07 ||r(i)||/||b|| 1.506515465721e-03
>
> > > > >
>
> > > > > 35 KSP Residual norm 3.851871986267e-11
>
> > > > >
>
> > > > > 35 KSP preconditioned resid norm 3.851871986267e-11 true resid norm
>
> > > > >
>
> > > > > 4.364097171314e-07 ||r(i)||/||b|| 1.470920806730e-03
>
> > > > >
>
> > > > > 36 KSP Residual norm 3.526035391634e-11
>
> > > > >
>
> > > > > 36 KSP preconditioned resid norm 3.526035391634e-11 true resid norm
>
> > > > >
>
> > > > > 4.065382561286e-07 ||r(i)||/||b|| 1.370238920439e-03
>
> > > > >
>
> > > > > 37 KSP Residual norm 3.144895012756e-11
>
> > > > >
>
> > > > > 37 KSP preconditioned resid norm 3.144895012756e-11 true resid norm
>
> > > > >
>
> > > > > 3.810168560499e-07 ||r(i)||/||b|| 1.284218932985e-03
>
> > > > >
>
> > > > > 38 KSP Residual norm 2.735239124106e-11
>
> > > > >
>
> > > > > 38 KSP preconditioned resid norm 2.735239124106e-11 true resid norm
>
> > > > >
>
> > > > > 3.407338934429e-07 ||r(i)||/||b|| 1.148445036278e-03
>
> > > > >
>
> > > > > 39 KSP Residual norm 2.306734554159e-11
>
> > > > >
>
> > > > > 39 KSP preconditioned resid norm 2.306734554159e-11 true resid norm
>
> > > > >
>
> > > > > 2.940484127641e-07 ||r(i)||/||b|| 9.910914251948e-04
>
> > > > >
>
> > > > > 40 KSP Residual norm 1.924842110922e-11
>
> > > > >
>
> > > > > 40 KSP preconditioned resid norm 1.924842110922e-11 true resid norm
>
> > > > >
>
> > > > > 2.443693779909e-07 ||r(i)||/||b|| 8.236480273107e-04
>
> > > > >
>
> > > > > 41 KSP Residual norm 1.618672399310e-11
>
> > > > >
>
> > > > > 41 KSP preconditioned resid norm 1.618672399310e-11 true resid norm
>
> > > > >
>
> > > > > 2.034502467604e-07 ||r(i)||/||b|| 6.857299215548e-04
>
> > > > >
>
> > > > > 42 KSP Residual norm 1.378885267147e-11
>
> > > > >
>
> > > > > 42 KSP preconditioned resid norm 1.378885267147e-11 true resid norm
>
> > > > >
>
> > > > > 1.689714775892e-07 ||r(i)||/||b|| 5.695190834970e-04
>
> > > > >
>
> > > > > 43 KSP Residual norm 1.179601185051e-11
>
> > > > >
>
> > > > > 43 KSP preconditioned resid norm 1.179601185051e-11 true resid norm
>
> > > > >
>
> > > > > 1.483783124701e-07 ||r(i)||/||b|| 5.001097329234e-04
>
> > > > >
>
> > > > > 44 KSP Residual norm 9.928908801463e-12
>
> > > > >
>
> > > > > 44 KSP preconditioned resid norm 9.928908801463e-12 true resid norm
>
> > > > >
>
> > > > > 1.276885725440e-07 ||r(i)||/||b|| 4.303748765523e-04
>
> > > > >
>
> > > > > 45 KSP Residual norm 8.512555069524e-12
>
> > > > >
>
> > > > > 45 KSP preconditioned resid norm 8.512555069524e-12 true resid norm
>
> > > > >
>
> > > > > 1.020939171370e-07 ||r(i)||/||b|| 3.441079817022e-04
>
> > > > >
>
> > > > > 46 KSP Residual norm 7.600214409478e-12
>
> > > > >
>
> > > > > 46 KSP preconditioned resid norm 7.600214409478e-12 true resid norm
>
> > > > >
>
> > > > > 8.873369034852e-08 ||r(i)||/||b|| 2.990772805186e-04
>
> > > > >
>
> > > > > 47 KSP Residual norm 6.577382322417e-12
>
> > > > >
>
> > > > > 47 KSP preconditioned resid norm 6.577382322417e-12 true resid norm
>
> > > > >
>
> > > > > 8.285748008098e-08 ||r(i)||/||b|| 2.792714888326e-04
>
> > > > >
>
> > > > > 48 KSP Residual norm 5.701374824383e-12
>
> > > > >
>
> > > > > 48 KSP preconditioned resid norm 5.701374824383e-12 true resid norm
>
> > > > >
>
> > > > > 6.754179947516e-08 ||r(i)||/||b|| 2.276499222452e-04
>
> > > > >
>
> > > > > 49 KSP Residual norm 5.201909133133e-12
>
> > > > >
>
> > > > > 49 KSP preconditioned resid norm 5.201909133133e-12 true resid norm
>
> > > > >
>
> > > > > 5.807496359589e-08 ||r(i)||/||b|| 1.957419116714e-04
>
> > > > >
>
> > > > > 50 KSP Residual norm 4.803415857522e-12
>
> > > > >
>
> > > > > 50 KSP preconditioned resid norm 4.803415857522e-12 true resid norm
>
> > > > >
>
> > > > > 5.648916764539e-08 ||r(i)||/||b|| 1.903969796791e-04
>
> > > > >
>
> > > > > 51 KSP Residual norm 4.213404847970e-12
>
> > > > >
>
> > > > > 51 KSP preconditioned resid norm 4.213404847970e-12 true resid norm
>
> > > > >
>
> > > > > 5.312367187239e-08 ||r(i)||/||b|| 1.790535618699e-04
>
> > > > >
>
> > > > > 52 KSP Residual norm 3.583723287544e-12
>
> > > > >
>
> > > > > 52 KSP preconditioned resid norm 3.583723287544e-12 true resid norm
>
> > > > >
>
> > > > > 4.482581392980e-08 ||r(i)||/||b|| 1.510855964009e-04
>
> > > > >
>
> > > > > 53 KSP Residual norm 3.055433343791e-12
>
> > > > >
>
> > > > > 53 KSP preconditioned resid norm 3.055433343791e-12 true resid norm
>
> > > > >
>
> > > > > 3.733365588478e-08 ||r(i)||/||b|| 1.258332458617e-04
>
> > > > >
>
> > > > > 54 KSP Residual norm 2.632415195178e-12
>
> > > > >
>
> > > > > 54 KSP preconditioned resid norm 2.632415195178e-12 true resid norm
>
> > > > >
>
> > > > > 3.259461418510e-08 ||r(i)||/||b|| 1.098602856676e-04
>
> > > > >
>
> > > > > 55 KSP Residual norm 2.358953414259e-12
>
> > > > >
>
> > > > > 55 KSP preconditioned resid norm 2.358953414259e-12 true resid norm
>
> > > > >
>
> > > > > 2.755891784400e-08 ||r(i)||/||b|| 9.288744974367e-05
>
> > > > >
>
> > > > > 56 KSP Residual norm 2.131084837625e-12
>
> > > > >
>
> > > > > 56 KSP preconditioned resid norm 2.131084837625e-12 true resid norm
>
> > > > >
>
> > > > > 2.435335186857e-08 ||r(i)||/||b|| 8.208307599694e-05
>
> > > > >
>
> > > > > 57 KSP Residual norm 1.906113965754e-12
>
> > > > >
>
> > > > > 57 KSP preconditioned resid norm 1.906113965754e-12 true resid norm
>
> > > > >
>
> > > > > 2.234444331024e-08 ||r(i)||/||b|| 7.531204116139e-05
>
> > > > >
>
> > > > > 58 KSP Residual norm 1.666029207693e-12
>
> > > > >
>
> > > > > 58 KSP preconditioned resid norm 1.666029207693e-12 true resid norm
>
> > > > >
>
> > > > > 2.187608862971e-08 ||r(i)||/||b|| 7.373344971974e-05
>
> > > > >
>
> > > > > 59 KSP Residual norm 1.397594452516e-12
>
> > > > >
>
> > > > > 59 KSP preconditioned resid norm 1.397594452516e-12 true resid norm
>
> > > > >
>
> > > > > 1.884281030995e-08 ||r(i)||/||b|| 6.350977224879e-05
>
> > > > >
>
> > > > > 60 KSP Residual norm 1.000720808126e-12
>
> > > > >
>
> > > > > 60 KSP preconditioned resid norm 1.000720808126e-12 true resid norm
>
> > > > >
>
> > > > > 1.495264965711e-08 ||r(i)||/||b|| 5.039796923166e-05
>
> > > > >
>
> > > > > 61 KSP Residual norm 8.368281175845e-13
>
> > > > >
>
> > > > > 61 KSP preconditioned resid norm 8.368281175845e-13 true resid norm
>
> > > > >
>
> > > > > 1.198110007063e-08 ||r(i)||/||b|| 4.038234871863e-05
>
> > > > >
>
> > > > > 62 KSP Residual norm 6.975866756145e-13
>
> > > > >
>
> > > > > 62 KSP preconditioned resid norm 6.975866756145e-13 true resid norm
>
> > > > >
>
> > > > > 8.713139114636e-09 ||r(i)||/||b|| 2.936767242465e-05
>
> > > > >
>
> > > > > 63 KSP Residual norm 6.055685829863e-13
>
> > > > >
>
> > > > > 63 KSP preconditioned resid norm 6.055685829863e-13 true resid norm
>
> > > > >
>
> > > > > 7.208696201852e-09 ||r(i)||/||b|| 2.429694119186e-05
>
> > > > >
>
> > > > > 64 KSP Residual norm 5.321880026091e-13
>
> > > > >
>
> > > > > 64 KSP preconditioned resid norm 5.321880026091e-13 true resid norm
>
> > > > >
>
> > > > > 6.450668328122e-09 ||r(i)||/||b|| 2.174200502115e-05
>
> > > > >
>
> > > > > 65 KSP Residual norm 4.652520314729e-13
>
> > > > >
>
> > > > > 65 KSP preconditioned resid norm 4.652520314729e-13 true resid norm
>
> > > > >
>
> > > > > 5.767198491137e-09 ||r(i)||/||b|| 1.943836703022e-05
>
> > > > >
>
> > > > > 66 KSP Residual norm 4.095554125778e-13
>
> > > > >
>
> > > > > 66 KSP preconditioned resid norm 4.095554125778e-13 true resid norm
>
> > > > >
>
> > > > > 5.117612129931e-09 ||r(i)||/||b|| 1.724893343844e-05
>
> > > > >
>
> > > > > 67 KSP Residual norm 3.575261581129e-13
>
> > > > >
>
> > > > > 67 KSP preconditioned resid norm 3.575261581129e-13 true resid norm
>
> > > > >
>
> > > > > 4.479181564454e-09 ||r(i)||/||b|| 1.509710050359e-05
>
> > > > >
>
> > > > > 68 KSP Residual norm 3.109053579360e-13
>
> > > > >
>
> > > > > 68 KSP preconditioned resid norm 3.109053579360e-13 true resid norm
>
> > > > >
>
> > > > > 3.779777902208e-09 ||r(i)||/||b|| 1.273975748689e-05
>
> > > > >
>
> > > > > 69 KSP Residual norm 2.721884401030e-13
>
> > > > >
>
> > > > > 69 KSP preconditioned resid norm 2.721884401030e-13 true resid norm
>
> > > > >
>
> > > > > 3.393605957929e-09 ||r(i)||/||b|| 1.143816330711e-05
>
> > > > >
>
> > > > > Linear solve converged due to CONVERGED_RTOL iterations 69
>
> > > > >
>
> > > > > 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: 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=16384, cols=16384
>
> > > > >
>
> > > > > package used to perform factorization: petsc
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > 4)
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Linear solve converged due to CONVERGED_ITS iterations 1
>
> > > > >
>
> > > > > KSP Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: preonly
>
> > > > >
>
> > > > > maximum iterations=10000, initial guess is zero
>
> > > > >
>
> > > > > tolerances: relative=1e-05, absolute=1e-50, divergence=10000
>
> > > > >
>
> > > > > left preconditioning
>
> > > > >
>
> > > > > using NONE norm type for convergence test
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > PC Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: lu
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > LU: out-of-place factorization
>
> > > > >
>
> > > > > tolerance for zero pivot 2.22045e-14
>
> > > > >
>
> > > > > matrix ordering: nd
>
> > > > >
>
> > > > > factor fill ratio given 5, needed 9.6582
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > Factored matrix follows:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > package used to perform factorization: petsc
>
> > > > >
>
> > > > > total: nonzeros=791200, allocated nonzeros=791200
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > linear system matrix = precond matrix:
>
> > > > >
>
> > > > > Mat Object: 1 MPI processes
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > type: seqaij
>
> > > > >
>
> > > > > rows=16384, cols=16384
>
> > > > >
>
> > > > > total: nonzeros=81920, allocated nonzeros=81920
>
> > > > >
>
> > > > > total number of mallocs used during MatSetValues calls =0
>
> > > > >
>
> > > > >
>
> > > > >
>
> > > > > has attached null space
>
> > > > >
>
> > > > > not using I-node routines
>
> > > > >
>
> > > > > On Tuesday 01 September 2015 03:40:14 Matthew Knepley wrote:
>
> > > > > > On Tue, Sep 1, 2015 at 2:22 AM, Filippo Leonardi <
>
> > > > > >
>
> > > > > > filippo.leonardi at sam.math.ethz.ch> wrote:
>
> > > > > > > Dear PETSc Users,
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > I want to use multigrid to solve uniform (just Laplace)
> poisson in
>
> > > > >
>
> > > > > 2D/3D
>
> > > > >
>
> > > > > > > on cartesian, non-uniform meshes with a standard 5 (7)-points
>
> > >
>
> > > stencil
>
> > >
>
> > > > > FD.
>
> > > > >
>
> > > > > > > I always scaled my Poisson matrix like in the doc examples,
> i.e.
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > multiplying by dx*dy (so that in ComputeRHS I need to scale b,
> in
>
> > >
>
> > > A*x
>
> > >
>
> > > > > = b,
>
> > > > >
>
> > > > > > > as well). This always worked properly with both MG/GAMG and
> with
>
> > > > >
>
> > > > > galerkin
>
> > > > >
>
> > > > > > > matrices.
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > Now I'd like to use non-uniform meshes, therefore the scaling
> is
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > non-uniform. However I cannot get my matrices to scale properly
>
> > >
>
> > > with
>
> > >
>
> > > > > any
>
> > > > >
>
> > > > > > > sort of multigrid.
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > One would think that without scaling, i.e. solving the original
>
> > > > >
>
> > > > > system, at
>
> > > > >
>
> > > > > > > least MG+galerkin or GAMG should work anyways provided the
> matrix
>
> > > > > > > A
>
> > > > >
>
> > > > > and b
>
> > > > >
>
> > > > > > > are consistent.
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > I tried without boundaries (i.e. torus), so this is not the
>
> > >
>
> > > problem.
>
> > >
>
> > > > > > The point of that scaling is to make the boundary values the same
>
> > >
>
> > > size
>
> > >
>
> > > > > > as
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > the residuals. If you have
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > no boundary values, the problem is scale invariant since its
> linear.
>
> > >
>
> > > It
>
> > >
>
> > > > > > would be nice to scale it so everything
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > is about size 1.
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > When asking about convergence always run with -ksp_view
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > -ksp_monitor_true_residual -ksp_converged_reason
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > Thanks,
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > >
>
> > > > > > Matt
>
> > > > > >
>
> > > > > > > Anyone did/knows how to do this properly?
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > Thanks,
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > >
>
> > > > > > > F
>
>
>

-- 
What most experimenters take for granted before they begin their
experiments is infinitely more interesting than any results to which their
experiments lead.
-- Norbert Wiener
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