dokken:

mixed finite element

Dokken, thank you for your reply!

My problem is 1D, the functions are m(x), phi(x) and sigma(x). They are satisfying the the following equations,

```
m' = N * phi'**2 + mu**2 * phi**2 + omega**2 *phi**2 /(N*sigma**2)
sigma' /(sigma*x) = 2.0*(phi**2+ omega**2*phi**2/(N**2*sigma**2))*sigma
phi'' = (2.0/x+N_x/N+sigma.dx(0)/sigma)*phi.dx(0) + (omega**2/N/sigma**2-mu**2)*phi/N
```

where

```
N(x) = (1-2 m / x) , and N_x= N' = dN(x)/dx
```

The boundary condations are

```
m(0) = 0, sigma(100)= 1, phi(100)=0, phi'(0) = 0
```

My codes are

```
from fenics import *
# parmeters
r_0 = 1.0E-6
r_b = 100 # r的右边界值，the coordinate of right boundary
omega = 0.9
mu = 1.0
mesh=IntervalMesh(500,r_0,r_b)
# Define function space
P1 = FiniteElement( "Lagrange",mesh.ufl_cell(), 1)
element = MixedElement([P1, P1, P1])
V = FunctionSpace(mesh, element)
# Define functions
u = Function(V)
# Split system functions to access components
m, phi, sigma = split(u)
# def the_boundary(x, on_boundary):
# tol = 1E-6 # tolerance for coordinate comparisons
# return on_boundary and (near(x[0], 0, tol) or near(x[0], r_b, tol))
def the_boundary_R(x, on_boundary):
tol = 1E-6 # tolerance for coordinate comparisons
return on_boundary and near(x[0], r_b, tol)
def the_boundary_L(x, on_boundary):
tol = 1E-6 # tolerance for coordinate comparisons
return on_boundary and near(x[0], r_0, tol)
m_boundary = Expression ( "0.0", degree = 1 )
phi_boundary = Expression ( "0.0", degree = 1 )
sigma_boundary = Expression ( "1.0", degree = 1 )
bc_m = DirichletBC(V.sub(0), m_boundary, the_boundary_L)
bc_phi = DirichletBC(V.sub(1), phi_boundary, the_boundary_R)
bc_sigma = DirichletBC(V.sub(2), sigma_boundary, the_boundary_R)
# Define test functions
v_1, v_2, v_3 = TestFunctions(V)
# Define variational problem
x = Expression ("x[0]",degree = 1)
N = (x-2.0*m)/x
N_x = (2.0*m.dx(0)*x-2.0*m)/x**2
F1 = N*phi.dx(0)**2+ mu**2*phi**2+omega**2*phi**2/N/sigma**2 - m.dx(0)/x**2
F2 = 2.0*(phi.dx(0)**2+ omega**2*phi**2/(N**2*sigma**2))*sigma*x - sigma.dx(0)
F3 = (2.0/x+N_x/N+sigma.dx(0)/sigma)*phi.dx(0) + (omega**2/N/sigma**2-mu**2)*phi/N
F = (F1*v_1+F2*v_2+F3*v_3 - phi.dx(0)*v_3.dx(0))*dx
# Solve the system.
#
solve ( F == 0, u, [bc_m, bc_phi, bc_sigma] )
u_1_, u_2_, u_3_ = u.split()
plt.figure(1,dpi=300)
plt.plot (u_1_)
plt.plot (u_2_)
plt.plot (u_3_)
filename = 'solutions.png'
plt.savefig ( filename )
print ( ' Graphics saved as "%s"' % ( filename ) )
plt.show()
plt.close ( )
```

I got the following error messages,

```
No Jacobian form specified for nonlinear variational problem.
Differentiating residual form F to obtain Jacobian J = F'.
Solving nonlinear variational problem.
Newton iteration 0: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 1: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 2: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 3: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 4: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 5: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 6: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 7: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 8: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 9: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 10: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 11: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 12: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 13: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 14: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 15: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 16: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 17: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 18: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 19: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 20: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 21: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 22: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 23: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 24: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 25: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 26: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 27: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 28: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 29: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 30: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 31: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 32: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 33: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 34: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 35: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 36: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 37: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 38: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 39: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 40: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 41: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 42: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 43: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 44: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 45: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 46: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 47: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 48: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 49: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Newton iteration 50: r (abs) = -nan (tol = 1.000e-10) r (rel) = -nan (tol = 1.000e-09)
Traceback (most recent call last):
File "boson.py", line 56, in <module>
solve ( F == 0, u, [bc_m, bc_phi, bc_sigma] )
File "/usr/lib/petsc/lib/python3/dist-packages/dolfin/fem/solving.py", line 233, in solve
_solve_varproblem(*args, **kwargs)
File "/usr/lib/petsc/lib/python3/dist-packages/dolfin/fem/solving.py", line 314, in _solve_varproblem
solver.solve()
RuntimeError:
*** -------------------------------------------------------------------------
*** 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 nonlinear system with NewtonSolver.
*** Reason: Newton solver did not converge because maximum number of iterations reached.
*** Where: This error was encountered inside NewtonSolver.cpp.
*** Process: 0
***
*** DOLFIN version: 2019.2.0.dev0
*** Git changeset: unknown
*** -------------------------------------------------------------------------
```

Can you help me to find out the bugs in my codes?