# [petsc-users] Overcoming slow convergence with GMRES+Hypre BoomerAMG

Christopher, Joshua jchristopher at anl.gov
Wed Mar 1 18:17:03 CST 2023

```Hello,

I am trying to solve the leaky-dielectric model equations with PETSc using a second-order discretization scheme (with limiting to first order as needed) using the finite volume method. The leaky dielectric model is a coupled system of two equations, consisting of a Poisson equation and a convection-diffusion equation.  I have tested on small problems with simple geometry (~1000 DoFs) using:

-ksp_type gmres
-pc_type hypre
-pc_hypre_type boomeramg

and I get RTOL convergence to 1.e-5 in about 4 iterations. I tested this in parallel with 2 cores, but also previously was able to use successfully use a direct solver in serial to solve this problem. When I scale up to my production problem, I get significantly worse convergence. My production problem has ~3 million DoFs, more complex geometry, and is solved on ~100 cores across two nodes. The boundary conditions change a little because of the geometry, but are of the same classifications (e.g. only Dirichlet and Neumann). On the production case, I am needing 600-4000 iterations to converge. I've attached the output from the first solve that took 658 iterations to converge, using the following output options:

-ksp_view_pre
-ksp_view
-ksp_converged_reason
-ksp_monitor_true_residual
-ksp_test_null_space

My matrix is non-symmetric, the condition number can be around 10e6, and the eigenvalues reported by PETSc have been real and positive (using -ksp_view_eigenvalues).

I have tried using other preconditions (superlu, mumps, gamg, mg) but hypre+boomeramg has performed the best so far. The literature seems to indicate that AMG is the best approach for solving these equations in a coupled fashion.

Do you have any advice on speeding up the convergence of this system?

Thank you,
Joshua
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