A more general solver function¶

We consider the flat program ft01_poisson.py for solving the Poisson problem developed in the chapter Fundamentals: Solving the Poisson equation. Some of the code in this program is needed to solve any Poisson problem \(-\nabla^2 u=f\) on \([0,1]\times [0,1]\) with \(u=u_{_\mathrm{D}}\) on the boundary, while other statements arise from our simple test problem. Let us collect the general, reusable code in a function called solver. Our special test problem will then just be an application of our solver with some additional statements. We limit the solver function to just compute the numerical solution. Plotting and comparing the solution with the exact solution are considered to be problem-specific activities to be performed elsewhere.

We parameterize solver by \(f\), \(u_{_\mathrm{D}}\), and the resolution of the mesh. Since it is so trivial to use higher-order finite element functions by changing the third argument to FunctionSpace, we also add the polynomial degree of the finite element function space as an argument to solver.

from fenics import *
import numpy as np

def solver(f, u_D, Nx, Ny, degree=1):
    """
    Solve -Laplace(u) = f on [0,1] x [0,1] with 2*Nx*Ny Lagrange
    elements of specified degree and u = u_D (Expresssion) on
    the boundary.
    """

    # Create mesh and define function space
    mesh = UnitSquareMesh(Nx, Ny)
    V = FunctionSpace(mesh, 'P', degree)

    # Define boundary condition
    def boundary(x, on_boundary):
        return on_boundary

    bc = DirichletBC(V, u_D, boundary)

    # Define variational problem
    u = TrialFunction(V)
    v = TestFunction(V)
    a = dot(grad(u), grad(v))*dx
    L = f*v*dx

    # Compute solution
    u = Function(V)
    solve(a == L, u, bc)

    return u

The remaining tasks of our initial program, such as calling the solver function with problem-specific parameters and plotting, can be placed in a separate function. Here we choose to put this code in a function named run_solver:

def run_solver():
    "Run solver to compute and post-process solution"

    # Set up problem parameters and call solver
    u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', degree=2)
    f = Constant(-6.0)
    u = solver(f, u_D, 8, 8, 1)

    # Plot solution and mesh
    plot(u)
    plot(u.function_space().mesh())

    # Save solution to file in VTK format
    vtkfile = File('poisson_solver/solution.pvd')
    vtkfile << u

The solution can now be computed, plotted, and saved to file by simply calling the run_solver function.

Writing the solver as a Python module¶

The refactored code is placed in a file ft12_poisson_solver.py. We should make sure that such a file can be imported (and hence reused) in other programs. This means that all statements in the main program that are not inside functions should appear within a test if __name__ == '__main__':. This test is true if the file is executed as a program, but false if the file is imported. If we want to run this file in the same way as we can run ft01_poisson.py, the main program is simply a call to run_solver followed by a call to interactive to hold the plot:

if __name__ == '__main__':
    run_solver()
    interactive()

This complete program can be found in the file ft12_poisson_solver.py.

Verification and unit tests¶

The remaining part of our first program is to compare the numerical and the exact solutions. Every time we edit the code we must rerun the test and examine that error_max is sufficiently small so we know that the code still works. To this end, we shall adopt unit testing, meaning that we create a mathematical test and corresponding software that can run all our tests automatically and check that all tests pass. Python has several tools for unit testing. Two very popular ones are pytest and nose. These are almost identical and very easy to use. More classical unit testing with test classes is offered by the built-in module unittest, but here we are going to use pytest (or nose) since that will result in shorter and clearer code.

Mathematically, our unit test is that the finite element solution of our problem when \(f=-6\) equals the exact solution \(u=u_{_\mathrm{D}}=1+x^2+2y^2\) at the vertices of the mesh. We have already created a code that finds the error at the vertices for our numerical solution. Because of rounding errors, we cannot demand this error to be zero, but we have to use a tolerance, which depends on the number of elements and the degrees of the polynomials in the finite element basis. If we want to test that the solver function works for meshes up to \(2\times(20\times 20)\) elements and cubic Lagrange elements, \(10^{-10}\) is an appropriate tolerance for testing that the maximum error vanishes.

To make our test case work together with pytest and nose, we have to make a couple of small adjustments to our program. The simple rule is that each test must be placed in a function that

• has a name starting with test_,
• has no arguments, and
• implements a test expressed as assert success, msg.

Regarding the last point, success is a boolean expression that is False if the test fails, and in that case the string msg is written to the screen. When the test fails, assert raises an AssertionError exception in Python, and otherwise runs silently. The msg string is optional, so assert success is the minimal test. In our case, we will write assert error_max < tol, where tol is the tolerance mentioned above.

A proper test function for implementing this unit test in the pytest or nose testing frameworks has the following form. Note that we perform the test for different mesh resolutions and degrees of finite elements.

def test_solver():
    "Test solver by reproducing u = 1 + x^2 + 2y^2"

    # Set up parameters for testing
    tol = 1E-10
    u_D = Expression('1 + x[0]*x[0] + 2*x[1]*x[1]', degree=2)
    f = Constant(-6.0)

    # Iterate over mesh sizes and degrees
    for Nx, Ny in [(3, 3), (3, 5), (5, 3), (20, 20)]:
        for degree in 1, 2, 3:
            print('Solving on a 2 x (%d x %d) mesh with P%d elements.'
                  % (Nx, Ny, degree))

            # Compute solution
            u = solver(f, u_D, Nx, Ny, degree)

            # Extract the mesh
            mesh = u.function_space().mesh()

            # Compute maximum error at vertices
            vertex_values_u_D = u_D.compute_vertex_values(mesh)
            vertex_values_u  = u.compute_vertex_values(mesh)
            error_max = np.max(np.abs(vertex_values_u_D - \
                                      vertex_values_u))

            # Check maximum error
            msg = 'error_max = %g' % error_max
            assert error_max < tol, msg

To run the test, we type the following command:

Terminal> py.test ft12_poisson_solver.py

This will run all functions named test_* (currently only the test_solver function) found in the file and report the results. For more verbose output, add the flags -s -v.

We shall make it a habit to encapsulate numerical test problems in unit tests as above, and we strongly encourage the reader to create similar unit tests whenever a FEniCS solver is implemented.

from IPython import embed; embed()

Parameterizing the number of space dimensions¶

FEniCS makes it is easy to write a unified simulation code that can operate in 1D, 2D, and 3D. As an appetizer, go back to the previous programs ft01_poisson.py or ft12_poisson_solver.py and change the mesh construction from UnitSquareMesh(8, 8) to UnitCubeMesh(8, 8, 8). Now the domain is the unit cube partitioned into \(8\times 8\times 8\) boxes, and each box is divided into six tetrahedron-shaped finite elements for computations. Run the program and observe that we can solve a 3D problem without any other modifications! (In 1D, expressions must be modified to not depend on x[1].) The visualization allows you to rotate the cube and observe the function values as colors on the boundary.

If we want to parameterize the creation of unit interval, unit square, or unit cube over dimension, we can do so by encapsulating this part of the code in a function. Given a list or tuple specifying the division into cells in the spatial coordinates, the following function returns the mesh for a \(d\)-dimensional cube:

def UnitHyperCube(divisions):
    mesh_classes = [UnitIntervalMesh, UnitSquareMesh, UnitCubeMesh]
    d = len(divisions)
    mesh = mesh_classes[d - 1](*divisions)
    return mesh

The construction mesh_class[d - 1] will pick the right name of the object used to define the domain and generate the mesh. Moreover, the argument *divisions sends all the components of the list divisions as separate arguments to the constructor of the mesh construction class picked out by mesh_class[d - 1]. For example, in a 2D problem where divisions has two elements, the statement

mesh = mesh_classes[d - 1](*divisions)

is equivalent to

mesh = UnitSquareMesh(divisions[0], divisions[1])

The solver function from ft12_poisson_solver.py may be modified to solve \(d\)-dimensional problems by replacing the Nx and Ny parameters by divisions, and calling the function UnitHyperCube to create the mesh. Note that UnitHyperCube is a function and not a class, but we have named it using so-called CamelCase notation to make it look like a class:

mesh = UnitHyperCube(divisions)

Choosing a linear solver and preconditioner¶

Preconditioned Krylov solvers is a type of popular iterative methods that are easily accessible in FEniCS programs. The Poisson equation results in a symmetric, positive definite system matrix, for which the optimal Krylov solver is the Conjugate Gradient (CG) method. For non-symmetric problems, a Krylov solver for non-symmetric systems, such as GMRES, is a better choice. Incomplete LU factorization (ILU) is a popular and robust all-round preconditioner, so let us try the GMRES-ILU pair:

solve(a == L, u, bc,
      solver_parameters={'linear_solver': 'gmres',
                         'preconditioner': 'ilu'})
# Alternative syntax
solve(a == L, u, bc,
      solver_parameters=dict(linear_solver='gmres',
                             preconditioner='ilu'))

the section List of linear solver methods and preconditioners lists the most popular choices of Krylov solvers and preconditioners available in FEniCS.

Choosing a linear algebra backend¶

The actual GMRES and ILU implementations that are brought into action depend on the choice of linear algebra package. FEniCS interfaces several linear algebra packages, called linear algebra backends in FEniCS terminology. PETSc is the default choice if FEniCS is compiled with PETSc. If PETSc is not available, then FEniCS falls back to using the Eigen backend. The linear algebra backend in FEniCS can be set using the following command:

parameters.linear_algebra_backend = backendname

where backendname is a string. To see which linear algebra backends are available, you can call the FEniCS function list_linear_algebra_backends. Similarly, one may check which linear algebra backend is currently being used by the following command:

print(parameters.linear_algebra_backend)

Setting solver parameters¶

We will normally want to control the tolerance in the stopping criterion and the maximum number of iterations when running an iterative method. Such parameters can be controlled at both a global and a local level. We will start by looking at how to set global parameters. For more advanced programs, one may want to use a number of different linear solvers and set different tolerances and other parameters. Then it becomes important to control the parameters at a local level. We will return to this issue in the section Linear variational problem and solver objects.

Changing a parameter in the global FEniCS parameter database affects all linear solvers (created after the parameter has been set). The global FEniCS parameter database is simply called parameters and it behaves as a nested dictionary. Write

info(parameters, verbose=True)

to list all parameters and their default values in the database. The nesting of parameter sets is indicated through indentation in the output from info. According to this output, the relevant parameter set is named 'krylov_solver', and the parameters are set like this:

prm = parameters.krylov_solver  # short form
prm.absolute_tolerance = 1E-10
prm.relative_tolerance = 1E-6
prm.maximum_iterations = 1000

Stopping criteria for Krylov solvers usually involve some norm of the residual, which must be smaller than the absolute tolerance parameter or smaller than the relative tolerance parameter times the initial residual.

We remark that default values for the global parameter database can be defined in an XML file. To generate such a file from the current set of parameters in a program, run

File('parameters.xml') << parameters

If a dolfin_parameters.xml file is found in the directory where a FEniCS program is run, this file is read and used to initialize the parameters object. Otherwise, the file .config/fenics/dolfin_parameters.xml in the user’s home directory is read, if it exists. Another alternative is to load the XML file (with any name) manually in the program:

File('parameters.xml') >> parameters

The XML file can also be in gzip’ed form with the extension .xml.gz.

An extended solver function¶

We may extend the previous solver function from ft12_poisson_solver.py in the section A more general solver function such that it also offers the GMRES+ILU preconditioned Krylov solver:

This new solver function, found in the file ft10_poisson_extended.py, replaces the one in ft12_poisson_solver.py. It has all the functionality of the previous solver function, but can also solve the linear system with iterative methods.

A remark regarding unit tests¶

Regarding verification of the new solver function in terms of unit tests, it turns out that unit testing for a problem where the approximation error vanishes gets more complicated when we use iterative methods. The problem is to keep the error due to iterative solution smaller than the tolerance used in the verification tests. First of all, this means that the tolerances used in the Krylov solvers must be smaller than the tolerance used in the assert test, but this is no guarantee to keep the linear solver error this small. For linear elements and small meshes, a tolerance of \(10^{-11}\) works well in the case of Krylov solvers too (using a tolerance \(10^{-12}\) in those solvers). The interested reader is referred to the demo_solvers function in ft10_poisson_extended.py for details: this function tests the numerical solution for direct and iterative linear solvers, for different meshes, and different degrees of the polynomials in the finite element basis functions.

List of linear solver methods and preconditioners¶

Which linear solvers and preconditioners that are available in FEniCS depends on how FEniCS has been configured and which linear algebra backend is currently active. The following table shows an example of which linear solvers that can be available through FEniCS when the PETSc backend is active:

The set of available preconditioners also depends on configuration and linear algebra backend. The following table shows an example of which preconditioners may be available:

An up-to-date list of the available solvers and preconditioners for your FEniCS installation can be produced by

list_linear_solver_methods()
list_krylov_solver_preconditioners()

Linear variational problem and solver objects¶

In this section, we will look at an alternative interface to solving linear variational problems in FEniCS, which may be preferable in many situations compared to the high-level solve function interface. This interface uses the two classes LinearVariationalProblem and LinearVariationalSolver. Using this interface, the equivalent of solve(a == L, u, bc) looks as follows:

u = Function(V)
problem = LinearVariationalProblem(a, L, u, bc)
solver = LinearVariationalSolver(problem)
solver.solve()

Many FEniCS objects have an attribute parameters, similar to the global parameters database, but local to the object. Here, solver.parameters play that role. Setting the CG method with ILU preconditioning as the solution method and specifying solver-specific parameters can be done like this:

solver.parameters.linear_solver = 'gmres'
solver.parameters.preconditioner = 'ilu'
prm = solver.parameters.krylov_solver  # short form
prm.absolute_tolerance = 1E-7
prm.relative_tolerance = 1E-4
prm.maximum_iterations = 1000

Settings in the global parameters database are propagated to parameter sets in individual objects, with the possibility of being overwritten as above. Note that global parameter values can only affect local parameter values if set before the time of creation of the local object. Thus, changing the value of the tolerance in the global parameter database will not affect the parameters for already created solvers.

Explicit assembly and solve¶

As we saw already in the section The Navier - Stokes equations, linear variational problems can be assembled explicitly in FEniCS into matrices and vectors using the assemble function. This allows even more fine-grained control of the solution process compared to using the high-level solve function or using the classes LinearVariationalProblem and LinearVariationalSolver. We will now look more closely into how to use the assemble function and how to combine this with low-level calls for solving the assembled linear systems.

Given a variational problem \(a(u,v)=L(v)\), the discrete solution \(u\) is computed by inserting \(u=\sum_{j=1}^N U_j \phi_j\) into \(a(u,v)\) and demanding \(a(u,v)=L(v)\) to be fulfilled for \(N\) test functions \(\hat\phi_1,\ldots,\hat\phi_N\). This implies

which is nothing but a linear system,

where the entries of \(A\) and \(b\) are given by

The examples so far have specified the left- and right-hand sides of the variational formulation and then asked FEniCS to assemble the linear system and solve it. An alternative is to explicitly call functions for assembling the coefficient matrix \(A\) and the right-hand side vector \(b\), and then solve the linear system \(AU=b\) for the vector \(U\). Instead of solve(a == L, U, b) we now write

A = assemble(a)
b = assemble(L)
bc.apply(A, b)
u = Function(V)
U = u.vector()
solve(A, U, b)

The variables a and L are the same as before; that is, a refers to the bilinear form involving a TrialFunction object u and a TestFunction object v, and L involves the same TestFunction object v. From a and L, the assemble function can compute \(A\) and \(b\).

Creating the linear system explicitly in a program can have some advantages in more advanced problem settings. For example, \(A\) may be constant throughout a time-dependent simulation, so we can avoid recalculating \(A\) at every time level and save a significant amount of simulation time.

The matrix \(A\) and vector \(b\) are first assembled without incorporating essential (Dirichlet) boundary conditions. Thereafter, the call bc.apply(A, b) performs the necessary modifications of the linear system such that u is guaranteed to equal the prescribed boundary values. When we have multiple Dirichlet conditions stored in a list bcs, we must apply each condition in bcs to the system:

for bc in bcs:
    bc.apply(A, b)

# Alternative syntax using list comprehension
[bc.apply(A, b) for bc in bcs]

Alternatively, we can use the function assemble_system, which takes the boundary conditions into account when assembling the linear system. This method preserves the symmetry of the linear system for a symmetric bilinear form. Even if the matrix A that comes out of the call to assemble is symmetric (for a symmetric bilinear form a), the call to bc.apply will break the symmetry. Preserving the symmetry of a variational problem is important when using particular linear solvers designed for symmetric systems, such as the conjugate gradient method.

Once the linear system has been assembled, we need to compute the solution \(U=A^{-1}b\) and store the solution \(U\) in the vector U = u.vector(). In the same way as linear variational problems can be programmed using different interfaces in FEniCS—the high-level solve function, the class LinearVariationalSolver, and the low-level assemble function—linear systems can also be programmed using different interfaces in FEniCS. The high-level interface to solving a linear system in FEniCS is also named solve:

solve(A, U, b)

By default, solve(A, U, b) uses sparse LU decomposition to compute the solution. Specification of an iterative solver and preconditioner can be made through two optional arguments:

solve(A, U, b, 'cg', 'ilu')

Appropriate names of solvers and preconditioners are found in the section List of linear solver methods and preconditioners.

This high-level interface is useful for many applications, but sometimes more fine-grained control is needed. One can then create one or more KrylovSolver objects that are then used to solve linear systems. Each different solver object can have its own set of parameters and selection of iterative method and preconditioner. Here is an example:

solver = KrylovSolver('cg', 'ilu')
prm = solver.parameters
prm.absolute_tolerance = 1E-7
prm.relative_tolerance = 1E-4
prm.maximum_iterations = 1000
u = Function(V)
U = u.vector()
solver.solve(A, U, b)

The function solver_linalg in the program file ft10_poisson_extended.py implements such a solver.

The choice of start vector for the iterations in a linear solver is often important. By default, the values of u and thus the vector U = u.vector() will be initialized to zero. If we instead wanted to initialize U with random numbers in the interval \([-100,100]\) this can be done as follows:

n = u.vector().array().size
U = u.vector()
U[:] = numpy.random.uniform(-100, 100, n)
solver.parameters.nonzero_initial_guess = True
solver.solve(A, U, b)

Note that we must both turn off the default behavior of setting the start vector (“initial guess”) to zero, and also set the values of the vector U to nonzero values. This is useful if we happen to know a good initial guess for the solution.

Using a nonzero initial guess can be particularly important for time-dependent problems or when solving a linear system as part of a nonlinear iteration, since then the previous solution vector U will often be a good initial guess for the solution in the next time step or iteration. In this case, the values in the vector U will naturally be initialized with the previous solution vector (if we just used it to solve a linear system), so the only extra step necessary is to set the parameter nonzero_initial_guess to True.

Examining matrix and vector values¶

When calling A = assemble(a) and b = assemble(L), the object A will be of type Matrix, while b and u.vector() are of type Vector. To examine the values, we may convert the matrix and vector data to numpy arrays by calling the array method as shown before. For example, if you wonder how essential boundary conditions are incorporated into linear systems, you can print out A and b before and after the bc.apply(A, b) call:

A = assemble(a)
b = assemble(L)
if mesh.num_cells() < 16:  # print for small meshes only
    print(A.array())
    print(b.array())
bc.apply(A, b)
if mesh.num_cells() < 16:
    print(A.array())
    print(b.array())

With access to the elements in A through a numpy array, we can easily perform computations on this matrix, such as computing the eigenvalues (using the eig function in numpy.linalg). We can alternatively dump A.array() and b.array() to file in MATLAB format and invoke MATLAB or Octave to analyze the linear system. Dumping the arrays to MATLAB format is done by

import scipy.io
scipy.io.savemat('Ab.mat', {'A': A.array(), 'b': b.array()})

Writing load Ab.mat in MATLAB or Octave will then make the array variables A and b available for computations.

Matrix processing in Python or MATLAB/Octave is only feasible for small PDE problems since the numpy arrays or matrices in MATLAB file format are dense matrices. FEniCS also has an interface to the eigensolver package SLEPc, which is the preferred tool for computing the eigenvalues of large, sparse matrices of the type encountered in PDE problems (see demo/documented/eigenvalue/python/ in the FEniCS/DOLFIN source code tree for a demo).

Examining the degrees of freedom¶

We have seen before how to grab the degrees of freedom array from a finite element function u:

nodal_values = u.vector().array()

For a finite element function from a standard continuous piecewise linear function space (\(\mathsf{P}_1\) Lagrange elements), these values will be the same as the values we get by the following statement:

vertex_values = u.compute_vertex_values(mesh)

Both nodal_values and vertex_values will be numpy arrays and they will be of the same length and contain the same values (for \(\mathsf{P}_1\) elements), but with possibly different ordering. The array vertex_values will have the same ordering as the vertices of the mesh, while nodal_values will be ordered in a way that (nearly) minimizes the bandwidth of the system matrix and thus improves the efficiency of linear solvers.

A fundamental question is: what are the coordinates of the vertex whose value is nodal_values[i]? To answer this question, we need to understand how to get our hands on the coordinates, and in particular, the numbering of degrees of freedom and the numbering of vertices in the mesh.

The function mesh.coordinates returns the coordinates of the vertices as a numpy array with shape \((M,d)\), \(M\) being the number of vertices in the mesh and \(d\) being the number of space dimensions:

>>> from fenics import *
>>> mesh = UnitSquareMesh(2, 2)
>>> coordinates = mesh.coordinates()
>>> coordinates
array([[ 0. ,  0. ],
       [ 0.5,  0. ],
       [ 1. ,  0. ],
       [ 0. ,  0.5],
       [ 0.5,  0.5],
       [ 1. ,  0.5],
       [ 0. ,  1. ],
       [ 0.5,  1. ],
       [ 1. ,  1. ]])

We see from this output that for this particular mesh, the vertices are first numbered along \(y=0\) with increasing \(x\) coordinate, then along \(y=0.5\), and so on.

Next we compute a function u on this mesh. Let’s take \(u=x+y\):

>>> V = FunctionSpace(mesh, 'P', 1)
>>> u = interpolate(Expression('x[0] + x[1]', degree=1), V)
>>> plot(u, interactive=True)
>>> nodal_values = u.vector().array()
>>> nodal_values
array([ 1. ,  0.5,  1.5,  0. ,  1. ,  2. ,  0.5,  1.5,  1. ])

We observe that nodal_values[0] is not the value of \(x+y\) at vertex number 0, since this vertex has coordinates \(x=y=0\). The numbering of the nodal values (degrees of freedom) \(U_1,\ldots,U_{N}\) is obviously not the same as the numbering of the vertices.

The vertex numbering may be examined by using the FEniCS plot command. To do this, plot the function u, press w to turn on wireframe instead of a fully colored surface, m to show the mesh, and then v to show the numbering of the vertices.

Let’s instead examine the values by calling u.compute_vertex_values:

>>> vertex_values = u.compute_vertex_values()
>>> for i, x in enumerate(coordinates):
...     print('vertex %d: vertex_values[%d] = %g\tu(%s) = %g' %
...           (i, i, vertex_values[i], x, u(x)))
vertex 0: vertex_values[0] = 0          u([ 0.  0.]) = 8.46545e-16
vertex 1: vertex_values[1] = 0.5        u([ 0.5  0. ]) = 0.5
vertex 2: vertex_values[2] = 1          u([ 1.  0.]) = 1
vertex 3: vertex_values[3] = 0.5        u([ 0.   0.5]) = 0.5
vertex 4: vertex_values[4] = 1          u([ 0.5  0.5]) = 1
vertex 5: vertex_values[5] = 1.5        u([ 1.   0.5]) = 1.5
vertex 6: vertex_values[6] = 1          u([ 0.  1.]) = 1
vertex 7: vertex_values[7] = 1.5        u([ 0.5  1. ]) = 1.5
vertex 8: vertex_values[8] = 2          u([ 1.  1.]) = 2

We can ask FEniCS to give us the mapping from vertices to degrees of freedom for a certain function space \(V\):

v2d = vertex_to_dof_map(V)

Now, nodal_values[v2d[i]] will give us the value of the degree of freedom corresponding to vertex i (v2d[i]). In particular, nodal_values[v2d] is an array with all the elements in the same (vertex numbered) order as coordinates. The inverse map, from degrees of freedom number to vertex number is given by dof_to_vertex_map(V). This means that we may call coordinates[dof_to_vertex_map(V)] to get an array of all the coordinates in the same order as the degrees of freedom. Note that these mappings are only available in FEniCS for \(\mathsf{P}_1\) elements.

For Lagrange elements of degree larger than 1, there are degrees of freedom (nodes) that do not correspond to vertices. For these elements, we may get the vertex values by calling u.compute_vertex_values(mesh), and we can get the degrees of freedom by the call u.vector().array(). To get the coordinates associated with all degrees of freedom, we need to iterate over the elements of the mesh and ask FEniCS to return the coordinates and dofs associated with each element (cell). This information is stored in the FiniteElement and DofMap object of a FunctionSpace. The following code illustrates how to iterate over all elements of a mesh and print the coordinates and degrees of freedom associated with the element.

element = V.element()
dofmap = V.dofmap()
for cell in cells(mesh):
    print(element.tabulate_dof_coordinates(cell))
    print(dofmap.cell_dofs(cell.index()))

Setting the degrees of freedom¶

We have seen how to extract the nodal values in a numpy array. If desired, we can adjust the nodal values too. Say we want to normalize the solution such that \(\max_j |U_j| = 1\). Then we must divide all \(U_j\) values by \(\max_j |U_j|\). The following function performs the task:

def normalize_solution(u):
    "Normalize u: return u divided by max(u)"
    u_array = u.vector().array()
    u_max = np.max(np.abs(u_array))
    u_array /= u_max
    u.vector()[:] = u_array
    #u.vector().set_local(u_array)  # alternative
    return u

When using Lagrange elements, this (approximately) ensures that the maximum value of the function \(u\) is \(1\).

The /= operator implies an in-place modification of the object on the left-hand side: all elements of the array nodal_values are divided by the value u_max. Alternatively, we could do nodal_values = nodal_values / u_max, which implies creating a new array on the right-hand side and assigning this array to the name nodal_values.

Function evaluation¶

A FEniCS Function object is uniquely defined in the interior of each cell of the finite element mesh. For continuous (Lagrange) function spaces, the function values are also uniquely defined on cell boundaries. A Function object u can be evaluated by simply calling

u(x)

where x is either a Point or a Python tuple of the correct space dimension. When a Function is evaluated, FEniCS must first find which cell of the mesh that contains the given point (if any), and then evaluate a linear combination of basis functions at the given point inside the cell in question. FEniCS uses efficient data structures (bounding box trees) to quickly find the point, but building the tree is a relatively expensive operation so the cost of evaluating a Function at a single point is costly. Repeated evaluation will reuse the computed data structures and thus be relatively less expensive.

To demonstrate the use of point evaluation of Function objects, we print the value of the computed finite element solution u for the Poisson problem at the center point of the domain and compare it with the exact solution:

center = (0.5, 0.5)
error = u_D(center) - u(center)
print('Error at %s: %g' % (center, error))

For a \(2\times(3\times 3)\) mesh, the output from the previous snippet becomes

Error at (0.5, 0.5): -0.0833333

The discrepancy is due to the fact that the center point is not a node in this particular mesh, but a point in the interior of a cell, and u varies linearly over the cell while u_D is a quadratic function. When the center point is a node, as in a \(2\times(2\times 2)\) or \(2\times(4\times 4)\) mesh, the error is of the order \(10^{-15}\).

Test problem¶

As a test problem, we consider again the variable-coefficient Poisson problem with a single Dirichlet boundary condition:

Let us continue to use our favorite solution \(u(x,y)=1+x^2+2y^2\) and then prescribe \(\kappa(x,y)=x+y\). It follows that \(u_{_\mathrm{D}}(x,y) = 1 + x^2 + 2y^2\) and \(f(x,y)=-8x-10y\).

As before, the variational formulation for this model problem can be specified in FEniCS as

a = kappa*dot(grad(u), grad(v))*dx
L = f*v*dx

with the coefficient \(\kappa\) and right-hand side \(f\) given by

kappa = Expression('x[0] + x[1]', degree=1)
f = Expression('-8*x[0] - 10*x[1]', degree=1)

Flux computations¶

It is often of interest to compute the flux \(Q = -\kappa\nabla u\). Since \(u = \sum_{j=1}^N U_j \phi_j\), it follows that

We note that the gradient of a piecewise continuous finite element scalar field is a discontinuous vector field since the basis functions \(\{\phi_j\}\) have discontinuous derivatives at the boundaries of the cells. For example, using Lagrange elements of degree 1, \(u\) is linear over each cell, and the gradient becomes a piecewise constant vector field. On the contrary, the exact gradient is continuous. For visualization and data analysis purposes, we often want the computed gradient to be a continuous vector field. Typically, we want each component of \(\nabla u\) to be represented in the same way as \(u\) itself. To this end, we can project the components of \(\nabla u\) onto the same function space as we used for \(u\). This means that we solve \(w = \nabla u\) approximately by a finite element method, using the same elements for the components of \(w\) as we used for \(u\). This process is known as projection.

Projection is a common operation in finite element analysis and, as we have already seen, FEniCS has a function for easily performing the projection: project(expression, W), which returns the projection of some expression into the space W.

In our case, the flux \(Q = -\kappa\nabla u\) is vector-valued and we need to pick W as the vector-valued function space of the same degree as the space V where u resides:

V = u.function_space()
mesh = V.mesh()
degree = V.ufl_element().degree()
W = VectorFunctionSpace(mesh, 'P', degree)

grad_u = project(grad(u), W)
flux_u = project(-k*grad(u), W)

The applications of projection are many, including turning discontinuous gradient fields into continuous ones, comparing higher- and lower-order function approximations, and transforming a higher-order finite element solution down to a piecewise linear field, which is required by many visualization packages.

Plotting the flux vector field is naturally as easy as plotting anything else:

plot(flux_u, title='flux field')

flux_x, flux_y = flux_u.split(deepcopy=True)  # extract components
plot(flux_x, title='x-component of flux (-kappa*grad(u))')
plot(flux_y, title='y-component of flux (-kappa*grad(u))')

The deepcopy=True argument signifies a deep copy, which is a general term in computer science implying that a copy of the data is returned. (The opposite, deepcopy=False, means a shallow copy, where the returned objects are just pointers to the original data.)

For data analysis of the nodal values of the flux field, we can grab the underlying numpy arrays (which demands a deepcopy=True in the split of flux):

flux_x_nodal_values = flux_x.vector().dofs()
flux_y_nodal_values = flux_y.vector().dofs()

The degrees of freedom of the flux_u vector field can also be reached by

flux_u_nodal_values = flux_u.vector().array()

However, this is a flat numpy array containing the degrees of freedom for both the \(x\) and \(y\) components of the flux and the ordering of the components may be mixed up by FEniCS in order to improve computational efficiency.

The function demo_flux in the program ft10_poisson_extended.py demonstrates the computations described above.

Computing functionals¶

After the solution \(u\) of a PDE is computed, we occasionally want to compute functionals of \(u\), for example,

which often reflects some energy quantity. Another frequently occurring functional is the error

where \({u_{\small\mbox{e}}}\) is the exact solution. The error is of particular interest when studying convergence properties of finite element methods. Other times, we may instead be interested in computing the flux out through a part \(\Gamma\) of the boundary \(\partial\Omega\),

where \(n\) is the outward-pointing unit normal on \(\Gamma\).

All these functionals are easy to compute with FEniCS, as we shall see in the examples below.

energy = 0.5*dot(grad(u), grad(u))*dx
E = assemble(energy)

error = (u_e - u)**2*dx
E = sqrt(abs(assemble(error)))

E = errornorm(u_e, u)

n = FacetNormal(mesh)
flux = -k*dot(grad(u), n)*ds
total_flux = assemble(flux)

mesh = UnitIntervalMesh(1)
I = assemble(Expression('cos(x[0])', degree=degree)*dx(domain=mesh))

mesh = UnitIntervalMesh(1)
x = SpatialCoordinate(mesh)
I = assemble(cos(x[0])*dx(degree=degree))

Computing convergence rates¶

A central question for any numerical method is its convergence rate: how fast does the error approach zero when the resolution is increased? For finite element methods, this typically corresponds to proving, theoretically or empirically, that the error \(e = {u_{\small\mbox{e}}} - u\) is bounded by the mesh size \(h\) to some power \(r\); that is, \(\|e\| \leq C h^r\) for some constant \(C\). The number \(r\) is called the convergence rate of the method. Note that different norms, like the \(L^2\)-norm \(\|e\|\) or \(H^1_0\)-norm \(\|\nabla e\|\) typically have different convergence rates.

To illustrate how to compute errors and convergence rates in FEniCS, we have included the function compute_convergence_rates in the tutorial program ft10_poisson_extended.py. This is a tool that is very handy when verifying finite element codes and will therefore be explained in detail here.

error = (u_e - u)**2*dx
E = sqrt(abs(assemble(error)))

u_e = Expression('sin(x[0])', degree=1)

u_e = Expression('sin(x[0])', degree=3)

E = errornorm(u_e, u, normtype='L2')

def errornorm(u_e, u):
    V = u.function_space()
    mesh = V.mesh()
    degree = V.ufl_element().degree()
    W = FunctionSpace(mesh, 'P', degree + 3)
    u_e_W = interpolate(u_e, W)
    u_W = interpolate(u, W)
    e_W = Function(W)
    e_W.vector()[:] = u_e_W.vector().array() - u_W.vector().array()
    error = e_W**2*dx
    return sqrt(abs(assemble(error)))

E = errornorm(u_e, u, norm_type='H10')

def boundary(x, on_boundary):
    return on_boundary

bc = DirichletBC(V, Constant(0), boundary)

omega = 1.0
u_e = Expression('sin(omega*pi*x[0])*sin(omega*pi*x[1])',
                 degree=6, omega=omega)

f = 2*pi**2*omega**2*u_e

Taking advantage of structured mesh data¶

Many readers have extensive experience with visualization and data analysis of 1D, 2D, and 3D scalar and vector fields on uniform, structured meshes, while FEniCS solvers exclusively work with unstructured meshes. Since it can many times be practical to work with structured data, we discuss in this section how to extract structured data for finite element solutions computed with FEniCS.

A necessary first step is to transform our Mesh object to an object representing a rectangle (or a 3D box) with equally-shaped rectangular cells. The second step is to transform the one-dimensional array of nodal values to a two-dimensional array holding the values at the corners of the cells in the structured mesh. We want to access a value by its \(i\) and \(j\) indices, \(i\) counting cells in the \(x\) direction, and \(j\) counting cells in the \(y\) direction. This transformation is in principle straightforward, yet it frequently leads to obscure indexing errors, so using software tools to ease the work is advantageous.

In the directory of example programs included with this book, we have included the Python module boxfield which provides utilities for working with structured mesh data in FEniCS. Given a finite element function u, the following function returns a BoxField object that represents u on a structured mesh:

from boxfield import *
u_box = FEniCSBoxField(u, (nx, ny))

The u_box object contains several useful data structures:

• u_box.grid: object for the structured mesh
• u_box.grid.coor[X]: grid coordinates in X=0 direction
• u_box.grid.coor[Y]: grid coordinates in Y=1 direction
• u_box.grid.coor[Z]: grid coordinates in Z=2 direction
• u_box.grid.coorv[X]: vectorized version of u_box.grid.coor[X]
• u_box.grid.coorv[Y]: vectorized version of u_box.grid.coor[Y]
• u_box.grid.coorv[Z]: vectorized version of u_box.grid.coor[Z]
• u_box.values: numpy array holding the u values; u_box.values[i,j] holds u at the mesh point with coordinates

u = solver(p, f, u_b, nx, ny, 1, linear_solver='direct')
u_box = structured_mesh(u, (nx, ny))
u_ = u_box.values

# Iterate over 2D mesh points (i, j)
for j in range(u_.shape[1]):
    for i in range(u_.shape[0]):
        print('u[%d, %d] = u(%g, %g) = %g' %
              (i, j,
               u_box.grid.coor[X][i], u_box.grid.coor[Y][j],
               u_[i, j]))

x = u_box.grid.coor[X]
dx = x[1] - x[0]
u_xx = (u_[i - 1, j] - 2*u_[i, j] + u_[i + 1, j]) / dx**2

Making surface plots¶

The ability to access a finite element field as structured data is handy in many occasions, e.g., for visualization and data analysis. Using Matplotlib, we can create a surface plot, as shown in Figure Various plots of the solution on a structured mesh (upper left):

import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D  # necessary for 3D plotting
from matplotlib import cm
fig = plt.figure()
ax = fig.gca(projection='3d')
cv = u_box.grid.coorv  # vectorized mesh coordinates
ax.plot_surface(cv[X], cv[Y], u_, cmap=cm.coolwarm,
                rstride=1, cstride=1)
plt.title('Surface plot of solution')

The key issue is to know that the coordinates needed for the surface plot is in u_box.grid.coorv and that the values are in u_.

Various plots of the solution on a structured mesh

fig = plt.figure()
ax = fig.gca()
levels = [1.5, 2.0, 2.5, 3.5]
cs = ax.contour(cv[X], cv[Y], u_, levels=levels)
plt.clabel(cs)  # add labels to contour lines
plt.axis('equal')
plt.title('Contour plot of solution')

start = (0, 0.4)
x, u_val, y_fixed, snapped = u_box.gridline(start, direction=X)

# Plot u along a line y = const and compare with exact solution
start = (0, 0.4)
x, u_val, y_fixed, snapped = u_box.gridline(start, direction=X)
u_e_val = [u_D((x_, y_fixed)) for x_ in x]
plt.figure()
plt.plot(x, u_val, 'r-')
plt.plot(x, u_e_val, 'bo')
plt.legend(['P1 elements', 'exact'], loc='best')
plt.title('Solution along line y=%g' % y_fixed)
plt.xlabel('x');  plt.ylabel('u')

# Plot the numerical and exact flux along the same line
flux_u = flux(u, kappa)
flux_u_x, flux_u_y = flux_u.split(deepcopy=True)
flux2_x = flux_u_x if flux_u_x.ufl_element().degree() == 1 \
          else interpolate(flux_x,
               FunctionSpace(u.function_space().mesh(), 'P', 1))
flux_u_x_box = FEniCSBoxField(flux_u_x, (nx,ny))
x, flux_u_val, y_fixed, snapped = \
   flux_u_x_box.gridline(start, direction=X)
y = y_fixed
plt.figure()
plt.plot(x, flux_u_val, 'r-')
plt.plot(x, flux_u_x_exact(x, y_fixed), 'bo')
plt.legend(['P1 elements', 'exact'], loc='best')
plt.title('Flux along line y=%g' % y_fixed)
plt.xlabel('x');  plt.ylabel('u')

from sympy import exp, sin, pi  # for use in math formulas
import sympy as sym

H = lambda x: exp(-16*(x-0.5)**2)*sin(3*pi*x)
x, y = sym.symbols('x[0], x[1]')
u = H(x)*H(y)

u_code = sym.printing.ccode(u)

-exp(-16*pow(x[0] - 0.5, 2) - 16*pow(x[1] - 0.5, 2))*
sin(3*M_PI*x[0])*sin(3*M_PI*x[1])

u_code = u_code.replace('M_PI', 'pi')
u_b = Expression(u_code, degree=1)

kappa = 1
f = sym.diff(-kappa*sym.diff(u, x), x) + \
    sym.diff(-kappa*sym.diff(u, y), y)
f = sym.simplify(f)
f_code = sym.printing.ccode(f)
f_code = f_code.replace('M_PI', 'pi')
f = Expression(f_code, degree=1)

flux_u_x_exact = sym.lambdify([x, y], -kappa*sym.diff(u, x),
                              modules='numpy')