[petsc-users] convergence problem in spherical coordinates

[Matthew Knepley]

On Mon, Feb 20, 2012 at 4:52 PM, Patrick Alken
<patrick.alken at colorado.edu>wrote:

>  Hello all,
>
>   I am having great difficulty solving a 3D finite difference equation in
> spherical coordinates. I am solving the equation in a spherical shell
> region S(a,b), with the boundary conditions being that the function is 0 on
> both boundaries (r = a and r = b). I haven't imposed any boundary
> conditions on theta or phi which may be a reason its not converging. The
> phi boundary condition would be that the function is periodic in phi, but I
> don't know if this needs to be put into the matrix somehow?
>

1) The periodicity appears in the definition of the FD derivative in phi.
Since this is Cartesian, you can use a DA in 3D, and make one
direction periodic.

2) Don't you have a coordinate singularity at the pole? This is why every
code I know of uses something like a Ying-Yang grid.

   Matt


> I nondimensionalized the equation before solving which helped a little
> bit. I've also scaled the matrix and RHS vectors by their maximum element
> to make all entries <= 1.
>
> I've tried both direct and iterative solvers. The direct solvers give a
> fairly accurate solution for small grids but seem unstable for larger
> grids. The PETSc iterative solvers converge for very small grids but for
> medium to large grids don't converge at all.
>
> When running with the command (for a small grid):
>
> *> ./main -ksp_converged_reason -ksp_monitor_true_residual -pc_type svd
> -pc_svd_monitor*
>
> I get the output:
>
>     SVD: condition number 5.929088512946e+03, 0 of 1440 singular values
> are (nearly) zero
>     SVD: smallest singular values: 2.742809162118e-04 2.807446554985e-04
> 1.548488288425e-03 1.852332719983e-03 2.782708934678e-03
>     SVD: largest singular values : 1.590835571953e+00 1.593368145758e+00
> 1.595771695877e+00 1.623691828398e+00 1.626235829632e+00
>   0 KSP preconditioned resid norm 2.154365616645e+03 true resid norm
> 8.365589263063e+00 ||r(i)||/||b|| 1.000000000000e+00
>   1 KSP preconditioned resid norm 4.832753933427e-10 true resid norm
> 4.587845792963e-12 ||r(i)||/||b|| 5.484187244549e-13
> Linear solve converged due to CONVERGED_RTOL iterations 1
>
> When plotting the output of this SVD solution, it looks pretty good, but
> svd isn't practical for larger grids.
>
> Using the command (on the same grid):
>
> *> ./main -ksp_converged_reason -ksp_monitor_true_residual
> -ksp_compute_eigenvalues -ksp_gmres_restart 1000 -pc_type none*
>
> The output is attached. There do not appear to be any 0 eigenvalues. The
> solution here is much less accurate than the SVD case since it didn't
> converge.
>
> I've also tried the -ksp_diagonal_scale -ksp_diagonal_scale_fix options
> which don't help very much.
>
> Any advice on how to trouble shoot this would be greatly appreciated.
>
> Some things I've checked already:
>
> 1) there aren't any 0 rows in the matrix
> 2) using direct solvers on very small grids seems to give decent solutions
> 3) there don't appear to be any 0 singular values or eigenvalues
>
> Perhaps the matrix has a null space, but I don't know how I would find out
> what the null space is? Is there a tutorial on how to do this?
>
> Thanks in advance!
>
>


-- 
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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