Simple 1D advection-diffusion problem in FEniCS.

1d_advection_diffusion.py

"""Steady-state diffusion equation,

    - div grad u(x) = 0

and boundary conditions given by

	u(0) = 1 	(inflow)
	u(1) = 0	(outflow).
"""

from dolfin import *
import numpy as np

# Create mesh and define function space
mesh = UnitIntervalMesh(50)
V = FunctionSpace(mesh, "CG", 1)

# Location of Dirichlet boundaries,
# x = 0, x = 1.
def left_boundary(x):
	return np.isclose(x[0], 0.0)

def right_boundary(x):
	return np.isclose(x[0], 1.0)

# Define boundary condition
# u(0) = 1, u(1) = 0
bc_left = DirichletBC(V, Constant(1.0), left_boundary)
bc_right = DirichletBC(V, Constant(0.0), right_boundary)

# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Expression("0")
g = Expression("0")
#      advection term 						diffusion term
a = div(Constant(100)*grad(u))*v*dx + inner(Constant(1.0)*grad(u), grad(v))*dx
L = f*v*dx + g*v*ds

# Compute solution
u = Function(V)
solve(a == L, u, bcs=[bc_left, bc_right])

# Save solution in VTK format
file = File("diffusion.pvd")
file << u

# Plot solution
plot(u, interactive=True)