[petsc-users] convergence problem in spherical coordinates

[Patrick Alken]

On 02/20/2012 04:37 PM, Matthew Knepley wrote:
> On Mon, Feb 20, 2012 at 4:52 PM, Patrick Alken 
> <patrick.alken at colorado.edu <mailto: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.

I've made this change in the FD derivatives of phi. Unfortunately PETSc 
still does not converge properly. In the meantime I've tried two 
different direct solver libraries, both of which find correct solutions 
to the matrix problem, so I don't think I can use petsc for this problem.

>
> 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.

Yes but my grid points start a little away from the poles to avoid this.

>
>    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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