def test_kmv_quad_tri_schemes(p_d):
fct = np.math.factorial
p, d = p_d
q = create_quadrature(T, p, "KMV")
for i in range(d + 1):
for j in range(d + 1 - i):
trueval = fct(i) * fct(j) / fct(i + j + 2)
assert (np.abs(trueval - q.integrate(lambda x: x[0]**i * x[1]**j))
< 1.0e-10)
def test_Kronecker_property_tets(element_degree):
"""
Evaluating the nodal basis at the special quadrature points should
have a Kronecker property. Also checks that the basis functions
and quadrature points are given the same ordering.
"""
element, degree = element_degree
qr = create_quadrature(Te, degree, scheme="KMV")
(basis, ) = element.tabulate(0, qr.get_points()).values()
assert np.allclose(basis, np.eye(*basis.shape))
def test_kmv_quad_tet_schemes(p_d): # noqa: W503
fct = np.math.factorial
p, d = p_d
q = create_quadrature(Te, p, "KMV")
for i in range(d + 1):
for j in range(d + 1 - i):
for k in range(d + 1 - i - j):
trueval = fct(i) * fct(j) * fct(k) / fct(i + j + k + 3)
assert (
np.abs(trueval -
q.integrate(lambda x: x[0]**i * x[1]**j * x[2]**k))
< 1.0e-10)
def test_interpolate_monomials_tris(element_degree):
element, degree = element_degree
# ordered the same way as KMV nodes
pts = create_quadrature(T, degree, "KMV").pts
Q = make_quadrature(T, 2 * degree)
phis = element.tabulate(0, Q.pts)[0, 0]
print("deg", degree)
for i in range(degree + 1):
for j in range(degree + 1 - i):
m = lambda x: x[0]**i * x[1]**j
dofs = np.array([m(pt) for pt in pts])
interp = phis.T @ dofs
matqp = np.array([m(pt) for pt in Q.pts])
err = 0.0
for k in range(phis.shape[1]):
err += Q.wts[k] * (interp[k] - matqp[k])**2
assert np.sqrt(err) <= 1.0e-12