f_hat = inner(v, fj, output_array=f_hat)
if family == 'legendre':
f_hat *= -1.
# Get left hand side of Poisson equation
if family == 'chebyshev':
A = inner(v, div(grad(u))) + alpha*inner(v, grad(u))
else:
A = inner(grad(v), grad(u)) - alpha*inner(v, grad(u))
if family == 'chebyshev':
f_hat[0] -= 0.5*np.pi*alpha*a
f_hat[0] += 0.5*np.pi*alpha*b
elif family == 'legendre':
f_hat[0] += alpha*a
f_hat[0] -= alpha*b
f_hat[:-2] = A.solve(f_hat[:-2])
f_hat[-2] = a
f_hat[-1] = b
uj = SD.backward(f_hat)
# Compare with analytical solution
ua = ul(X)
print("Error=%2.16e" %(np.linalg.norm(uj-ua)))
assert np.allclose(uj, ua)
point = np.array([0.1, 0.2])
p = SD.eval(point, f_hat)
assert np.allclose(p, ul(point))
uj = Array(ST, buffer=ue)
# 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()
plt.plot(X, uj.real)
plt.title("U")
plt.figure()
plt.plot(X, (uq - uj).real)
plt.title("Error")
plt.show()
