def compute_gauss_jacobi_points(a, b, m):
"""Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method.
The initial guesses are the Chebyshev points. Algorithm
implemented in Python from the pseudocode given by Karniadakis and
Sherwin"""
x = []
eps = 1.e-8
max_iter = 100
for k in range(0, m):
r = -math.cos((2.0 * k + 1.0) * math.pi / (2.0 * m))
if k > 0:
r = 0.5 * (r + x[k - 1])
j = 0
delta = 2 * eps
while j < max_iter:
s = 0
for i in range(0, k):
s = s + 1.0 / (r - x[i])
f = jacobi.eval_jacobi(a, b, m, r)
fp = jacobi.eval_jacobi_deriv(a, b, m, r)
delta = f / (fp - f * s)
r = r - delta
if math.fabs(delta) < eps:
break
else:
j = j + 1
x.append(r)
return x
def compute_gauss_jacobi_points(a, b, m):
"""Computes the m roots of P_{m}^{a,b} on [-1,1] by Newton's method.
The initial guesses are the Chebyshev points. Algorithm
implemented in Python from the pseudocode given by Karniadakis and
Sherwin"""
x = []
eps = 1.e-8
max_iter = 100
for k in range(0, m):
r = -math.cos((2.0 * k + 1.0) * math.pi / (2.0 * m))
if k > 0:
r = 0.5 * (r + x[k - 1])
j = 0
delta = 2 * eps
while j < max_iter:
s = 0
for i in range(0, k):
s = s + 1.0 / (r - x[i])
f = jacobi.eval_jacobi(a, b, m, r)
fp = jacobi.eval_jacobi_deriv(a, b, m, r)
delta = f / (fp - f * s)
r = r - delta
if math.fabs(delta) < eps:
break
else:
j = j + 1
x.append(r)
return x
def __init__(self, ref_el):
entity_ids = {}
nodes = []
cur = 0
# make nodes by getting points
# need to do this dimension-by-dimension, facet-by-facet
top = ref_el.get_topology()
verts = ref_el.get_vertices()
sd = ref_el.get_spatial_dimension()
if ref_el.get_shape() != TRIANGLE:
raise ValueError("Bell only defined on triangles")
pd = functional.PointDerivative
# get jet at each vertex
entity_ids[0] = {}
for v in sorted(top[0]):
nodes.append(functional.PointEvaluation(ref_el, verts[v]))
# first derivatives
for i in range(sd):
alpha = [0] * sd
alpha[i] = 1
nodes.append(pd(ref_el, verts[v], alpha))
# second derivatives
alphas = [[2, 0], [1, 1], [0, 2]]
for alpha in alphas:
nodes.append(pd(ref_el, verts[v], alpha))
entity_ids[0][v] = list(range(cur, cur + 6))
cur += 6
# we need an edge quadrature rule for the moment
from FIAT.quadrature_schemes import create_quadrature
from FIAT.jacobi import eval_jacobi
rline = ufc_simplex(1)
q1d = create_quadrature(rline, 8)
q1dpts = q1d.get_points()
leg4_at_qpts = eval_jacobi(0, 0, 4, 2.0 * q1dpts - 1)
imond = functional.IntegralMomentOfNormalDerivative
entity_ids[1] = {}
for e in sorted(top[1]):
entity_ids[1][e] = [18 + e]
nodes.append(imond(ref_el, e, q1d, leg4_at_qpts))
entity_ids[2] = {0: []}
super(BellDualSet, self).__init__(nodes, ref_el, entity_ids)
def __init__(self, ref_el):
entity_ids = {}
nodes = []
cur = 0
# make nodes by getting points
# need to do this dimension-by-dimension, facet-by-facet
top = ref_el.get_topology()
verts = ref_el.get_vertices()
sd = ref_el.get_spatial_dimension()
if ref_el.get_shape() != TRIANGLE:
raise ValueError("Bell only defined on triangles")
pd = functional.PointDerivative
# get jet at each vertex
entity_ids[0] = {}
for v in sorted(top[0]):
nodes.append(functional.PointEvaluation(ref_el, verts[v]))
# first derivatives
for i in range(sd):
alpha = [0] * sd
alpha[i] = 1
nodes.append(pd(ref_el, verts[v], alpha))
# second derivatives
alphas = [[2, 0], [1, 1], [0, 2]]
for alpha in alphas:
nodes.append(pd(ref_el, verts[v], alpha))
entity_ids[0][v] = list(range(cur, cur + 6))
cur += 6
# we need an edge quadrature rule for the moment
from FIAT.quadrature_schemes import create_quadrature
from FIAT.jacobi import eval_jacobi
rline = ufc_simplex(1)
q1d = create_quadrature(rline, 8)
q1dpts = q1d.get_points()
leg4_at_qpts = eval_jacobi(0, 0, 4, 2.0*q1dpts - 1)
imond = functional.IntegralMomentOfNormalDerivative
entity_ids[1] = {}
for e in sorted(top[1]):
entity_ids[1][e] = [18+e]
nodes.append(imond(ref_el, e, q1d, leg4_at_qpts))
entity_ids[2] = {0: []}
super(BellDualSet, self).__init__(nodes, ref_el, entity_ids)