"""
In this script, we implement the hMSEM for the Poisson problem with a
manufactured solution in the crazy_mesh.
⭕ To access the source code, click on the [source] button at the right
side or click on
:download:`[Poisson_problem_h.py]</contents/LIBRARY/ptc/mathischeap_ptc/Poisson_problem_h.py>`.
Dependence may exist. In case of error, check import and install required
packages or download required scripts. © mathischeap.com
"""
from numpy import pi, sin, cos
import numpy as np
from crazy_mesh_hybrid import CrazyMeshHybrid, CrazyMeshHybridGlobalNumbering, CrazyMeshHybridLocalBoundaryDOFs
from mimetic_basis_polynomials import MimeticBasisPolynomials
from incidence_matrices import E_div
from trace_matrices import TF
from mass_matrices import MassMatrices
from mass_matrices_trace import MassMatricesTrace
from projection import Reduction
from scipy import sparse as spspa
from scipy.sparse import linalg as spspalinalg
from assembly import assemble
from L2_error import L2Error
# the manufactured solutions
def phi_exact(x, y, z):
return sin(2 * pi * x) * sin(2 * pi * y) * sin(2 * pi * z)
def u_exact(x, y, z):
return 2 * pi * cos(2 * pi * x) * sin(2 * pi * y) * sin(2 * pi * z)
def v_exact(x, y, z):
return 2 * pi * sin(2 * pi * x) * cos(2 * pi * y) * sin(2 * pi * z)
def w_exact(x, y, z):
return 2 * pi * sin(2 * pi * x) * sin(2 * pi * y) * cos(2 * pi * z)
def f_exact(x,y,z):
return 12 * pi**2 * sin(2 * pi * x) * sin(2 * pi * y) * sin(2 * pi * z)
def zero(x, y, z): # div u + f = 0
return 0 * x * y * z
[docs]
def Poisson_h(K, N, c, save=False):
"""
:param int K: We use a crazy mesh of :math:`K^3` elements. The
domain decomposition is based on this crazy mesh.
:param int N: We use mimetic polynomials of degree :math:`N`.
:param float c: The deformation factor of the crazy mesh is
:math:`c,\ 0\\leq c\\leq 0.25`.
:param save: Bool. If we save the coefficients of the variables.
:return: A tuple of several outputs:
- The :math:`L^2\\text{-error}` of solution :math:`\\boldsymbol{u}^h`.
- The :math:`H(\\mathrm{div})\\text{-error}` of solution :math:`\\boldsymbol{u}^h`.
- The :math:`L^2\\text{-error}` of solution :math:`\\varphi^h`.
- The :math:`L^2\\text{-error}` of the projection, :math:`f^h`.
- The :math:`L^2\\text{-error}` of :math:`\\nabla\\cdot\\boldsymbol{u}^h+f^h`.
- The :math:`L^\\infty\\text{-error}` of :math:`\\nabla\\cdot\\boldsymbol{u}^h+f^h`.
:example:
>>> K = 2
>>> N = 3
>>> c = 0
>>> Poisson_h(K, N, c) # doctest: +ELLIPSIS
hMSEM
L^2-error of u^h: 0.153538250...
"""
K = int(K)
N = int(N)
c1000 = int(c*1000)
# define the crazy mesh ...
crazy_mesh = CrazyMeshHybrid(c, K)
# generate the global numbering (gathering matrix) and find boundary dofs.
GM_crazy_mesh = CrazyMeshHybridGlobalNumbering(K, N)
GM_TF = GM_crazy_mesh.TF
BD_crazy_mesh = CrazyMeshHybridLocalBoundaryDOFs(K, N)
B_dofs_FP_dict = BD_crazy_mesh.FP
B_dofs_FP = dict()
for bn in B_dofs_FP_dict:
if bn != 'x_minus':
dofs_on_side = B_dofs_FP_dict[bn]
for m in dofs_on_side:
if m not in B_dofs_FP:
B_dofs_FP[m] = list()
B_dofs_FP[m].extend(dofs_on_side[m])
B_dofs_TF_dict = BD_crazy_mesh.TF
B_dofs_TF_EN = dict()
B_dofs_TF_NA = dict()
for bn in B_dofs_TF_dict:
if bn == 'x_minus':
dofs_on_side = B_dofs_TF_dict[bn]
for m in dofs_on_side:
if m not in B_dofs_TF_EN:
B_dofs_TF_EN[m] = list()
B_dofs_TF_EN[m].extend(dofs_on_side[m])
else:
dofs_on_side = B_dofs_TF_dict[bn]
for m in dofs_on_side:
if m not in B_dofs_TF_NA:
B_dofs_TF_NA[m] = list()
B_dofs_TF_NA[m].extend(dofs_on_side[m])
# define the basis functions
_bfN_ = 'Lobatto-' + str(N)
mbf = MimeticBasisPolynomials(_bfN_, _bfN_, _bfN_)
# generate incidence matrix, trace matrices and mass matrices
E = E_div(N, N, N)
T = TF(N, N, N)
MF = list()
MV = list()
MTF = list()
for k in range(K):
for j in range(K):
for i in range(K):
ct = crazy_mesh.CT_of_element_index(i, j, k)
MM = MassMatrices(mbf, ct)
MF.append(MM.FP)
MV.append(MM.VP)
tMM = MassMatricesTrace(mbf, ct)
MTF.append(tMM.TF)
# reduction of source term f_exact, and u_exact
f_exact_local = list()
u_exact_local = list()
f_L2 = list()
for k in range(K):
for j in range(K):
for i in range(K):
ct = crazy_mesh.CT_of_element_index(i, j, k)
RD = Reduction(mbf, ct)
f_dofs_local = RD.VP(f_exact)
f_exact_local.append(f_dofs_local)
L2e = L2Error(mbf, ct)
f_L2_local = L2e.VP(f_dofs_local, f_exact)
f_L2.append(f_L2_local**2)
_temp_ = RD.FP((u_exact, v_exact, w_exact))
u_exact_local.append(spspa.csc_matrix(
_temp_[:,np.newaxis]))
f_L2 = np.sum(f_L2)**0.5
# generate local systems
invA_List = list()
g_List = list()
B_List = list()
RSA = list() # reduced systems Ax = b
RSb = list() # reduced systems Ax = b
num_basis_FP = 3*(N+1)*N**2
num_basis_VP = N**3
num_basis_TF = 6 * N**2
for k in range(K):
for j in range(K):
for i in range(K):
m = i + j * K + k * K**2
A00 = MF[m]
A10 = MV[m] @ E
A01 = A10.T
A02 = - T.T @ MTF[m]
A11 = spspa.csc_matrix((num_basis_VP, num_basis_VP))
A12 = spspa.csc_matrix((num_basis_VP, num_basis_TF))
A20 = A02.T.tolil()
A21 = spspa.csc_matrix((num_basis_TF, num_basis_VP))
A22 = spspa.lil_matrix((num_basis_TF, num_basis_TF))
b0 = spspa.csc_matrix((num_basis_FP, 1))
b1 = spspa.csc_matrix(- MV[m] @ f_exact_local[m][:,np.newaxis])
b2 = spspa.lil_matrix((num_basis_TF, 1))
# now apply local boundary condition
if m in B_dofs_TF_EN:
EN_dofs = B_dofs_TF_EN[m]
A20[EN_dofs, :] = 0
A22[EN_dofs, EN_dofs] = 1
if m in B_dofs_TF_NA:
NA_dofs = B_dofs_TF_NA[m]
A20[NA_dofs, :] = 0
A20[NA_dofs, B_dofs_FP[m]] = 1
b2[NA_dofs] = u_exact_local[m][B_dofs_FP[m]]
A = spspa.bmat(([A00, A01],
[A10, A11]), format='csc')
invA = spspalinalg.inv(A)
invA_List.append(invA)
B = spspa.bmat(([A02],[A12]), format='csc')
B_List.append(B)
C = spspa.bmat(([A20, A21],), format='csc')
D = A22.tocsc()
g = spspa.bmat(([b0],[b1]), format='csc')
g_List.append(g)
h = b2.tocsc()
__ = C @ invA
rsA = D - __ @ B
rsb = h - __ @ g
RSA.append(rsA)
RSb.append(rsb)
# assemble the reduce systems to a global system
A = assemble(RSA, GM_TF, GM_TF)
b = assemble(RSb, GM_TF)
shape_S = A.shape[0]
# solve the global system using the direct solver provided by scipy
lamb = spspalinalg.spsolve(A, b) # solve Ax=b, obtain x
del A, b
lambda_local = lamb[GM_TF]
u_local = list()
phi_local = list()
for k in range(K):
for j in range(K):
for i in range(K):
m = i + j * K + k * K**2
u_phi_local_m = invA_List[m] @ (g_List[m].toarray().ravel('F') - B_List[m] @ lambda_local[m])
u_local_m = u_phi_local_m[:num_basis_FP]
phi_local_m = u_phi_local_m[num_basis_FP:]
u_local.append(u_local_m)
phi_local.append(phi_local_m)
u_local = np.array(u_local)
phi_local = np.array(phi_local)
div_u_local = (- E @ u_local.T).T
div_u_plus_f = (E @ u_local.T + np.array(f_exact_local).T).T
u_L2 = list()
div_u_L2 = list()
phi_L2 = list()
div_L2 = list()
div_Linf = list()
for k in range(K):
for j in range(K):
for i in range(K):
m = i + j * K + k * K**2
ct = crazy_mesh.CT_of_element_index(i, j, k)
L2E = L2Error(mbf, ct)
u_l2_m = L2E.FP(u_local[m], (u_exact, v_exact, w_exact))
div_u_l2_m = L2E.VP(div_u_local[m], f_exact)
phi_l2_m = L2E.VP(phi_local[m], phi_exact)
div_L2_m = L2E.VP(div_u_plus_f[m], zero)
div_Linf_m = L2E.VP(div_u_plus_f[m], zero, n='infty')
u_L2.append(u_l2_m**2)
div_u_L2.append(div_u_l2_m**2)
phi_L2.append(phi_l2_m**2)
div_L2.append(div_L2_m**2)
div_Linf.append(div_Linf_m)
u_L2 = np.sum(u_L2)**0.5
div_u_L2 = np.sum(div_u_L2)**0.5
phi_L2 = np.sum(phi_L2)**0.5
div_L2 = np.sum(div_L2)**0.5
div_Linf = np.max(div_Linf)
u_Hdiv = np.sqrt(u_L2**2 + div_u_L2**2)
# print info and return
print('hMSEM')
print("L^2-error of u^h: ", u_L2)
print("H(div)-error of u^h: ", u_Hdiv)
print("L^2-error of phi^h: ", phi_L2)
print("L^2-error of projection f^h-f: ", f_L2)
print("L^2-error of div(u^h)+f^h: ", div_L2)
print("L^inf-error of div(u^h)+f^h: ", div_Linf)
I = 3*K**2*(K-1)
b = 6 * K**2
M = K**3
shape_S_1 = (I + b) * N**2
print(f'I={I}, b={b}, 6M-2I-b={6*M-2*I-b}, shape_S = {shape_S}, {shape_S_1}')
X = I/b
ratio = (6*X + 6) / (8*N*X + 4*N + 3)
print(f'system size ratio = {ratio, (3*K+3)/(4*N*K+3)}')
if save:
name_temp = f'results/hMSEM_K{K}_N{N}_c{c1000}_'
# noinspection PyTypeChecker
np.savetxt(name_temp + 'u.txt', u_local)
# noinspection PyTypeChecker
np.savetxt(name_temp + 'phi.txt', phi_local)
# noinspection PyTypeChecker
np.savetxt(name_temp + 'du_plus_f.txt', div_u_plus_f)
# noinspection PyTypeChecker
np.savetxt(name_temp + 'lambda.txt', lambda_local)
return u_L2, u_Hdiv, phi_L2, f_L2, div_L2, div_Linf
if __name__ == '__main__':
import doctest
doctest.testmod()
K = 4
N = 4
c = 0.25
Poisson_h(K, N, c, save=False)
