  import numpy as np
import matplotlib.pyplot as plt
from fealpy import MeshFactory, FirstKindNedelecFiniteElementSpace2d, DirichletBC, CosSinData
from scipy.sparse import spsolve

# Define functions to recover the curl of a given solution and to calculate the least squares matrix for each node of a mesh
def curl_recovery(solution, mesh):
    # TODO: Implement curl recovery function
    pass

def least_squares_matrix(mesh):
    # TODO: Implement least squares matrix calculation function
    pass

# Parse command-line arguments to set the degree of the first kind Nedelec element, the initial mesh size, the maximum number of adaptive iterations, and the theta parameter for adaptive iteration
# TODO: Implement argument parsing

# Initialize the problem using the CosSinData function from the fealpy library
data = CosSinData(N=100, M=100)

# Create a 2D box mesh using the MeshFactory class from the fealpy library and remove the fourth quadrant of the mesh
mesh = MeshFactory.create_box_mesh(data.domain, 10, 10, 0, 0, 0, 0)
mesh.remove_cells(mesh.cells[mesh.cells[:, 0] < 0])

# Iterate over the maximum number of adaptive iterations
for i in range(max_iterations):
    # Define the function space using the FirstKindNedelecFiniteElementSpace2d class from the fealpy library
    V = FirstKindNedelecFiniteElementSpace2d(mesh, N)

    # Apply Dirichlet boundary conditions using the DirichletBC class from the fealpy library
    bc = DirichletBC(V, data.boundary_conditions, mesh.boundary_facets)

    # Solve the system of equations using the scipy library's spsolve function
    solution = spsolve(V.assemble_matrix(), V.assemble_rhs())

    # Calculate the L2 error between the solution and the exact solution, the curl of the solution and the exact curl, and the curl of the solution and the recovered curl
    error = np.linalg.norm(solution - data.exact_solution)
    curl_error = np.linalg.norm(curl_recovery(solution, mesh) - data.exact_curl)
    recovered_curl_error = np.linalg.norm(curl_recovery(solution, mesh) - curl_recovery(data.exact_solution, mesh))

    # If not the last iteration, mark the cells for refinement based on the recovery error and refine the mesh
    if i < max_iterations - 1:
        mesh.mark_cells(recovery_error > theta * error, 1)
        mesh.refine_cells()

# Plot the error rates using the showmultirate function from the fealpy library
plt.showmultirate(error, curl_error, recovered_curl_error)