def P1_solution():
plt.figure()
from fe1D import mesh_uniform, u_glob
N_e_values = [2, 4, 8]
d = 1
legends = []
for N_e in N_e_values:
vertices, cells, dof_map = mesh_uniform(N_e=N_e,
d=d,
Omega=[0, 2],
symbolic=False)
h = vertices[1] - vertices[0]
Ae = 1. / h * np.array([[1, -1], [-1, 1]])
N = N_e + 1
A = np.zeros((N, N))
b = np.zeros(N)
for e in range(N_e):
if vertices[e] >= 1:
be = -h / 2. * np.array([1, 1])
else:
be = h / 2. * np.array([0, 0])
for r in range(d + 1):
for s in range(d + 1):
A[dof_map[e][r], dof_map[e][s]] += Ae[r, s]
b[dof_map[e][r]] += be[r]
# Enforce boundary conditions
A[0, :] = 0
A[0, 0] = 1
b[0] = 0
A[-1, :] = 0
A[-1, -1] = 1
b[-1] = 0
c = np.linalg.solve(A, b)
# Plot solution
print(('c:', c))
print(('vertices:', vertices))
print(('cells:', cells))
print(('len(cells):', len(cells)))
print(('dof_map:', dof_map))
xc, u, nodes = u_glob(c, vertices, cells, dof_map)
plt.plot(xc, u)
legends.append('$N_e=%d$' % N_e)
plt.plot(xc, exact_solution(xc), '--')
legends.append('exact')
plt.legend(legends, loc='lower left')
plt.savefig('tmp3.png')
plt.savefig('tmp3.pdf')
def P1_solution():
plt.figure()
from fe1D import mesh_uniform, u_glob
N_e_values = [2, 4, 8]
d = 1
legends = []
for N_e in N_e_values:
vertices, cells, dof_map = mesh_uniform(
N_e=N_e, d=d, Omega=[0,2], symbolic=False)
h = vertices[1] - vertices[0]
Ae = 1./h*np.array(
[[1, -1],
[-1, 1]])
N = N_e + 1
A = np.zeros((N, N))
b = np.zeros(N)
for e in range(N_e):
if vertices[e] >= 1:
be = -h/2.*np.array(
[1, 1])
else:
be = h/2.*np.array(
[0, 0])
for r in range(d+1):
for s in range(d+1):
A[dof_map[e][r], dof_map[e][s]] += Ae[r,s]
b[dof_map[e][r]] += be[r]
# Enforce boundary conditions
A[0,:] = 0; A[0,0] = 1; b[0] = 0
A[-1,:] = 0; A[-1,-1] = 1; b[-1] = 0
c = np.linalg.solve(A, b)
# Plot solution
xc, u, nodes = u_glob(c, vertices, cells, dof_map)
plt.plot(xc, u)
legends.append('$N_e=%d$' % N_e)
plt.plot(xc, exact_solution(xc), '--')
legends.append('exact')
plt.legend(legends, loc='lower left')
plt.savefig('tmp3.png'); plt.savefig('tmp3.pdf')
c, A, b, timing = finite_element1D(
vertices, cells, dof_map,
essbc,
ilhs=lambda e, phi, r, s, X, x, h:
phi[1][r](X, h)*phi[1][s](X, h),
irhs=lambda e, phi, r, X, x, h:
x**m*phi[0][r](X),
blhs=lambda e, phi, r, s, X, x, h: 0,
brhs=lambda e, phi, r, X, x, h:
-C*phi[0][r](-1) if e == 0 else 0,
intrule='GaussLegendre')
# Visualize
# (Recall that x is a symbol, use xc for coordinates)
xc, u, nodes = u_glob(c, vertices, cells, dof_map)
u_e = u_exact(xc)
print('Max diff at nodes, d=%d:' % d, \
np.abs(u_exact(nodes) - c).max())
plt.figure()
plt.plot(xc, u, 'b-', xc, u_e, 'r--')
plt.legend(['finite elements, d=%d' %d, 'exact'],
loc='lower left')
figname = 'tmp_%d_%d' % (m, d)
plt.savefig(figname + '.png'); plt.savefig(figname + '.pdf')
for ext in 'pdf', 'png':
cmd = 'doconce combine_images -2 '
cmd += ' '.join(['tmp_%d_%d.' % (m, d) + ext
for d in d_values])
cmd += ' u_xx_xm%d_P1to4.' % m + ext
print(cmd)
vertices[1] = 3
essbc = {}
essbc[dof_map[-1][-1]] = D
c, A, b, timing = finite_element1D_naive(
vertices, cells, dof_map,
essbc,
ilhs=lambda e, phi, r, s, X, x, h:
phi[1][r](X, h)*phi[1][s](X, h),
irhs=lambda e, phi, r, X, x, h:
x**m*phi[0][r](X),
blhs=lambda e, phi, r, s, X, x, h: 0,
brhs=lambda e, phi, r, X, x, h:
-C*phi[0][r](-1) if e == 0 else 0,
intrule='GaussLegendre',
verbose=False,
)
# Visualize
from fe1D import u_glob
x, u, nodes = u_glob(c, cells, vertices, dof_map)
u_e = u_exact(x, C, D, L)
print u_exact(nodes, C, D, L) - c # difference at the nodes
import matplotlib.pyplot as plt
plt.plot(x, u, 'b-', x, u_e, 'r--')
plt.legend(['finite elements, d=%d' %d, 'exact'], loc='upper left')
plt.savefig('tmp.png'); plt.savefig('tmp.pdf')
plt.show()
