On Feb 9, 2011, at 9:58 AM, Peter Wang 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.
You have a little confusion here. Multigrid (in the context of PETSc and
numerical solvers) is ONLY an efficient way to solve a set of linear equations
arising from discretizing a PDE. It is not a different way of discretizing the
PDEs or giving a different or better solution. It is only a way of getting the
same solution (potentially much) faster than running the slower convergent
solver.
Definitely configure PETSc with --download-ml --download-hypre and make
runs using -pc_type hypre and then -pc_type ml to see how algebraic multigrid
works, it should work fine for your problem.
Barry
>
>
> > 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?
> > >
> >
