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a516f1b Nov 24, 2016
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"""
FEniCS tutorial demo program: Heat equation with Dirichlet conditions.
Test problem is chosen to give an exact solution at all nodes of the mesh.
u'= Laplace(u) + f in the unit square
u = u_D on the boundary
u = u_0 at t = 0
u = 1 + x^2 + alpha*y^2 + \beta*t
f = beta - 2 - 2*alpha
"""
from __future__ import print_function
from fenics import *
import numpy as np
T = 2.0 # final time
num_steps = 10 # number of time steps
dt = T / num_steps # time step size
alpha = 3 # parameter alpha
beta = 1.2 # parameter beta
# Create mesh and define function space
nx = ny = 8
mesh = UnitSquareMesh(nx, ny)
V = FunctionSpace(mesh, 'P', 1)
# Define boundary condition
u_D = Expression('1 + x[0]*x[0] + alpha*x[1]*x[1] + beta*t',
degree=2, alpha=alpha, beta=beta, t=0)
def boundary(x, on_boundary):
return on_boundary
bc = DirichletBC(V, u_D, boundary)
# Define initial value
u_n = interpolate(u_D, V)
#u_n = project(u_D, V)
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
f = Constant(beta - 2 - 2*alpha)
F = u*v*dx + dt*dot(grad(u), grad(v))*dx - (u_n + dt*f)*v*dx
a, L = lhs(F), rhs(F)
# Time-stepping
u = Function(V)
t = 0
for n in range(num_steps):
# Update current time
t += dt
u_D.t = t
# Compute solution
solve(a == L, u, bc)
# Plot solution
plot(u)
# Compute error at vertices
u_e = interpolate(u_D, V)
error = np.abs(u_e.vector().array() - u.vector().array()).max()
print('t = %.2f: error = %.3g' % (t, error))
# Update previous solution
u_n.assign(u)
# Hold plot
interactive()