 Here is a simplified version of the code based on your instructions. Please note that this is a basic example and might need adjustments to fit your specific needs. Also, fealpy library might require installation.

```python
from fealpy.meshes import ATriMesher, IntervalMesh, TetrahedronMesh
from fealpy.finiteelements import Lagrange
from fealpy.linear_elasticity import LinearElasticityLFEMFastSolver
from fealpy.materials import LinearElasticityMaterial
from fealpy.numerical_integration import GaussQuadrature
from fealpy.boundary_conditions import DirichletBC, NeumannBC
from fealpy.plots import plot_mesh
from fealpy.solvers import CGSolver
from fealpy.expressions import Expression
import numpy as np
import time

class BoxDomain:
    def __init__(self, Lx, Ly, Lz):
        self.Lx = Lx
        self.Ly = Ly
        self.Lz = Lz

    def create_mesh(self):
        return TetrahedronMesh([(0, 0, 0), (self.Lx, 0, 0), (0, self.Ly, 0), (self.Lx, self.Ly, 0),
                                (0, 0, self.Lz), (self.Lx, 0, self.Lz), (0, self.Ly, self.Lz),
                                (self.Lx, self.Ly, self.Lz)])

    def check_boundary(self, point):
        if point[0] == 0 or point[0] == self.Lx or point[1] == 0 or point[1] == self.Ly or point[2] == 0 or point[2] == self.Lz:
            return True
        return False

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

    def increment(self):
        self.iteration += 1

class FastSolver:
    def __init__(self, element, order):
        self.element = element
        self.order = order
        self.solver = LinearElasticityLFEMFastSolver(element, order)

    def precondition(self, A, b):
        pass  # Implement preconditioning if needed

    def solve(self, A, b):
        self.solver.solve(A, b)

class MyProblem:
    def __init__(self, domain, element, order):
        self.domain = domain
        self.element = element
        self.order = order
        self.mesh = domain.create_mesh()
        self.space = Lagrange(self.element, self.order, self.mesh)
        self.solver = FastSolver(self.element, self.order)
        self.counter = IterationCounter()

    def define_displacement(self, u):
        u.x = Expression(0, self.mesh)
        u.y = Expression(0, self.mesh)
        u.z = Expression(0, self.mesh)

    def define_strain(self, e):
        e.xx = self.space.d1_grad(self.space.u).xx
        e.yy = self.space.d1_grad(self.space.u).yy
        e.zz = self.space.d1_grad(self.space.u).zz
        e.xy = self.space.d1_grad(self.space.u).xy
        e.xz = self.space.d1_grad(self.space.u).xz
        e.yz = self.space.d1_grad(self.space.u).yz

    def define_stress(self, s):
        material = LinearElasticityMaterial(YoungsModulus=1, PoissonRatio=0.3)
        s.xx = material.lambda_ * self.space.u.xx + material.mu_ * (self.space.u.xx + self.space.u.yy + self.space.u.zz)
        s.yy = material.lambda_ * self.space.u.yy + material.mu_ * (self.space.u.xx + self.space.u.yy + self.space.u.zz)
        s.zz = material.lambda_ * self.space.u.zz + material.mu_ * (self.space.u.xx + self.space.u.yy + self.space.u.zz)
        s.xy = material.mu_ * (self.space.u.xy + self.space.u.yx)
        s.xz = material.mu_ * (self.space.u.xz + self.space.u.zx)
        s.yz = material.mu_ * (self.space.u.yz + self.space.u.zy)

    def define_source(self, f):
        f.xx = Expression(0, self.mesh)
        f.yy = Expression(0, self.mesh)
        f.zz = Expression(0, self.mesh)

    def define_boundary_conditions(self):
        self.dirichlet = DirichletBC(self.space, self.space.u, self.mesh, self.domain.check_boundary)
        self.neumann = NeumannBC(self.space, self.space.u.x, self.space.u.y, self.space.u.z, self.mesh, self.domain.check_boundary)

    def define_function(self, u):
        u.x = Expression(0, self.mesh)
        u.y = Expression(np.sin, degrees=90, self.mesh)
        u.z = Expression(0, self.mesh)

    def setup(self):
        self.define_displacement(self.space.u)
        self.define_strain(self.space.e)
        self.define_stress(self.space.s)
        self.define_source(self.space.f)
        self.define_boundary_conditions()
        self.define_function(self.space.u)

    def compute_matrices_and_vectors(self):
        A = matrix_A(self.space.e, self.space.s, self.space, self.mesh)
        b = a_u(self.space.f, self.space.u, self.space, self.mesh)
        return A, b

    def apply_boundary_conditions(self, A, b):
        self.dirichlet.apply(A, b)
        self.neumann.apply(A, b)

    def solve(self):
        A, b = self.compute_matrices_and_vectors()
        self.apply_boundary_conditions(A, b)
        start_time = time.time()
        self.solver.solve(A, b)
        end_time = time.time()
        print("Time to solve the system: ", end_time - start_time)
        return self.solver.solver.solution

    def plot_mesh(self):
        plot_mesh(self.mesh)

if __name__ == "__main__":
    domain = BoxDomain(Lx=1, Ly=1, Lz=1)
    problem = MyProblem(domain, element=Lagrange(2, 2), order=2)
    problem.setup()
    A, b = problem.compute_matrices_and_vectors()
    print("Shape of the stiffness matrix: ", A.shape)
    solution = problem.solve()
    problem.plot_mesh()
```

This code creates a 2D box domain, defines a class for the problem, and sets up a linear elasticity problem using the fealpy library. It also includes methods for creating a mesh, defining displacement, strain, stress, source, Dirichlet and Neumann boundary conditions, and solving the system of equations. The main part of the code 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, applies the boundary conditions to the system of equations, and prints the shape of the matrix. Finally, it 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.