Time dependent problem in mixed elements

1

Hi, I have a simple time dependent equation set here:
image
and I’m trying to solve this equation set with reference to the mixed Poisson demo Bitbucket. Following is my MWE:

from dolfin import *

# timestep
dt = 1

# define mesh and FunctionSpace
mesh = UnitSquareMesh(32,32)
V1 = FiniteElement('CG',mesh.ufl_cell(),1)
V2 = FiniteElement('CG',mesh.ufl_cell(),1)
V = FunctionSpace(mesh,V1*V2)

# initial condition projection
w_n = Function(V)
u10,u20 = w_n.split()
u10 = project(Expression('1+x[1]+x[0]',degree=1),V.sub(0).collapse())

# Define TrialFunctions and TestFunctions
u1,u2 = TrialFunctions(V)
v1,v2 = TestFunctions(V)

# Define variational problem
a = u1*v1*dx+dt*inner(grad(u2),grad(v1))*dx+u1*v2*dx-(1+u2)**(-1.5)*v2*dx
L = u10*v1*dx

# Solve and time stepping
w=Function(V)
out1=File('mixed/u1.pvd')
out2=File('mixed/u2.pvd')
t=0
for t in range(0,40):
    A = assemble(a)
    b = assemble(L)
    solve(A,w.vector(),b)
    u1,u2 = w.split()
    out1 << u1
    out2 << u2
    w_n.assign(w)
    t += 1

When I run this MWE, an error occurs as:
ufl.algorithms.check_arities.ArityMismatch: Applying nonlinear operator Power to expression depending on form argument v_1.
So what should I do to eliminate this error?
And also, as I tried to dodge the error by simplly replacing the (1+u2)**(-1.5)*v2*dx with u2*v2*dx, the u1 and u2 seems can’t update. Is there something wrong in my initial condition projection or somewhere else? Thanks!

2

Your project call overwrites the u10 variable.

And your problem isn’t linear given the right hand side of the second equation \frac{1}{(1 + u_2)^{1.5}} so I’d recommend consulting the nonlinear Poisson demo.

Also I’m a bit rushed here and may have made a stupid mistake, but can’t you reformulate this as:

- \frac{3}{2} (1 + u)^{-\frac{5}{2}} \frac{\partial u}{\partial t} = \nabla^2 u
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