Update ft03

[OsbertWang]

I use FEniCS installed in WSL of Windows 10.
The version is 2019.01.
Then I update demo files in the tutorial.
I change the plot code because WSL cannot use GUI.

"""
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 dolfin import *
import numpy as np
import matplotlib.pyplot as plt

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)
 plt.savefig('u'+str(n)+'.png')
 plt.cla 

 # Compute error at vertices
 u_e = interpolate(u_D, V)
 error = np.abs(u_e.vector().get_local() - u.vector().get_local()).max()
 print('t = %.2f: error = %.3g' % (t, error))

 # Update previous solution
 u_n.assign(u)

# Hold plot
# interactive()