def energy_inner_pjkpmn(j, k, m, n, lam=np.array([1])):
'''
Calculates energy inner product between monomials P_{j,k} and P_{m, n}.
Uses the Gauss-Green formula so that
<P_{j,k}, P_{m,n}>_energy = <P_{j,k}, P_{m,n}>_L2 - lam*<P_{j-1,k}, P_{m,n}>_L2 + lam*sum_V0{P_{m,n}*d_n(P_{j,k})}
Args:
j, k, m, n: Represent monomial indices P_{j,k} and P_{m, n}
lam: represents weight given to the energy part of the inner product
Returns:
<P_{j,k}, P_{m,n}>_energy
'''
lam = lam[0]
res = Polynomial.basis_inner(j, k, m, n, lam=np.array([0]))
res -= lam * Polynomial.basis_inner(j - 1, k, m, n, lam=np.array([0]))
for i in range(2):
with HiddenPrints():
res += lam * norm_p_jk(str(i), j, k) * p_jk(str(i), m, n)
return res
def ints_vec(j):
'''
This function computes the integrals of {P_01, P_02, P_03, ..., P_j2, P_j3}.
Args:
j: maximum degree of polynomial
Returns:
np.array of integrals of {P_01, P_02, P_03, ..., P_j2, P_j3}
'''
res = np.zeros(3 * j + 3)
for ind in range(3 * j + 3):
jj = int(np.floor(ind / 3))
kk = int(ind % 3 + 1)
res[ind] = Polynomial.basis_inner(jj, kk, 0, 1)
return res
