element size and convergence

Question

Hi All,
I want to solve the steady-state PNP (Poisson-Nernst-Planck) equation using FEniCS. The domain is nano-scale 2D square domain with 2 ion species: K+ and Cl-.
I first solve the Poission equation to get electric potential ( concentrations of K+ and Cl- are fixed ), then put this electric potential into Nernst-Planck equation and solve for new concentrations of K+ and Cl-, respectively. Repeat above cycle iteratively until convergence is reached. Following is the pseudo code I used:

def solvePBproblem(cationconc,anionconc):

         ......... receive concentrations of K+ and Cl-, 
         solve for electric potential phi...........

         return phi

def solveNPproblem(electric_potential,charge):

            ......... receive electric potential, charge,
             solve for new concentration ...........

          return new_concentration

while True: #  do iterations until convergent
        phi = solvePBproblem(cp1,cp2)
        cp1 = solveNPproblem(phi,z1)
        cp2 = solveNPproblem(phi,z2)

In my calculations, the convergence was reached only when I set the element size to be extremely small. If I slightly increased the element size, the calculation would abort after several iterations ( the electric potential suddenly became unexpectedly large ). I also tried to change the configurations of the solver from default to

solver.parameters["linear_solver"] = "minres" # also tried umfpack and gmres
solver.parameters["preconditioner"] = "amg" # also tried ilu as well

All above tests failed to reach convergence with the slightly increased element size. My question is : Is using the extremely small element size the only way to reach convergence or could I use some more advanced approaches with a fairly small element size to achieve convergence ?

Any suggestions would be highly appreciated !

Answer 1

Hi,
how is your matrix?
The algorithm you should use depends on the kind of matrix you have:

1. Symmetric Positive Definite (SPD): Use Conjugate Gradient with an SPD preconditioner
2. Symmetric and indefinite: Use Minimal Residual method with and SPD preconditioner
3. Positive: GMRES and ILU (or AMG) are often good, but you might need to experiment

Reference: notes from MEK4250 course at UiO.