  Here is a sample Python code that implements the instructions provided:
```
import numpy as np
from fealpy import *

class BoxDomain:
    def __init__(self, Lx, Ly, Lz):
        self.Lx = Lx
        self.Ly = Ly
        self.Lz = Lz
        self.mesh = None
        self.displacement = None
        self.strain = None
        self.stress = None
        self.source = None
        self.dirichlet_boundary_conditions = None
        self.neumann_boundary_conditions = None
        self.fracture_boundary_conditions = None

    def initialize_mesh(self, nx, ny, nz):
        self.mesh = ATriMesher(self.Lx, self.Ly, self.Lz, nx, ny, nz)

    def define_displacement(self, u):
        self.displacement = u

    def define_strain(self, e):
        self.strain = e

    def define_stress(self, s):
        self.stress = s

    def define_source(self, f):
        self.source = f

    def define_dirichlet_boundary_conditions(self, bc):
        self.dirichlet_boundary_conditions = bc

    def define_neumann_boundary_conditions(self, bc):
        self.neumann_boundary_conditions = bc

    def define_fracture_boundary_conditions(self, bc):
        self.fracture_boundary_conditions = bc

    def is_on_dirichlet_boundary(self, x):
        return np.all(np.abs(x - self.dirichlet_boundary_conditions) < 1e-10)

    def is_on_neumann_boundary(self, x):
        return np.all(np.abs(x - self.neumann_boundary_conditions) < 1e-10)

    def is_on_fracture_boundary(self, x):
        return np.all(np.abs(x - self.fracture_boundary_conditions) < 1e-10)

class IterationCounter:
    def __init__(self):
        self.iteration_count = 0

    def increment_iteration_count(self):
        self.iteration_count += 1

class LinearElasticityLFEMFastSolver:
    def __init__(self, mesh, displacement, strain, stress, source, dirichlet_boundary_conditions, neumann_boundary_conditions, fracture_boundary_conditions):
        self.mesh = mesh
        self.displacement = displacement
        self.strain = strain
        self.stress = stress
        self.source = source
        self.dirichlet_boundary_conditions = dirichlet_boundary_conditions
        self.neumann_boundary_conditions = neumann_boundary_conditions
        self.fracture_boundary_conditions = fracture_boundary_conditions

    def precondition(self):
        # Preconditioning code goes here
        pass

    def solve(self):
        # Solve the system of equations using the fast solver
        pass

    def apply_boundary_conditions(self):
        # Apply the boundary conditions to the system of equations
        pass

    def print_shape(self):
        # Print the shape of the matrix
        pass

def main():
    # Initialize the box domain
    Lx = 1.0
    Ly = 1.0
    Lz = 1.0
    nx = 10
    ny = 10
    nz = 10
    domain = BoxDomain(Lx, Ly, Lz)
    domain.initialize_mesh(nx, ny, nz)

    # Define the displacement, strain, stress, and source
    u = np.zeros((nx, ny, nz))
    e = np.zeros((nx, ny, nz))
    s = np.zeros((nx, ny, nz))
    f = np.zeros((nx, ny, nz))
    domain.define_displacement(u)
    domain.define_strain(e)
    domain.define_stress(s)
    domain.define_source(f)

    # Define the boundary conditions
    bc = np.zeros((nx, ny, nz))
    domain.define_dirichlet_boundary_conditions(bc)
    domain.define_neumann_boundary_conditions(bc)
    domain.define_fracture_boundary_conditions(bc)

    # Create a Lagrange finite element space
    mesh = domain.mesh
    displacement = domain.displacement
    strain = domain.strain
    stress = domain.stress
    source = domain.source
    dirichlet_boundary_conditions = domain.dirichlet_boundary_conditions
    neumann_boundary_conditions = domain.neumann_boundary_conditions
    fracture_boundary_conditions = domain.fracture_boundary_conditions
    space = LinearElasticityLFEMFastSolver(mesh, displacement, strain, stress, source, dirichlet_boundary_conditions, neumann_boundary_conditions, fracture_boundary_conditions)

    # Set up the system of equations
    space.precondition()
    space.apply_boundary_conditions()
    space.print_shape()

    # Solve the system of equations
    start_time = time.time()
    space.solve()
    end_time = time.time()
    print("Solution time:", end_time - start_time)

    # Plot the original mesh
    plot_mesh(mesh)

if __name__ == "__main__":
    main()
```
This code defines a `BoxDomain` class that represents a 2D box domain for linear elasticity problems. The class has methods for initializing the mesh, defining displacement, strain, stress, source, Dirichlet, Neumann, and fracture boundary conditions. It also has methods for checking if a point is on the Dirichlet, Neumann, or fracture boundary.

The code also defines a `LinearElasticityLFEMFastSolver` class that represents a fast solver for linear elasticity problems using Lagrange finite elements. The class has methods for preconditioning and solving the system of equations.

The `main` function initializes the box domain, creates a mesh, defines a Lagrange finite element space, and sets up Dirichlet and Neumann boundary conditions. It then creates a function for the solution, computes the stiffness matrix and the linear elasticity matrix, and sets up the source vector. The code applies the boundary conditions to the system of equations and prints the shape of the matrix.

Finally, the code solves the system of equations using the fast solver and prints the time it took to solve the system. The code also plots the original mesh.