Source code for Poisson_problem

"""
In this script, we implement the MSEM for the Poisson problem with a
manufactured solution in the crazy_mesh.

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:download:`[Poisson_problem.py]</contents/LIBRARY/ptc/mathischeap_ptc/Poisson_problem.py>`.
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"""

from numpy import pi, sin, cos
import numpy as np
from crazy_mesh import CrazyMesh, CrazyMeshGlobalNumbering, CrazyMeshGlobalBoundaryDOFs
from mimetic_basis_polynomials import MimeticBasisPolynomials
from incidence_matrices import E_div
from mass_matrices import MassMatrices
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(K, N, c, save=False): """ :param int K: We use a crazy mesh of :math:`K^3` elements. :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(K, N, c) # doctest: +ELLIPSIS MSEM L^2-error of u^h: 0.1535... """ K = int(K) N = int(N) c1000 = int(c*1000) # define the crazy mesh ... crazy_mesh = CrazyMesh(c, K) # generate the global numbering (gathering matrix) and find boundary dofs. GM_crazy_mesh = CrazyMeshGlobalNumbering(K, N) BD_crazy_mesh = CrazyMeshGlobalBoundaryDOFs(K, N) GM_FP = GM_crazy_mesh.FP GM_VP = GM_crazy_mesh.VP B_dofs_FP_dict = BD_crazy_mesh.FP B_dofs_FP = list() for bn in B_dofs_FP_dict: if bn != 'x_minus': B_dofs_FP.extend(B_dofs_FP_dict[bn]) # define the basis functions _bfN_ = 'Lobatto-' + str(N) mbf = MimeticBasisPolynomials(_bfN_, _bfN_, _bfN_) # generate incidence matrix and mass matrices E = E_div(N, N, N) MF = list() MV = 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) # 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])) u_exact_global = assemble(u_exact_local, GM_FP) f_L2 = np.sum(f_L2)**0.5 # generate local systems A_m x = b_m for all elements. A00, A01, A10 = list(), list(), list() # store blocks in list b0, b1 = list(), list() # store vectors in list for k in range(K): for j in range(K): for i in range(K): m = i + j * K + k * K**2 A00_m = MF[m] A10_m = MV[m] @ E A01_m = A10_m.T b0_m = spspa.csc_matrix((3*(N+1)*N**2, 1)) b1_m = - MV[m] @ f_exact_local[m] b1_m = spspa.csc_matrix(b1_m[:,np.newaxis]) A00.append(A00_m) A01.append(A01_m) A10.append(A10_m) b0.append(b0_m) b1.append(b1_m) # ( A00[m] A01[m] ) (b0[m]) # ( A01[m] ) (b1[m]) # # refers to the local system in element m del A00_m, A01_m, A10_m # assemble local systems into global system A00 = assemble(A00, GM_FP, GM_FP) A01 = assemble(A01, GM_FP, GM_VP) A10 = assemble(A10, GM_VP, GM_FP) A = spspa.bmat([(A00, A01 ), # left hand side matrix A (A10, None)], format='lil')# of global system Ax = b del A00, A01, A10 b0 = assemble(b0, GM_FP) b1 = assemble(b1, GM_VP) b = spspa.vstack((b0, b1), format='lil') # right hand side vector b # we apply the boundary condition. A[B_dofs_FP, :] = 0 A[B_dofs_FP, B_dofs_FP] = 1 b[B_dofs_FP] = u_exact_global[B_dofs_FP] A = A.tocsc() # scipy spsolve handles csc or csr matrix well shape_F = A.shape[0] # solve the global system using the direct solver provided by scipy x = spspalinalg.spsolve(A, b) # solve Ax=b, obtain x del A, b # post-process x into u, and phi, compute div u + f u_global = x[:int(np.max(GM_FP)+1)] phi_global = x[int(np.max(GM_FP)+1):] u_local = u_global[GM_FP] phi_local = phi_global[GM_VP] div_u_local = (- E @ u_local.T).T div_u_plus_f = (E @ u_local.T + np.array(f_exact_local).T).T # measure the L2-error of u phi, and div u + f 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('MSEM') 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) M = K**3 I = 3*K**2*(K-1) shape_F_1 = M*(3*N**2*(N+1)+N**3) - I*N**2 print(f'M={M}, I={I}, shape_F = {shape_F}, {shape_F_1}') if save: name_temp = f'results/MSEM_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) return u_L2, u_Hdiv, phi_L2, f_L2, div_L2, div_Linf
if __name__ == '__main__': import doctest doctest.testmod() K = 2 N = 3 c = 0. Poisson(K, N, c, save=False)