```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 UniformTimeLine
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 the simplified friction problem
mesh = HalfEdgeMesh2d.from_polygons([[0, 0], [1, 0], [1, 1], [0, 1]])
space = LagrangeFiniteElementSpace(mesh, p=1)
dt = 0.01  # Time step
timeline = UniformTimeLine(0, 1, int(1/dt))
bc = DirichletBC(space, lambda x: np.zeros(x.shape[:-1]))

# Loop for the maximum number of iterations
for i in range(max_iterations):
    # Solve the problem
    uh = space.function()
    A = space.stiff_matrix(c=theta)
    F = space.source_vector(lambda x: np.ones(x.shape[:-1]))
    bc.apply(A, F, uh)
    
    # Calculate residuals and high order terms
    residual = np.linalg.norm(F - A@uh)
    high_order_term = k * np.max(np.abs(uh))
    
    # Save results and error data to a file
    with open(f"results_iteration_{i}.txt", "w") as file:
        file.write(f"Residual: {residual}\nHigh Order Term: {high_order_term}\n")
    
    # 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 not the last iteration, refine the mesh based on the residuals
    if i < max_iterations - 1:
        mesh.uniform_refine()
        space = LagrangeFiniteElementSpace(mesh, p=1)
        bc = DirichletBC(space, lambda x: np.zeros(x.shape[:-1]))

# Save the final error data to a file
with open("final_error_data.txt", "w") as file:
    file.write(f"Final Residual: {residual}\nFinal High Order Term: {high_order_term}\n")

# Display a multi-rate plot
showmultirate(plt, 0, timeline, [residual], ["Residual"], "Iteration", "Error")
```