v = TestFunction(SD)
# Get f on quad points
fj = Array(SD, buffer=fe)
# Compute right hand side of Poisson equation
f_hat = Function(SD)
f_hat = inner(v, -fj, output_array=f_hat)
# Get left hand side of Poisson equation
A = inner(grad(v), grad(u))
f_hat = A / f_hat
uj = f_hat.backward()
uh = uj.forward()
# Compare with analytical solution
ua = Array(SD, buffer=ue)
print("Error=%2.16e" % (np.linalg.norm(uj - ua)))
assert np.allclose(uj, ua, atol=1e-5)
point = np.array([0.1, 0.2])
p = SD.eval(point, f_hat)
assert np.allclose(p, lambdify(x, ue)(point), atol=1e-5)
if 'pytest' not in os.environ:
import matplotlib.pyplot as plt
xx = np.linspace(-8, 8, 100)
plt.plot(xx, lambdify(x, ue)(xx), 'r', xx, uh.eval(xx), 'bo', markersize=2)
plt.show()
# Compute right hand side
f_hat = Function(ST)
f_hat = inner(v, fj, output_array=f_hat)
# Solve Poisson equation
A = inner(grad(v), grad(u))
u_hat = Function(ST)
u_hat = A.solve(-f_hat, u_hat)
uq = ST.backward(u_hat)
u_hat = ST.forward(uq, u_hat, fast_transform=False)
uq = ST.backward(u_hat, uq, fast_transform=False)
assert np.allclose(uj, uq)
point = np.array([0.1, 0.2])
p = ST.eval(point, u_hat)
assert np.allclose(p, lambdify(x, ue)(point))
if 'pytest' not in os.environ:
import matplotlib.pyplot as plt
plt.figure()
X = ST.mesh()
plt.plot(X, uj.real)
plt.title("U")
plt.figure()
plt.plot(X, (uq - uj).real)
plt.title("Error")
plt.show()
