def test_Biharmonic(quad):
M = 128
SB = ShenBiharmonicBasis(M, quad=quad)
x = Symbol("x")
u = sin(6*pi*x)**2
a = 1.0
b = 1.0
f = -u.diff(x, 4) + a*u.diff(x, 2) + b*u
ul = lambdify(x, u, 'numpy')
fl = lambdify(x, f, 'numpy')
points, _ = SB.points_and_weights(M)
uj = ul(points)
fj = fl(points)
A = inner_product((SB, 0), (SB, 4))
B = inner_product((SB, 0), (SB, 0))
C = inner_product((SB, 0), (SB, 2))
AA = -A.diags() + C.diags() + B.diags()
f_hat = np.zeros(M)
f_hat = SB.scalar_product(fj, f_hat)
u_hat = np.zeros(M)
u_hat[:-4] = la.spsolve(AA, f_hat[:-4])
u1 = np.zeros(M)
u1 = SB.backward(u_hat, u1)
#from IPython import embed; embed()
assert np.allclose(u1, uj)
def assemble(self):
N = self.N
SB = ShenBiharmonicBasis(N, quad=self.quad)
SB.plan((N, N), 0, np.float, {})
x, w = self.x, self.w = SB.points_and_weights(N)
V = SB.vandermonde(x)
# Trial function
P4 = SB.get_vandermonde_basis(V)
# Second derivatives
T2x = SB.get_vandermonde_basis_derivative(V, 2)
# (u'', v)
K = np.zeros((N, N))
K[:-4, :-4] = inner_product((SB, 0), (SB, 2)).diags().toarray()
# ((1-x**2)u, v)
xx = np.broadcast_to((1 - x**2)[:, np.newaxis], (N, N))
#K1 = np.dot(w*P4.T, xx*P4) # Alternative: K1 = np.dot(w*P4.T, ((1-x**2)*P4.T).T)
K1 = np.zeros((N, N))
K1 = SB.scalar_product(xx * P4, K1)
K1 = extract_diagonal_matrix(
K1).diags().toarray() # For improved roundoff
# ((1-x**2)u'', v)
K2 = np.zeros((N, N))
K2 = SB.scalar_product(xx * T2x, K2)
K2 = extract_diagonal_matrix(
K2).diags().toarray() # For improved roundoff
# (u'''', v)
Q = np.zeros((self.N, self.N))
Q[:-4, :-4] = inner_product((SB, 0), (SB, 4)).diags().toarray()
# (u, v)
M = np.zeros((self.N, self.N))
M[:-4, :-4] = inner_product((SB, 0), (SB, 0)).diags().toarray()
Re = self.Re
a = self.alfa
B = -Re * a * 1j * (K - a**2 * M)
A = Q - 2 * a**2 * K + a**4 * M - 2 * a * Re * 1j * M - 1j * a * Re * (
K2 - a**2 * K1)
return A, B