Solving PDEs

Introduction

Now that we have learnt how to define weak variational problems, we will move on to how to actually solve them using Firedrake. Let us consider a weak variational problem

\[ \begin{align}\begin{aligned}a(u, v) = L(v) \; \forall v \in V \mathrm{on}\: \Omega\\u = u_0 \; \mathrm{on}\: \partial\Omega\end{aligned}\end{align} \]

we will call the bilinear and linear parts of this form a and L respectively. The strongly imposed boundary condition, \(u = u_0 \;\mathrm{on}\:\partial\Omega\) will be represented by a variable of type DirichletBC, bc.

Now that we have all the pieces of our variational problem, we can move forward to solving it.

Solving the variational problem

The function used to solve PDEs defined as above is solve(). This is a unified interface for solving both linear and non-linear variational problems along with linear systems (where the arguments are already assembled matrices and vectors, rather than UFL forms). We will treat the variational interface first.

Linear variational problems

If the problem is linear, that is a is linear in both the test and trial functions and L is linear in the test function, we can use the linear variational problem interface to solve. To start, we need a Function to hold the value of the solution:

s = Function(V)

We can then solve the problem, placing the solution in s with:

solve(a == L, s)

To apply boundary conditions, one passes a list of DirichletBC objects using the bcs keyword argument. For example, if there are two boundary conditions, in bc1 and bc2, we write:

solve(a == L, s, bcs=[bc1, bc2])

Nonlinear variational problems

For nonlinear problems, the interface is similar. In this case, we solve a problem:

\[ \begin{align}\begin{aligned}F(u; v) = 0 \; \forall v \in V \mathrm{on}\: \Omega\\u = u_0 \; \mathrm{on}\: \partial\Omega\end{aligned}\end{align} \]

where the residual \(F(u; v)\) is linear in the test function \(v\) but possibly non-linear in the unknown Function \(u\). To solve such a problem we write, if F is the residual form:

solve(F == 0, u)

to apply strong boundary conditions, as before, we provide a list of DirichletBC objects using the bcs keyword:

solve(F == 0, u, bcs=[bc1, bc2])

Nonlinear problems in Firedrake are solved using Newton-like methods. That is, we compute successive approximations to the solution using

\[u_{k+1} = u_{k} - J(u_k)^{-1} F(u_k) \; k = 0, 1, \dots\]

where \(u_0\) is an initial guess for the solution and \(J(u_k) = \frac{\partial F(u_k)}{\partial u_k}\) is the Jacobian of the residual, which should be non-singular at each iteration. Notice how in the above examples, we did not explicitly supply a Jacobian. If it is not supplied, it will be computed by automatic differentiation of the residual form F with respect to the solution variable u. However, we may also supply the Jacobian explicitly, using the keyword argument J:

solve(F == 0, u, J=user_supplied_jacobian_form)

The initial guess for the Newton iterations is provided in u, for example, to provide a non-zero guess that the solution is the value of the x coordinate everywhere:

x = SpatialCoordinate(m)
u.interpolate(x[0])

solve(F == 0, u)

Solving linear systems

Often, we might be solving a time-dependent linear system. In this case, the bilinear form a does not change between timesteps, whereas the linear form L does. Since assembly of the bilinear form is a potentially costly process, Firedrake offers the ability to “pre-assemble” forms in such systems and then reuse the assembled operator in successive linear solves. Again, we use the same solve interface to do this, but must build slightly different objects to pass in. In the pre-assembled case, we are solving a linear system:

\[A\vec{x} = \vec{b}\]

Where \(A\) is a known matrix, \(\vec{b}\) is a known right hand side vector and \(\vec{x}\) is the unknown solution vector. In Firedrake, \(A\) is represented as a Matrix, while \(\vec{b}\) and \(\vec{x}\) are both Functions. We build these values by calling assemble on the UFL forms that define our problem, which, as before are denoted a and L. Similarly to the linear variational case, we first need a function in which to place our solution:

x = Function(V)

We then assemble() the left hand side matrix A and known right hand side b from the bilinear and linear forms respectively:

A = assemble(a)
b = assemble(L)

Finally, we can solve the problem placing the solution in x:

solve(A, x, b)

to apply boundary conditions to the problem, we can assemble the linear operator A with boundary conditions using the bcs keyword argument to assemble() (and then not supply them in solve call):

A = assemble(a, bcs=[bc1, bc2])
b = assemble(L)
solve(A, x, b)

alternately, we can supply boundary conditions in solve() as before:

A = assemble(a)
b = assemble(L)
solve(A, x, b, bcs=[bc1, bc2])

If boundary conditions have been supplied both in the assemble and solve calls, then those provided for the solve take precedence, for example, in the following, the system is solved only applying bc1:

A = assemble(a, bcs=[bc1, bc2])
b = assemble(L)
solve(A, x, b, bcs=[bc1])

Note that after the call to solve, A will be an assembled system with only bc1 applied, hence subsequent calls to solve that do not change the boundary conditions again will not require a further re-assembly.

Specifying solution methods

Not all linear and non-linear systems defined by PDEs are created equal, and we therefore need ways of specifying which solvers to use and options to pass to them. Firedrake uses PETSc to solve both linear and non-linear systems and presents a uniform interface in solve to set PETSc solver options. In all cases, we set options in the solve call by passing a dictionary to the solver_parameters keyword argument. To set options we use the same names that PETSc uses in its command-line option setting interface (having removed the leading -). For more complete details on PETSc option naming we recommend looking in the PETSc manual. We describe some of the more common options here.

Configuring solvers from the commandline

As well as specifying solver options in a parameters dict at the call site for each solve, one can configure solvers by passing options in the normal PETSc style via the commandline. To do this, we need to specify the options_prefix for each solver that we wish to configure via the commandline. This is done by providing a non-None argument as the options_prefix keyword argument to the solver. The separator between the prefix and the subsequent options is an underscore, which is automatically appended if the provided options prefix does end in one.

When using an options prefix, we do not need to specify the prefix in the solver parameters dictionary (it is automatically added in the appropriate way). Also to note is that command line options override parameters set in the dictionary. This way we can provide good defaults for solvers, and override them on a case-by-case basis.

For example, suppose we have a file pde.py that contains

...
solve(F == 0, u, options_prefix="pde",
      solver_parameters={"ksp_type": "gmres"})

If we run this code as:

python pde.py

Then the KSP solver will be GMRES. Conversely, when running

python pde.py -pde_ksp_type cg

we will use conjugate gradients as the KSP solver.

Linear solver options

We use a PETSc KSP object to solve linear systems. This is a uniform interface for solving linear systems using Krylov subspace methods. By default, the solve call will use GMRES using an incomplete LU factorisation to precondition the problem. To change the Krylov method used in solving the problem, we set the 'ksp_type' option. For example, if we want to solve a modified Helmholtz equation, we know the operator is symmetric positive definite, and therefore can choose the conjugate gradient method, rather than GMRES.

solve(a == L, solver_parameters={'ksp_type': 'cg'})

To change the preconditioner used, we set the 'pc_type' option. For example, if PETSc has been installed with the Hypre package, we can use its algebraic multigrid preconditioner, BoomerAMG, to precondition the system with:

solve(a == L,
      solver_parameters={'pc_type': 'hypre',
                         'pc_hypre_type': 'boomeramg'})

Although the KSP name suggests that only Krylov methods are supported, this is not the case. We may, for example, solve the system directly by computing an LU factorisation of the problem. To do this, we set the pc_type to 'lu' and tell PETSc to use a “preconditioner only” Krylov method:

solve(a == L,
      solver_parameters={'ksp_type': 'preonly',
                         'pc_type': 'lu'})

In a similar manner, we can use Jacobi preconditioned Richardson iterations with:

solve(a == L,
      solver_parameters={'ksp_type': 'richardson',
                         'pc_type': 'jacobi'}

Note

We note in passing that the method Firedrake utilises internally for applying strong boundary conditions does not destroy the symmetry of the linear operator. If the system without boundary conditions is symmetric, it will continue to be so after the application of any boundary conditions.

Setting solver tolerances

In an iterative solver, such as Krylov method, we iterate until some specified tolerance is reached. The measure of how much the current solution \(\vec{x}_i\) differs from the true solution is called the residual and is calculated as:

\[r = |\vec{b} - A \vec{x}_i|\]

PETSc allows us to set three different tolerance options for solving the system. The absolute tolerance tells us we should stop if \(r\) drops below some given value. The relative tolerance tells us we should stop if \(\frac{r}{|\vec{b}|}\) drops below some given value. Finally, PETSc can detect divergence in a linear solve, that is, if \(r\) increases above some specified value. These values are set with the options 'ksp_atol' for the absolute tolerance, 'ksp_rtol' for the relative tolerance, and 'ksp_divtol' for the divergence tolerance. The values provided to these options should be floats. For example, to set the absolute tolerance to \(10^{-30}\), the relative tolerance to \(10^{-9}\) and the divergence tolerance to \(10^4\) we would use:

solver_parameters={'ksp_atol': 1e-30,
                   'ksp_rtol': 1e-9,
                   'ksp_divtol': 1e4}

Note

By default, PETSc (and hence Firedrake) check for the convergence in the preconditioned norm, that is, if the system is preconditioned with a matrix \(P\) the residual is calculated as:

\[r = |P^{-1}(\vec{b} - A \vec{x}_i)|\]

to check for convergence in the unpreconditioned norm set the 'ksp_norm_type' option to 'unpreconditioned'.

Finally, we can set the maximum allowed number of iterations for the Krylov method by using the 'ksp_max_it' option.

Preconditioning mixed finite element systems

PETSc provides an interface to composing “physics-based” preconditioners for mixed systems which Firedrake exploits when it assembles linear systems. In particular, for systems with two variables (for example Navier-Stokes where we solve for the velocity and pressure of the fluid), we can exploit PETSc’s ability to build preconditioners from Schur complements. This is one type of preconditioner based on PETSc’s fieldsplit technology. To take a concrete example, let us consider solving the dual form of the modified Helmholtz equation:

\[\langle p, q \rangle - \langle q, \mathrm{div} u \rangle + \lambda \langle v, u \rangle + \langle \mathrm{div}v, p \rangle = \langle f, q \rangle \; \forall v \in V_1, q \in V_2\]

This has a stable solution if, for example, \(V_1\) is the lowest order Raviart-Thomas space and \(V_2\) is the lowest order discontinuous space.

V1 = FunctionSpace(mesh, 'RT', 1)
V2 = FunctionSpace(mesh, 'DG', 0)
W = V1 * V2
lmbda = 1
u, p = TrialFunctions(W)
v, q = TestFunctions(W)
f = Function(V2)

a = (p*q - q*div(u) + lmbda*inner(v, u) + div(v)*p)*dx
L = f*q*dx

u = Function(W)
solve(a == L, u,
      solver_parameters={'ksp_type': 'cg',
                         'pc_type': 'fieldsplit',
                         'pc_fieldsplit_type': 'schur',
                         'pc_fieldsplit_schur_fact_type': 'FULL',
                         'fieldsplit_0_ksp_type': 'cg',
                         'fieldsplit_1_ksp_type': 'cg'})

We refer to section 4.5 of the PETSc manual for more complete details, but briefly describe the options in use here. The monolithic system is conceptually a \(2\times2\) block matrix:

\[\begin{split}\left(\begin{matrix} \lambda \langle v, u \rangle & -\langle q, \mathrm{div} u \rangle \\ \langle \mathrm{div} v, p \rangle & \langle p, q \rangle \end{matrix} \right) = \left(\begin{matrix} A & B \\ C & D \end{matrix}\right).\end{split}\]

We can factor this block matrix in the following way:

\[\begin{split}\left(\begin{matrix} I & 0 \\ C A^{-1} & I\end{matrix}\right) \left(\begin{matrix}A & 0 \\ 0 & S\end{matrix}\right) \left(\begin{matrix} I & A^{-1} B \\ 0 & I\end{matrix}\right).\end{split}\]

This is the Schur complement factorisation of the block system, its inverse is:

\[\begin{split}P = \left(\begin{matrix} I & -A^{-1}B \\ 0 & I \end{matrix}\right) \left(\begin{matrix} A^{-1} & 0 \\ 0 & S^{-1}\end{matrix}\right) \left(\begin{matrix} I & 0 \\ -CA^{-1} & I\end{matrix}\right).\end{split}\]

Where \(S\) is the Schur complement:

\[S = D - C A^{-1} B.\]

The options in the example above use an approximation to \(P\) to precondition the system. To do so, we tell PETSc that the preconditioner should be of type 'fieldsplit', and the the fieldsplit’s type should be 'schur'. We then select a factorisation type for the Schur complement. The option 'FULL' as used above preconditions using an approximation to \(P\). We can also use 'diag' which uses an approximation to:

\[\begin{split}\left(\begin{matrix} A^{-1} & 0 \\ 0 & -S^{-1} \end{matrix}\right).\end{split}\]

Note the minus sign in front of \(S^{-1}\) which is there such that this preconditioner is positive definite. Two other options are 'lower', where the preconditioner is an approximation to:

\[\begin{split}\left(\begin{matrix}A & 0 \\ C & S\end{matrix}\right)^{-1} = \left(\begin{matrix}A^{-1} & 0 \\ 0 & S^{-1}\end{matrix}\right) \left(\begin{matrix}I & 0 \\ -C A^{-1} & I\end{matrix}\right)\end{split}\]

and 'upper' which uses:

\[\begin{split}\left(\begin{matrix}A & B \\ 0 & S\end{matrix}\right)^{-1} = \left(\begin{matrix}I & -A^{-1}B \\ 0 & I\end{matrix}\right) \left(\begin{matrix}A^{-1} & 0 \\ 0 & S^{-1}\end{matrix}\right).\end{split}\]

Note that the inverses of \(A\) and \(S\) are never formed explicitly by PETSc, instead their actions are computed approximately using a Krylov method. The choice of method is selected using the 'fieldsplit_0_ksp_type' option (for the Krylov solver computing \(A^{-1}\)) and 'fieldsplit_1_ksp_type' (for the Krylov solver computing \(S^{-1}\)).

Note

If you have given your FunctionSpaces names, then instead of 0 and 1, you should use the name of the function space in these options.

By default PETSc uses an approximation to \(D^{-1}\) to precondition the Krylov system solving for \(S\), you can also use a least squares commutator, see the relevant section of the PETSc manual pages for more details.

Specifying assembled matrix types

Firedrake supports the assembly of linear operators in a number of different formats. Either as assembled sparse matrices, or as matrix-free operators that only provide matrix-vector products. Since matrix-free actions require special preconditioning, they have their own section in the manual. Even in the sparse matrix case there are a few options. Firedrake can build nested block matrices, or monolithic sparse matrices. The latter admit a wider range of preconditioners, but are memory-inefficient when using fieldsplit preconditioning. Even monolithic matrices have choices, specifically, if there is a block structure, whether that should be exploited or not. Again the trade-off is between memory efficiency (in the block case) and access to a slightly smaller range of preconditioners.

The default matrix type can be set with the global parameter parameters["default_matrix_type"]. In the case where the matrix is assembled as a nested matrix, there is a choice as to the type of the blocks (they may be “aij” or “baij”). The default choice can be controlled with parameters["default_sub_matrix_type"]. For finer-grained control over the matrix type, one can provide it when calling assemble() through the mat_type and sub_mat_type keyword arguments. When using variational solvers, the matrix type is controlled through use of the solver_parameters dictionary by specifying the "mat_type" entry.

Note

It is not currently possible to control the matrix type of sub-matrices through the solving interface. If you need this functionality, please get in touch

More block preconditioners

As well as physics-based Schur complement preconditioners for block systems, PETSc also allows us to use preconditioners formed from block Jacobi ('pc_fieldsplit_type': 'additive') and block Gauss-Seidel ('multiplicative' or 'symmetric_multiplicative') inverses of the block system. These work for any number of blocks, whereas the Schur complement approach mentioned above only works for two by two blocks. There is also a separate manual section on specifying preconditioners that require auxiliary operators.

Recursive fieldsplits

If your system contains more than two fields, it is possible to recursively define block preconditioners by specifying the fields which should belong to each split. Note that at present this only works for “monolithically assembled” matrices, so you should set the solver parameter "mat_type" to "aij" when solving your system or assembling your matrix. To change the default assembly from nested matrices to monolithically assembled matrices, set the global parameter parameters["default_matrix_type"] = "aij".

As an example, consider a three field system which we wish to precondition by forming a schur complement of the first two fields into the third, and then using a multiplicative fieldsplit with LU on each split for the approximation to \(A^{-1}\) and ILU to precondition the schur complement. The solver parameters we need are as follows:

parameters = {"pc_type": "fieldsplit",
              "pc_fieldsplit_type": "schur",
              # first split contains first two fields, second
              # contains the third
              "pc_fieldsplit_0_fields": "0, 1",
              "pc_fieldsplit_1_fields": "2",
              # Multiplicative fieldsplit for first field
              "fieldsplit_0_pc_type": "fieldsplit",
              "fieldsplit_0_pc_fieldsplit_type": "multiplicative",
              # LU on each field
              "fieldsplit_0_fieldsplit_0_pc_type": "lu",
              "fieldsplit_0_fieldsplit_1_pc_type": "lu",
              # ILU on the schur complement block
              "fieldsplit_1_pc_type": "ilu"}

In this example, none of the FunctionSpaces used had names, and hence we referred to the fields by number. If the function spaces are named, then any time a single field appears as a split, its options prefix is referred to by the space’s name (rather than a number). Concretely, if the previous example had use a set of FunctionSpace definitions:

V = FunctionSpace(..., name="V")
P = FunctionSpace(..., name="P")
T = FunctionSpace(..., name="T")
W = V*P*T

Then we would have referred to the single (field 1) split using fieldsplit_T_pc_type, rather than fieldsplit_1_pc_type.

Specifying nested options blocks

For complex nested preconditioners, it can be tedious to write out the same prefix over and over. Moreover, we may have a block system where multiple blocks use the same preconditioning options. It is then error-prone to type these options out twice. To alleviate these problems, one can describe the nesting in the solver parameters dictionary by using a nested dict as the value. In this case, the key is used as an options prefix to all of the key-value pairs in the nested dictionary. As an example, the following two parameter sets are equivalent:

{"ksp_type": "cg",
 "pc_type": "fieldsplit",
 "fieldsplit_0": {"ksp_type": "gmres",
                  "pc_type": "hypre",
                   "ksp_rtol": 1e-5},
 "fieldsplit_1": {"ksp_type": "richardson",
                  "pc_type": "ilu"}}

and

{"ksp_type": "cg",
 "pc_type": "fieldsplit",
 "fieldsplit_0_ksp_type": "gmres",
 "fieldsplit_0_pc_type": "hypre",
 "fieldsplit_0_ksp_rtol": 1e-5,
 "fieldsplit_1_ksp_type": "richardson",
 "fieldsplit_1_pc_type": "ilu"}

PETSc uses an underscore as a separator between option names, and we do the same. For convenience, the prefix key to a nested dict can omit the trailing underscore, it will be added automatically if missing. Hence

{"a": {"b": "foo"}}

and

{"a_": {"b": "foo"}}

both expand to

{"a_b": "foo"}

Nonlinear solver options

As for linear systems, we use a PETSc object to solve nonlinear systems. This time it is a SNES. This offers a uniform interface to Newton-like and quasi-Newton solution schemes. To select the SNES type to use, we use the 'snes_type' option. Recall that each Newton iteration is the solution of a linear system, options for the inner linear solve may be set in the same way as described above for linear problems. For example, to solve a nonlinear problem using Newton-Krylov iterations using a line search and direct factorisation to solve the linear system we would write:

solve(F == 0, u,
      solver_parameters={'snes_type': 'newtonls',
                         'ksp_type': 'preonly',
                         'pc_type': 'lu'}

Note

Not all of PETSc’s SNES types are currently supported by Firedrake, since some of them require extra information which we do not currently provide.

Setting convergence criteria

In addition to setting the tolerances for the inner, linear solve in a nonlinear system, which is done in exactly the same way as for linear problems, we can also set convergence tolerances on the outer SNES object. These are the absolute tolerance ('snes_atol'), relative tolerance ('snes_rtol'), step tolerance ('snes_stol') along with the maximum number of nonlinear iterations ('snes_max_it') and the maximum number of allowed function evaluations ('snes_max_func'). The step tolerance checks for convergence due to:

\[|\Delta x_k| < \mathrm{stol} \, |x_k|\]

The maximum number of allowed function evaluations limits the number of times the residual may be evaluated before returning a non-convergence error, and defaults to 1000.

Providing an operator for preconditioning

By default, Firedrake uses the Jacobian of the residual (or equally the bilinear form for linear problems) to construct preconditioners for the linear systems it solves. That is, it does not directly solve:

\[A \vec{x} = \vec{b}\]

but rather

\[\tilde{A}^{-1} A \vec{x} = \tilde{A}^{-1} \vec{b}\]

where \(\tilde{A}^{-1}\) is an approximation to \(A^{-1}\). If we know something about the structure of our problem, we may be able to construct an operator \(P\) explicitly which is “easy” to invert, and whose inverse approximates \(A^{-1}\) well. Firedrake allows you to provide this operator when solving variational problems by passing an explicit Jp keyword argument to the solve call, the provided form will then be used to construct an approximate inverse when preconditioning the problem, rather than the form we’re solving with.

a = ...
L = ...
Jp = ...
# Use the approximate inverse of Jp to precondition solves
solve(a == L, ..., Jp=Jp)

Default solver options

If no parameters are passed to a solve call, we use, in most cases, the defaults that PETSc supplies for solving the linear or nonlinear system. We describe the most commonly modified options (along with their defaults in Firedrake) here. For linear variational solves we use:

  • ksp_type: GMRES, with a restart (ksp_gmres_restart) of 30
  • ksp_rtol: 1e-7
  • ksp_atol: 1e-50
  • ksp_divtol 1e4
  • ksp_max_it: 10000
  • pc_type: ILU (Jacobi preconditioning for mixed problems)

For nonlinear variational solves we have:

  • snes_type: Newton linesearch
  • ksp_type: GMRES, with a restart (ksp_gmres_restart) of 30
  • snes_rtol: 1e-8
  • snes_atol: 1e-50
  • snes_stol: 1e-8
  • snes_max_it: 50
  • ksp_rtol: 1e-5
  • ksp_atol: 1e-50
  • ksp_divtol: 1e4
  • ksp_max_it: 10000
  • pc_type: ILU (Jacobi preconditioning for mixed problems)

To see the full view that PETSc has of solver objects, you can pass a view flag to the solve call. For linear solves pass:

solver_parameters={'ksp_view': True}

For nonlinear solves use:

solver_parameters={'snes_view': True}

PETSc will then print its view of the solver objects that Firedrake has constructed. This is especially useful for debugging complicated preconditioner setups for mixed problems.

Solving singular systems

Some systems of PDEs, for example the Poisson equation with pure Neumann boundary conditions, have an operator which is singular. That is, we have \(Ae = 0\) with \(e \neq 0\). The vector space spanned by the set of vectors \({e}\) for which \(Ae = 0\) is termed the null space of \(A\). If we wish to solve such a system, we must remove the null space from the solution. To do this in Firedrake, we first must define the null space, and then inform the solver of its existance. We use a VectorSpaceBasis to hold the vectors which span the null space. We must provide a list of Functions or Vectors spanning the space. Additionally, since removing a constant null space is such a common operation, we can pass constant=True to the constructor (rather than constructing the constant vector by hand). Note that the vectors we pass in must be orthonormal. Once the null space is built, we just need to inform the solve about it (using the nullspace keyword argument).

As an example, consider the Poisson equation with pure Neumann boundary conditions:

\[\begin{split}-\nabla^2 u &= 0 \quad \mathrm{in}\;\Omega\\ \nabla u \cdot n &= g \quad \mathrm{on}\;\Gamma.\end{split}\]

We will solve this problem on the unit square applying homogeneous Neumann boundary conditions on the planes \(x = 0\) and \(x = 1\). On \(y = 0\) we set \(g = -1\) while on \(y = 1\) we set \(g = 1\). The null space of the operator we form is the set of constant functions, and thus the problem has solution \(u(x, y) = y + c\) where \(c\) is a constant. To solve the problem, we will inform the solver of this constant null space, fixing the solution to be \(u(x, y) = y - 0.5\).

m = UnitSquareMesh(25, 25)
V = FunctionSpace(m, 'CG', 1)
u = TrialFunction(V)
v = TestFunction(V)

a = inner(grad(u), grad(v))*dx
L = -v*ds(3) + v*ds(4)

nullspace = VectorSpaceBasis(constant=True)
u = Function(V)
solve(a == L, u, nullspace=nullspace)
x = SpatialCoordinate(m)
exact = Function(V).interpolate(x[1] - 0.5)
print sqrt(assemble((u - exact)*(u - exact)*dx))

For this to work, the provided right hand side must be orthogonal to the transpose nullspace of the operator as well. In many cases, we can arrange for this to occur by careful choice of initial conditions. Sometimes this is not possible. In this case, you can ask Firedrake to remove the component of the right hand side that is in the transpose nullspace by providing a VectorSpaceBasis with the transpose_nullspace keyword argument to solve().

Singular operators in mixed spaces

If you have an operator in a mixed space, you may well precondition the system using a Schur complement. If the operator is singular, you will therefore have to tell the solver about the null space of each diagonal block separately. To do this in Firedrake, we build a MixedVectorSpaceBasis instead of a VectorSpaceBasis and then inform the solver about it as before. A MixedVectorSpaceBasis takes a list of VectorSpaceBasis objects defining the null spaces of each of the diagonal blocks in the mixed operator. In addition, as a first argument, you must provide the MixedFunctionSpace you’re building a basis for. You do not have to provide a null space for all blocks. For those you don’t care about, you can pass an indexed function space at the appropriate position. For example, imagine we have a mixed space \(W = V \times Q\) and an operator which has a null space of constant functions in \(V\) (this occurs, for example, for a discretisation of the mixed poisson problem on the surface of a sphere). We can specify the null space (indicating that we only really care about the constant function) as:

V = ...
Q = ...
W = V*Q
v_basis = VectorSpaceBasis(constant=True)
nullspace = MixedVectorSpaceBasis(W, [v_basis, W.sub(1)])

Debugging convergence failures

Occasionally, we will set up a problem and call solve only to be confronted with an error that the solve failed to converge. Here, we discuss some useful techniques to try and understand the reason. Much of the advice in the PETSc FAQ is useful here, especially the sections on SNES nonconvergence and KSP nonconvergence. We first consider linear problems.

Linear convergence failures

If the linear operator is correct, but the solve fails to converge, it is likely the case that the problem is badly conditioned (leading to slow convergence) or a symmetric method is being used (such as conjugate gradient) where the problem is non-symmetric. The first thing to check is what happened to the residual (error) term. To monitor this in the solution we pass the “flag” options 'ksp_converged_reason' and 'ksp_monitor_true_residual', additionally, we pass ksp_view so that PETSc prints its idea of what the solver object contains (this is useful to debug the where options are not being passed in correctly):

solver_parameters={'ksp_converged_reason': True,
                   'ksp_monitor_true_residual': True,
                   'ksp_view': True}

If the problem is converging, but only slowly, it may be that it is badly conditioned. If the problem is small, we can try using a direct solve to see if the solution obtained is correct:

solver_parameters={'ksp_type': 'preonly', 'pc_type': 'lu'}

If this approach fails with a “zero-pivot” error, it is likely that the equations are singular, or nearly so, check to see if boundary conditions have been imposed correctly.

If the problem converges with a direct method to the correct solution but does not converge with a Krylov method, it’s probable that the conditioning is bad. If it’s a mixed problem, try using a physics-based preconditioner as described above, if not maybe try using an algebraic multigrid preconditioner. If PETSc was installed with Hypre use:

solver_parameters={'pc_type': 'hypre', 'pc_hypre_type': 'boomeramg'}

If you’re using a symmetric method, such as conjugate gradient, check that the linear operator is actually symmetric, which you can compute with the following:

A = assemble(a)  # use bcs keyword if there are boundary conditions
print A.M.handle.isSymmetric(tol=1e-13)

If the problem is not symmetric, try using a method such as GMRES instead. PETSc uses restarted GMRES with a default restart of 30, for difficult problems this might be too low, in which case, you can increase the restart length with:

solver_parameters={'ksp_gmres_restart': 100}

Nonlinear convergence failures

Much of the advice for linear systems applies to nonlinear systems as well. If you have a convergence failure for a nonlinear problem, the first thing to do is run with monitors to see what is going on, and view the SNES object with snes_view to ensure that PETSc is seeing the correct options:

solver_parameters={'snes_monitor': True,
                   'snes_view': True,
                   'ksp_monitor_true_residual': True,
                   'snes_converged_reason': True,
                   'ksp_converged_reason': True}

If the linear solve fails to converge, debug the problem as above for linear systems. If the linear solve converges but the outer Newton iterations do not, the problem is likely a bad Jacobian. If you provided the Jacobian by hand, is it correct? If no Jacobian was provided in the solve call, it is likely a bug in Firedrake and you should report it to us.

Checking the provided Jacobian

It is possible to verify that the provided Jacobian is consistent with the residual we are trying to minimise by comparing it with a finite differenced Jacobian computed by PETSc. This is possible using only a few extra options to the call to solve(). We just need to specify that the nonlinear solver we want PETSc to employ should be of type test. PETSc will then go away, compute an approximate Jacobian by finite differencing the residual and compare it to our provided exact Jacobian. The only thing we need to be aware of is that if the problem to be solved is in a mixed space, we need to set the solver parameter "mat_type" to "aij" in the solve call.

To make things concrete, consider the following, somewhat contrived, example where we attempt to solve a Galerkin projection in a mixed space, but provide an incorrectly scaled Jacobian to the solve.

from firedrake import *
mesh = UnitSquareMesh(1, 1)
V = FunctionSpace(mesh, "CG", 1)
W = V*V
f = Function(W)
v = TestFunction(W)
u = TrialFunction(W)

F = dot(f, v)*dx - dot(Constant((1, 2)), v)*dx

J = Constant(4)*dot(u, v)*dx

solve(F == 0, f, J=J)

When run, this produces the following output:

pyop2:INFO Solving nonlinear variational problem...
Traceback (most recent call last):
    solve(F == 0, u, J=J)
  File "firedrake/solving.py", line 120, in solve
    _solve_varproblem(*args, **kwargs)
  File "firedrake/solving.py", line 162, in _solve_varproblem
    solver.solve()
  File "<string>", line 2, in solve
  File "pyop2/profiling.py", line 203, in wrapper
    return f(*args, **kwargs)
  File "firedrake/variational_solver.py", line 175, in solve
    solving_utils.check_snes_convergence(self.snes)
  File "firedrake/solving_utils.py", line 62, in check_snes_convergence
    """%s""" % (snes.getIterationNumber(), msg))
RuntimeError: Nonlinear solve failed to converge after 50 nonlinear iterations.
Reason:
    DIVERGED_MAX_IT

In this example we can notice by inspection of the code that the provided Jacobian is incorrect. The Gateaux derivative of \(F\) with respect to \(f\) is \(\langle u, v \rangle\), not \(4\langle u, v \rangle\). In the more general case, it may be that there is a bug in the assembly of the Jacobian, even if the symbolic form is correct. To verify the Jacobian we rerun the solve, but pass some additional options:

solve(F == 0, f, J=J,
      solver_parameters={'snes_type': 'test',
                         'mat_type': 'aij'})

This time we get the following output

pyop2:INFO Solving nonlinear variational problem...
Testing hand-coded Jacobian, if the ratio is
O(1.e-8), the hand-coded Jacobian is probably correct.
Run with -snes_test_display to show difference
of hand-coded and finite difference Jacobian.
Norm of matrix ratio 0.75, difference 1.32288 (user-defined state)
Norm of matrix ratio 0.75, difference 1.32288 (constant state -1.0)
Norm of matrix ratio 0.75, difference 1.32288 (constant state 1.0)
Traceback (most recent call last):
    solve(F == 0, u, J=J, solver_parameters={'snes_type': 'test', 'mat_type': 'aij'})
  File "firedrake/solving.py", line 120, in solve
    _solve_varproblem(*args, **kwargs)
  File "firedrake/solving.py", line 162, in _solve_varproblem
    solver.solve()
  File "<string>", line 2, in solve
  File "pyop2/profiling.py", line 203, in wrapper
    return f(*args, **kwargs)
  File "firedrake/variational_solver.py", line 173, in solve
    self.snes.solve(None, v)
  File "PETSc/SNES.pyx", line 520, in petsc4py.PETSc.SNES.solve (src/petsc4py.PETSc.c:165224)
petsc4py.PETSc.Error: error code 73
[0] SNESSolve() line 3907 in petsc/src/snes/interface/snes.c
[0] SNESSolve_Test() line 127 in petsc/src/snes/impls/test/snestest.c
[0] Object is in wrong state
[0] SNESTest aborts after Jacobian test: it is NORMAL behavior.

The important lines are:

Testing hand-coded Jacobian, if the ratio is
O(1.e-8), the hand-coded Jacobian is probably correct.
Run with -snes_test_display to show difference
of hand-coded and finite difference Jacobian.
Norm of matrix ratio 0.75, difference 1.32288 (user-defined state)
Norm of matrix ratio 0.75, difference 1.32288 (constant state -1.0)
Norm of matrix ratio 0.75, difference 1.32288 (constant state 1.0)

Here PETSc is printing information about the difference between the finite difference and provided Jacobians. We can see that these differences are large. Therefore, we conclude the the provided “exact” Jacobian is not consistent with the residual, and likely incorrect.

For comparison, here are the same relevant lines when running with the correct Jacobian:

solve(F == 0, f, solver_parameters={'snes_type': 'test', 'mat_type': 'aij'})

Testing hand-coded Jacobian, if the ratio is
O(1.e-8), the hand-coded Jacobian is probably correct.
Run with -snes_test_display to show difference
of hand-coded and finite difference Jacobian.
Norm of matrix ratio 4.98807e-08, difference 2.19953e-08 (user-defined state)
Norm of matrix ratio 2.91936e-08, difference 1.28732e-08 (constant state -1.0)
Norm of matrix ratio 1.51242e-08, difference 6.66915e-09 (constant state 1.0)

Notice how now the differences are small (within expected error tolerances) so we are happy that the Jacobian is correct.