  import numpy as np
import matplotlib.pyplot as plt
from fealpy import Lagrange, TriangleMesh

# Define the problem parameters
N = 10  # Number of spatial divisions
T = 10  # Number of temporal divisions
dt = 0.01  # Time step size
t_end = 10  # End time

# Define the mesh parameters
mesh = TriangleMesh(N, N)
mesh.set_boundary_conditions(Dirichlet=True)

# Define the finite element space
V = Lagrange(mesh, 2)

# Define the adaptive refinement parameters
refine_threshold = 0.01
refine_factor = 2
coarsen_factor = 0.5

# Define the time stepping parameters
t = np.linspace(0, t_end, T)

# Define the initial conditions
u0 = np.zeros((N, N))
u0[0, :] = 1
u0[-1, :] = 0
u0[:, 0] = 0
u0[:, -1] = 0

# Define the boundary conditions
def boundary_conditions(x, y):
    if x[0] == 0 or x[0] == 1:
        return 0
    elif x[1] == 0 or x[1] == 1:
        return 0
    else:
        return 0

# Define the source term
def source_term(x, y):
    return 0

# Define the time stepping function
def time_stepping(u, t):
    # Compute the time derivative
    du = np.zeros((N, N))
    for i in range(1, N-1):
        for j in range(1, N-1):
            du[i, j] = (u[i+1, j] - 2*u[i, j] + u[i-1, j])/dt
    return du

# Define the error function
def error(u, t):
    return np.max(np.abs(u - u0))

# Initialize the mesh and the solution
mesh.set_mesh(u0)
u = u0

# Iteratively refine the mesh and solve the problem
for i in range(T):
    # Refine the mesh if necessary
    if error(u, t[i]) > refine_threshold:
        mesh.refine(refine_factor)
        u = mesh.solve(V, boundary_conditions, source_term, time_stepping, t[i])
    # Coarsen the mesh if necessary
    if error(u, t[i]) < refine_threshold:
        mesh.coarsen(coarsen_factor)
        u = mesh.solve(V, boundary_conditions, source_term, time_stepping, t[i])
    # Save a plot of the mesh and the solution
    mesh.plot(u, t[i])
    plt.savefig('plot_%d.png' % i)
    plt.close()
    # Print the error at each time step
    print('Error at time step %d: %f' % (i, error(u, t[i])))

# Plot the final solution
mesh.plot(u, t_end)
plt.savefig('plot_final.png')
plt.close()