Mixed problem where one unknown function is defined on the boundary

It seems that the answer given to question #136409 could really be useful. In the code which is refered to there ( http://arxiv.org/abs/1010.1873), one can find the definition of the function space

 V_cg = FunctionSpace(mesh, "CG", order, "facet")

which is restricted to facets. Where could I have found documentation on the option "facet"?

Thanks.

Xavier.

I actually found a way to deal with this question but it does not seem natural: I apply zero dirichlet boundary condition on the interior (!) of the domain. Is there any more natural way in Fenics of defining spaces of functions defined on the boundary?

You can find here a description of the problem
http://dl.dropbox.com/u/4003624/testcase.pdf

Here is the code:

from dolfin import *

mesh = UnitSquare(32,32)

movie=False

# Time step, Time interval
dt=0.1
T=80

# Choose finite element vector space
U = FunctionSpace(mesh, "Lagrange", 1)
P = FunctionSpace(mesh, "Lagrange", 1)

W=U*P

(u, p) = TrialFunctions(W)
(v, q) = TestFunctions(W)

def boundary(x):
   if x[0]<0.0001 or x[0]>1-0.0001 or x[1]<0.0001 or x[1]>1-0.0001:
      return False
   else:
      return True

bc = DirichletBC(W.sub(1), Constant(1.0), boundary)

#dissipation coefficient
alpha=0.001

uprev=interpolate(Constant(0.0),U)
pprev=interpolate(Constant(0.0),P)

# visualization parameters
h=plot(uprev)
# h.rescale=True
h.rescale=False
h.set_min_max(0,100)
h.init_writer(".tmp")

# Construct bilinear and linear form
(u,p) = TrialFunction(W)
(v,q) = TestFunction(W)

# k1=0
k2=1
gamma=1

# a = u*v*dx+dt*alpha*inner(grad(u), grad(v))*dx-(-k1*gamma*u*p*v+k2*gamma*(1-p))*ds+(p-dt*(-k1*gamma*u*p*v+k2*gamma*(1-p)))*w*ds

a = u*v*dx+dt*alpha*inner(grad(u),grad(v))*dx+k2*gamma*p*v*ds+(1+dt*k2)*p*q*ds
L = uprev*v*dx+k2*gamma*v*ds+(pprev+dt*k2)*q*ds

A=assemble(a)

w=Function(W)

# debug_here()

# Time integration
t=0
i=1
while t<=T:

   b=assemble(L)
   bc.apply(A,b)
   solve(A,w.vector(),b)

   (u,p)=w.split()
   h.update(data=u)
   if movie:
      h.write_png()

   i+=1
   t+=dt
   uprev.assign(u)
   pprev.assign(p)

if movie:
   h.movie("movie2.avi", fps=10, cleanup=True)

This answers my question. Thank you very much!

Thanks Garth Wells, that solved my question.