Problem in solve linear system for a saddle point problem with preconditioners

Linear Algebra
1

Good morning,
I am solving the Poisson with pure Neumann bounday condtions. I formulate the problem as a saddle point problem, in order to deal with the singular system. The code is the following:

from dolfin import *
from mpi4py import MPI
import matplotlib.pyplot as plt
import time

# For parallelization
comm = MPI.COMM_WORLD # comm in the comunicator objec, its methods will be used for comunication
my_rank = comm.Get_rank()
p = comm.Get_size() #number of processes

#Starting CPU time
st_time = time.time()

# Create mesh
N = 40
mesh = UnitCubeMesh(N,N,N)
#mesh = UnitSquareMesh(N, N)

# Build function space with Lagrange multiplier
P1 = FiniteElement("CG", mesh.ufl_cell(), 1)
R = FiniteElement("Real", mesh.ufl_cell(), 0)
W = FunctionSpace(mesh, P1 * R)
# Concerning the number of integration points
q_degree = 4
dx = dx(metadata={'quadrature_degree': q_degree})


# Define variational problem
(u, c) = TrialFunction(W)
(v, d) = TestFunctions(W)
f = Expression('4*pi*pi*cos(2*pi*x[0])', degree=5)
#g = Expression('-4*pi*pi*(2*pi*x[0])', degree=5)
g = Constant(0.0)
a = (inner(grad(u), grad(v)) + c*v + u*d)*dx
L = f*v*dx + g*v*ds

# Whit iterative solver
# Set linear solver parameters
abs_tol = 1E-10
rel_tol = 1E-10
max_iter = 1000

prm = parameters['krylov_solver']
prm['absolute_tolerance'] = abs_tol
prm['relative_tolerance'] = rel_tol
prm['maximum_iterations'] = max_iter
prm["monitor_convergence"] = True
#info(parameters, True)

# Compute solution
w = Function(W)
solve(a == L, w, solver_parameters={'linear_solver':'cg','preconditioner':'hypre_euclid'})
#
(u, c) = w.split()

and the error that I have is the following:

*** -------------------------------------------------------------------------
*** DOLFIN encountered an error. If you are not able to resolve this issue
*** using the information listed below, you can ask for help at
***
***     fenics-support@googlegroups.com
***
*** Remember to include the error message listed below and, if possible,
*** include a *minimal* running example to reproduce the error.
***
*** -------------------------------------------------------------------------
*** Error:   Unable to solve linear system using PETSc Krylov solver.
*** Reason:  Solution failed to converge in 10 iterations (PETSc reason DIVERGED_INDEFINITE_PC, residual norm ||r|| = 1.063353e-04).
*** Where:   This error was encountered inside PETScKrylovSolver.cpp.
*** Process: 0
*** 
*** DOLFIN version: 2019.2.0.dev0
*** Git changeset:  ubuntu
*** -------------------------------------------------------------------------

can someone help me or have some hint on what I am doing wrong.
Thanks in advance.

2

I’m not sure incomplete LU is an appropriate preconditioner for an indefinite problem.

3

I tried all the possible preconditioner and none of those worked unfortunatly.