[petsc-users] questions about the multigrid framework

[Matthew Knepley]

On Wed, Feb 9, 2011 at 9:58 AM, Peter Wang <pengxwang at hotmail.com> wrote:

>  Thanks Barry,
>
>     I run the code with -ksp_monitor_true_residual  -ksp_converged_reason,
> and it turns out that the computation didn't get the real convergence.
> After I set the rtol and more iteration, the numerical solution get better.
> However, the computation converges very slowly with finer grid points. For
> example, with nx=2500 and ny=10000, (lx=2.5e-4,ly=1e-3, and the distribution
> varys mainly in y direction)
> at IT=72009, true resid norm 1.638857052871e-01 ||Ae||/||Ax||
> 9.159199925235e-07
>   IT=400000,true resid norm 1.638852449299e-01 ||Ae||/||Ax||
> 9.159174196917e-07.
> and it didn't converge yet.
>
>   I am wondering if the solver is changed, the convergency speed could get
> fater? Or, I should take anohte approach to use finer grids, like multigrid?
> Thanks for your help.
>

If you can get MG to work for your problem, its optimal. All the Krylov
methods alone will get worse with increasing grid size.

   Matt


>
> > From: bsmith at mcs.anl.gov
> > Date: Sun, 6 Feb 2011 21:30:56 -0600
> > To: petsc-users at mcs.anl.gov
> > Subject: Re: [petsc-users] questions about the multigrid framework
> >
> >
> > On Feb 6, 2011, at 5:00 PM, Peter Wang wrote:
> >
> > > Hello, I have some concerns about the multigrid framework in PETSc.
> > >
> > > We are trying to solve a two dimensional problem with a large variety
> in length scales. The length of computational domain is in order of 1e3 m,
> and the width is in 1 m, nevertheless, there is a tiny object with 1e-3 m in
> a corner of the domain.
> > >
> > > As a first thinking, we tried to solve the problem with a larger number
> of uniform or non-uniform grids. However, the error of the numerical
> solution increases when the number of the grid is too large. In order to
> test the effect of the grid size on the solution, a domain with regular
> scale of 1m by 1m was tried to solve. It is found that the extreme small
> grid size might lead to large variation to the exact solution. For example,
> the exact solution is a linear distribution in the domain. The numerical
> solution is linear as similar as the exact solution when the grid number is
> nx=1000 by ny=1000. However, if the grid number is nx=10000 by ny=10000, the
> numerical solution varies to nonlinear distribution which boundary is the
> only same as the exact solution.
> >
> > Stop right here. 99.9% of the time what you describe should not happen,
> with a finer grid your solution (for a problem with a known solution for
> example) will be more accurate and won't suddenly get less accurate with a
> finer mesh.
> >
> > Are you running with -ksp_monitor_true_residual -ksp_converged_reason to
> make sure that it is converging? and using a smaller -ksp_rtol <tol> for
> more grid points. For example with 10,000 grid points in each direction and
> no better idea of what the discretization error is I would use a tol of
> 1.e-12
> >
> > Barry
> >
> > We'll deal with the multigrid questions after we've resolved the more
> basic issues.
> >
> >
> > > The solver I used is a KSP solver in PETSc, which is set by calling :
> > > KSPSetOperators(ksp,A,A,DIFFERENT_NONZERO_PATTERN,ierr). Whether this
> solver is not suitable to the system with small size grid? Or, whether the
> problem crossing 6 orders of length scale is solvable with only one level
> grid system when the memory is enough for large matrix? Since there is less
> coding work for one level grid size, it would be easy to implement the
> solver.
> > >
> > > I did some research work on the website and found the slides by Barry
> on
> > >
> http://www.mcs.anl.gov/petsc/petsc-2/documentation/tutorials/Columbia04/DDandMultigrid.pdf
> > > It seems that the multigrid framework in PETSc is a possible approach
> to our problem. We are thinking to turn to the multigrid framework in PETSc
> to solve the problem. However, before we dig into it, there are some issues
> confusing us. It would be great if we can get any suggestion from you:
> > > 1 Whether the multigrid framework can handle the problem with a large
> variety in length scales (up to 6 orders)? Is DMMG is the best tool for our
> problem?
> > >
> > > 2 The coefficient matrix A and the right hand side vector b were
> created for the finite difference scheme of the domain and solved by KSP
> solver (callKSPSetOperators(ksp,A,A,DIFFERENT_NONZERO_PATTERN,ierr)). Is it
> easy to immigrate the created Matrix A and Vector b to the multigrid
> framework?
> > >
> > > 3 How many levels of the subgrid are needed to obtain a solution close
> enough to the exact solution for a problem with 6 orders in length scale?
> > >
> >
>



-- 
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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