def inner0_j3k3(j, k):
'''
Calculates the L2 inner product <P_j3, P_k3>
Args:
j, k: indices for the monomials P_j3, P_k3
Returns:
L2 inner product <P_j3, P_k3>
'''
ms = min(j, k)
s1 = 0
for l in range(j - ms, j + 1):
s1 += alpha(j - l + 1) * eta(k + l + 2) - alpha(k + l + 2) * eta(j -
l + 1)
return 18 * s1
def inner0_j1k1(j, k):
'''
Calculates the L2 inner product <P_j1, P_k1>
Args:
j, k: indices for the monomials P_j1, P_k1
Returns:
L2 inner product <P_j1, P_k1>
'''
ms = min(j, k)
s1 = 0
for l in range(j - ms, j + 1):
s1 += alpha(j - l) * eta(k + l + 1) - alpha(k + l + 1) * eta(j - l)
return 2 * s1
def d_pnj1(nn, j, i):
'''
This function calculates the value of the normal derivative of
P_{j,1}^(n) at q_i.
Args:
nn: corresponds to the value of (n) in P_{j,1}^(n)
j: corresponds to the value of j in P_{j,1}^(n)
i: corresponds to the boundary index q_i
Returns:
Value of the normal derivative of P_{j,1}^(n) at q_i
'''
return 0 if i == nn else eta(j)
def inner0_j1k2(j, k):
'''
Calculates the L2 inner product <P_j1, P_k2>
Args:
j, k: indices for the monomials P_j1, P_k2
Returns:
L2 inner product <P_j1, P_k2>
'''
s1 = 0
for l in range(j + 1):
s1 += alpha(j - l) * alpha(k + l + 1) + beta(k + l + 1) * eta(j - l)
return -2 * s1
def dnpj3(j):
return 3 * eta(j + 1)