```python
import sys
import numpy as np
import matplotlib.pyplot as plt
from fealpy.mesh import HalfEdgeMesh2d
from fealpy.functionspace import LagrangeFiniteElementSpace
from fealpy.boundarycondition import DirichletBC
from fealpy.timeintegratoralg import solve_linear_system
from fealpy.tools.show import showmultirate

# Command line arguments for maximum iterations, theta, and k
max_iterations = int(sys.argv[1])
theta = float(sys.argv[2])
k = float(sys.argv[3])

# Initialize variables for storing results
errors = []
residuals = []

# Create a simple halfedge polygon mesh
mesh = HalfEdgeMesh2d.from_polygon(polygon=np.array([[0, 0], [1, 0], [1, 1], [0, 1]]))

for i in range(max_iterations):
    # Create finite element space and define problem
    space = LagrangeFiniteElementSpace(mesh, p=1)
    A = space.stiff_matrix()
    b = np.zeros(space.number_of_global_dofs())
    bc = DirichletBC(space, lambda x: np.zeros_like(x))
    A, b = bc.apply(A, b)
    
    # Solve the problem
    x = solve_linear_system(A, b)
    
    # Calculate residuals and high order terms
    residual = np.max(np.abs(A.dot(x) - b))
    residuals.append(residual)
    
    # Save results and error data to a file
    np.savetxt(f'results_iteration_{i}.txt', x)
    np.savetxt(f'error_data_iteration_{i}.txt', np.array([residual]))
    
    # Plot the mesh and save as an image file
    fig, ax = plt.subplots()
    mesh.add_plot(ax)
    plt.savefig(f'mesh_iteration_{i}.png')
    plt.close(fig)
    
    if i < max_iterations - 1:
        # Refine the mesh based on residuals
        # This is a placeholder for mesh refinement based on residuals
        # Actual implementation depends on the problem specifics and fealpy capabilities
        mesh.uniform_refine()
    
# Save the final error data to a file
np.savetxt('final_error_data.txt', np.array(errors))

# Display a multi-rate plot
showmultirate(plt, 0, residuals, residuals)
plt.show()
```