Defining variational problems¶
Firedrake uses a highlevel language, UFL, to describe variational
problems. To do this, we need a number of pieces. We need a
representation of the domain we’re solving the PDE on: Firedrake uses a
Mesh()
for this. On top of this mesh,
we build FunctionSpace
s which
define the space in which the solutions to our equation live. Finally
we define Function
s in those
function spaces to actually hold the solutions.
Constructing meshes¶
Firedrake can read meshes in Gmsh, triangle, CGNS, and
Exodus formats. To build a mesh one uses the Mesh()
constructor, passing the name of the file as an argument, which see
for more details. The mesh type is determined by the file extension,
for example if the provided filename is coastline.msh
the mesh is
assumed to be in Gmsh format, in which case you can construct a mesh
object like so:
coastline = Mesh("coastline.msh")
This works in both serial and parallel, Firedrake takes care of decomposing the mesh among processors transparently.
Reordering meshes for better performance¶
Most mesh generators produce badly numbered meshes (with bad data
locality) which can reduce the performance of assembling and solving
finite element problems. By default then, Firedrake reorders input
meshes to improve data locality by performing reverse CuthillMcKee
reordering on the adjacency matrix of the input mesh. If you know
your mesh has a good numbering (perhaps your mesh generator uses space
filling curves to number entities) then you can switch off this
reordering by passing reorder=False
to the appropriate
Mesh()
constructor. You can control Firedrake’s default
behaviour in reordering meshes with the "reorder_meshes"
parameter. For example, to turn off mesh reordering globally:
from firedrake import *
parameters["reorder_meshes"] = False
The parameter passed in to the mesh constructor overrides this default value.
Note
Firedrake numbers degrees of freedom in a function space by visiting each cell in order and performing a depth first numbering of all degrees of freedom on that cell. Hence, if your mesh has a good numbering, the degrees of freedom will too.
Utility mesh functions¶
As well as offering the ability to read mesh information from a file,
Firedrake also provides a number of built in mesh types for a number
of standard shapes. 1dimensional intervals may be constructed with
IntervalMesh()
; 2dimensional rectangles with
RectangleMesh()
; and 3dimensional boxes with
BoxMesh()
. There are also more specific constructors (for
example to build unit square meshes). See
utility_meshes
for full details.
Immersed manifolds¶
In addition to the simple meshes described above, Firedrake also has support for solving problems on orientable immersed manifolds. That is, meshes in which the entities are immersed in a higher dimensional space. For example, the surface of a sphere in 3D.
If your mesh is such an immersed manifold, you need to tell Firedrake that the geometric dimension of the coordinate field (defining where the points in mesh are) is not the same as the topological dimension of the mesh entities. This is done by passing an optional second argument to the mesh constructor which specifies the geometric dimension. For example, for the surface of a sphere embedded in 3D we use:
sphere_mesh = Mesh('sphere_mesh.node', dim=3)
Firedrake provides utility meshes for the surfaces of spheres immersed
in 3D that are approximated using an icosahedral mesh. You can
either build a mesh of the unit sphere with
UnitIcosahedralSphereMesh()
, or a mesh of a
sphere with specified radius using
IcosahedralSphereMesh()
. The meshes are
constructed by recursively refining a regular icosahedron, you can specify the refinement level by passing a
nonzero refinement_level
to the constructor. For example, to
build a sphere mesh that approximates the surface of the Earth (with a
radius of 6371 km) that has subdivided the original icosahedron 7
times we would write:
earth = IcosahedralSphereMesh(radius=6371, refinement_level=7)
Ensuring consistent cell orientations¶
Variational forms that include particular function spaces (those
requiring a contravariant Piola transform), require information
about the orientation of the cells. For normal meshes, this can be
deduced automatically. However, when using immersed meshes, Firedrake
needs extra information to calculate the orientation of each cell
relative to some global orientation. This
is used by Firedrake to ensure that the cell normal on,
say, the surface of a sphere, uniformly points outwards. To do this,
after constructing an immersed mesh, we must initialise the cell
orientation information. This is carried out with the function
~.Mesh.init_cell_orientations
, which
takes a UFL expression used to produce
the reference normal direction. For example, on the sphere mesh of
the earth defined above we can initialise the cell orientations
relative to vector pointing out from the origin:
earth.init_cell_orientations(SpatialCoordinate(earth))
However, a more complicated expression would be needed to initialise the cell orientations on a toroidal mesh.
Semistructured extruded meshes¶
Firedrake has special support for solving PDEs on highaspect ratio domains, such as in the ocean or atmosphere, where the numerics dictate that the “short” dimension should be structured. These are termed extruded meshes and have a separate section in the manual.
Building function spaces¶
Now that we have a mesh of our domain, we need to build the function
spaces the solution to our PDE
will live in, along with the spaces for the trial and test functions.
To do so, we use the FunctionSpace()
constructor.
This is the only way to obtain a function space for a scalar variable,
such as pressure, which has a single value at each point in the
domain.
To construct a function space, you must specify its family and polynomial degree. To build a scalarvalued function space of continuous piecewisecubic polynomials, we write:
V = FunctionSpace(mesh, "Lagrange", 3)
There are three main routes to obtaining a function space for a
vectorvalued variable such as velocity. Firstly, you can pass the
FunctionSpace()
constructor a natively vectorvalued
family such as "RaviartThomas"
. Secondly, you may use the
VectorFunctionSpace()
constructor with a scalarvalued
family, which gives a vectorvalued space where each component is
identical to the appropriate scalarvalued
FunctionSpace
. Thirdly, you can create a
VectorElement
directly (which is itself
vectorvalued and pass that to the FunctionSpace()
constructor).
To build a vectorvalued function space using the lowestorder
RaviartThomas
elements, we write
V = FunctionSpace(mesh, "RaviartThomas", 1)
To build a vectorvalued function space for which each component is a discontinuous piecewisequadratic polynomial, we can write either
V = VectorFunctionSpace(mesh, "Discontinuous Lagrange", 2)
or
Vele = VectorElement("Discontinuous Lagrange", cell=mesh.ufl_cell(), degree=2)
V = FunctionSpace(mesh, Vele)
Advanced usage of VectorFunctionSpace
¶
By default, the number of components of a
VectorFunctionSpace()
is the geometric dimension of the
mesh (e.g. 3, if the mesh is 3D). However, sometimes we might want
the number of components in the vector to differ from the geometric
dimension of the mesh. We can do this by passing a value for the
dim
argument to the VectorFunctionSpace()
constructor.
For example, if we wanted a vectorvalued function space on the surface
of a unit sphere mesh with only 2 components, we might write:
mesh = UnitIcosahedralSphereMesh(refinement_level=3)
V = VectorFunctionSpace(mesh, "Lagrange", 1, dim=2)
Mixed function spaces¶
Many PDEs are posed in terms
of multiple, coupled, variables. The variational problem for such a
PDE uses a socalled mixed function space. In Firedrake, this is
represented by a MixedFunctionSpace
. We can either
build such a space by invoking the constructor directly
, or, more readably, by taking existing function
spaces and multiplying them together using the *
operator. For
example:
V = FunctionSpace(mesh, 'RT', 1)
Q = FunctionSpace(mesh, 'DG', 0)
W = V*Q
is equivalent to:
V = FunctionSpace(mesh, 'RT', 1)
Q = FunctionSpace(mesh, 'DG', 0)
W = MixedFunctionSpace([V, Q])
Function spaces on extruded meshes¶
On extruded meshes, we build function spaces by taking a tensor product of the base (“horizontal”) space and the extruded (“vertical”) space. Firedrake allows us to separately choose the horizontal and vertical spaces when building a function space on an extruded mesh. We refer the reader to the manual section on extrusion for details.
Supported finite elements¶
Firedrake supports the use of the following finite elements.
Name 
Short name 
Value shape 
Valid cells 

Bernstein 
scalar 
interval, triangle, tetrahedron 

BernardiRaugel 
BR 
vector 
triangle, tetrahedron 
BernardiRaugel Bubble 
BRB 
vector 
triangle, tetrahedron 
BrezziDouglasMarini 
BDM 
vector 
triangle, tetrahedron 
BrezziDouglasFortinMarini 
BDFM 
vector 
triangle, tetrahedron 
Bubble 
B 
scalar 
interval, triangle, tetrahedron 
FacetBubble 
FB 
scalar 
interval, triangle, tetrahedron 
CrouzeixRaviart 
CR 
scalar 
triangle, tetrahedron 
Discontinuous Lagrange 
DG 
scalar 
interval, triangle, tetrahedron, quadrilateral, hexahedron 
Discontinuous RaviartThomas 
DRT 
vector 
triangle, tetrahedron 
Discontinuous Taylor 
TDG 
scalar 
interval, triangle, tetrahedron 
GaussLegendre 
GL 
scalar 
interval 
GaussLobattoLegendre 
GLL 
scalar 
interval 
HDiv Trace 
HDivT 
scalar 
interval, triangle, tetrahedron, quadrilateral, hexahedron 
HellanHerrmannJohnson 
HHJ 
tensor 
triangle 
JohnsonMercier 
JM 
tensor 
triangle, tetrahedron 
Nonconforming ArnoldWinther 
AWnc 
tensor 
triangle, tetrahedron 
Conforming ArnoldWinther 
AWc 
tensor 
triangle, tetrahedron 
Hermite 
HER 
scalar 
interval, triangle, tetrahedron 
KongMulderVeldhuizen 
KMV 
scalar 
triangle, tetrahedron 
Argyris 
ARG 
scalar 
triangle 
HsiehCloughTocher 
HCT 
scalar 
triangle 
QuadraticPowellSabin6 
PS6 
scalar 
triangle 
QuadraticPowellSabin12 
PS12 
scalar 
triangle 
ReducedHsiehCloughTocher 
HCTred 
scalar 
triangle 
MardalTaiWinther 
MTW 
vector 
triangle 
AlfeldSorokina 
AS 
vector 
triangle, tetrahedron 
ArnoldQin 
AQ 
vector 
triangle 
ReducedArnoldQin 
AQred 
vector 
triangle 
ChristiansenHu 
CH 
vector 
triangle, tetrahedron 
GuzmanNeilan 
GN 
vector 
triangle, tetrahedron 
GuzmanNeilan Bubble 
GNB 
vector 
triangle, tetrahedron 
Morley 
MOR 
scalar 
triangle 
Bell 
BELL 
scalar 
triangle 
Lagrange 
CG 
scalar 
interval, triangle, tetrahedron, quadrilateral, hexahedron 
Nedelec 1st kind H(curl) 
N1curl 
vector 
triangle, tetrahedron 
Nedelec 2nd kind H(curl) 
N2curl 
vector 
triangle, tetrahedron 
RaviartThomas 
RT 
vector 
triangle, tetrahedron 
Regge 
tensor 
triangle, tetrahedron 

BDMCE 
vector 
quadrilateral 

BDMCF 
vector 
quadrilateral 

DQ 
scalar 
interval, quadrilateral, hexahedron 

Q 
scalar 
interval, quadrilateral, hexahedron 

RTCE 
vector 
quadrilateral 

RTCF 
vector 
quadrilateral 

NCE 
vector 
hexahedron 

NCF 
vector 
hexahedron 

Real 
R 
scalar 
interval, triangle, tetrahedron, quadrilateral, hexahedron 
DPC 
scalar 
interval, quadrilateral, hexahedron 

S 
scalar 
interval, quadrilateral, hexahedron 

SminusF 
vector 
quadrilateral 

SminusDiv 
vector 
quadrilateral, hexahedron 

SminusE 
vector 
quadrilateral, hexahedron 

SminusCurl 
vector 
quadrilateral, hexahedron 

DPC L2 
scalar 
interval, quadrilateral, hexahedron 

Discontinuous Lagrange L2 
DG L2 
scalar 
interval, triangle, tetrahedron, quadrilateral, hexahedron 
GaussLegendre L2 
GL L2 
scalar 
interval 
DQ L2 
scalar 
interval, quadrilateral, hexahedron 

Direct Serendipity 
Sdirect 
scalar 
quadrilateral 
In addition, the
TensorProductElement
operator can be used to create product elements on extruded meshes.
Element variants¶
Some finite element spaces offer more than one choice of nodes. For Q, DQ, DQ
L2, RTCE, RTCF, NCE, and NCF spaces on intervals, quadrilaterals and hexahedra,
Firedrake offers both equispaced points and better conditioned Legendre points.
For discontinuous elements these are the GaussLegendre points, and for
continuous elements these are the GaussLobattoLegendre points.
For CG and DG spaces on simplices, Firedrake offers both equispaced points and
the better conditioned recursive Legendre points from [Isa20] via the
recursivenodes module. These are selected by passing variant=”equispaced”
or variant=”spectral” to the FiniteElement
or
FunctionSpace()
constructors. For example:
fe = FiniteElement("RTCE", quadrilateral, 2, variant="equispaced")
The default is the spectral variant.
Expressing a variational problem¶
Firedrake uses the UFL language to express variational problems. For complete documentation, we refer the reader to the UFL package documentation and the description of the language in TOMS. We present a brief overview of the syntax here, for a more didactic introduction, we refer the reader to the Firedrake tutorial examples.
There are two ways to express a variational problem in UFL. A linear variational problem is defined in terms of a bilinear form \(a(u,v)\) and a linear form \(L[v]\), seeking \(u\in V\) such that \(a(u,v)=L[v]\,\forall v\in V\) for some finite element space \(V\). The following section of the notes describes how such problems can be expressed in UFL and solved using Firedrake. We shall see that the solve is invoked by writing
solve(a == L, s)
but solver reuse can be achieved using LinearVariationalSolver
,
which is usually the most efficient option for timestepping problems.
A nonlinear variational problem is defined in terms of a linear form
\(F[u;v]\) which is linear in the test function \(v\) but may
be nonlinear in the coefficient \(u\). The nonlinear variational
problem seeks \(u\in V\) such that \(F[u;v]=0\, \forall v\in
V\). In UFL, the solution variable should be of type Function
instead of TrialFunction
.
solve(F == 0, s)
but solver reuse can be achieved using
NonlinearVariationalSolver
, which is usually the most
efficient option for timestepping problems. The solution approach for
this problems is some form of Newton’s method. UFL automates the
symbolic differentiation of \(F\) to obtain the Jacobian expressed
as a bilinear form, which is then solved. Note that nonlinear problems
can be linear (Firedrake and UFL will make no effort to detect this),
in which case Newton’s method will converge in one iteration.
For more details about nonlinear variational problems, see the UFL manual as well as the Firedrake tutorials.
Building test and trial spaces for linear variational problems¶
Now that we have function spaces that our solution will live in, the
next step is to actually write down the variational form of the
problem we wish to solve. To do this, we will need a test function in
an appropriate space along with a function to hold the solution and
perhaps a trial function. Test functions are obtained via a call to
TestFunction
, trial functions via
TrialFunction
and functions with
Function
. The former two are purely
symbolic objects, the latter contains storage for the coefficients of
the basis functions in the function space. We use them as follows:
u = TrialFunction(V)
v = TestFunction(V)
f = Function(V)
Note
A newly allocated Function
has
coefficients which are all zero.
If V
above were a
MixedFunctionSpace
, the test and
trial functions we obtain are for the combined mixed space. Often, we
would like to have test and trial functions for the subspaces of the
mixed space. We can do this by asking for
TrialFunctions
and
TestFunctions
, which return an ordered
tuple of test and trial functions for the underlying spaces. For
example, if we write:
V = FunctionSpace(mesh, 'RT', 1)
Q = FunctionSpace(mesh, 'DG', 0)
W = V * Q
u, p = TrialFunctions(W)
v, q = TestFunctions(W)
then u
and v
will be, respectively, trial and test
functions for V
, while p
and q
will be trial and test
functions for Q
.
Note
If we intend to build a variational problem on a mixed space, we cannot build the individual test and trial functions on the function spaces that were used to construct the mixed space directly. The functions that we build must “know” that they come from a mixed space or else Firedrake will not be able to assemble the correct system of equations.
A first variational form¶
With our test and trial functions defined, we can write down our first variational form. Let us consider solving the identity equation:
where \(\Omega\) is the unit square, using piecewise linear polynomials for our solution. We start with a mesh and build a function space on it:
mesh = UnitSquareMesh(10, 10)
V = FunctionSpace(mesh, "CG", 1)
now we need a test function, and since u
is unknown, a trial
function:
u = TrialFunction(V)
v = TestFunction(V)
finally we need a function to hold the right hand side \(f\) which we will populate with the x component of the coordinate field.
f = Function(V)
x = SpatialCoordinate(mesh)
f.interpolate(x[0])
For details on how interpolate()
works, see the
appropriate section in the manual. The
variational problem is to find \(u \in V\) such that
we define the variational problem in UFL with:
a = u*v*dx
L = f*v*dx
Where the dx
indicates that the integration should be carried out
over the cells of the mesh. UFL can also express integrals over the
boundary of the domain, using ds
, and the interior facets of the
domain, using dS
.
How to solve such variational problems is the subject of the next section, but for completeness we show how to do it here. First we define a function to hold the solution
s = Function(V)
and call solve()
to solve the variational
problem:
solve(a == L, s)
Forms with constant coefficients¶
Many PDEs will contain values that are constant over the whole mesh,
but may vary in time. For example, a timevarying diffusivity, or a
timedependent forcing function. Although you can create a new form
for each new value of this constant, this will not be efficient, since
Firedrake must generate new code each time the value changes. A
better option is to use a Constant
coefficient. This
object behaves exactly like a Function
, except that it
has a single value over the whole mesh. One may assign a new value to
the Constant
using the assign()
method. As an example, let us consider a form which contains a time
varying constant which we wish to assemble in a time loop. We can use
a Constant
to do this:
...
t = 0
dt = 0.1
from math import exp
c = Constant(exp(t))
# Exponentially decaying RHS
L = f*v*c*dx
while t < tend:
solve(a == L, ...)
t += dt
c.assign(exp(t))
Warning
Although UFL supports computing the derivative of a form with
respect to a Constant
, the resulting form will have
an unknown in the reals, which is currently unsupported by
Firedrake.
Incorporating boundary conditions¶
Boundary conditions enter the variational problem in one of two ways.
Natural (often termed Neumann or weak) boundary conditions,
which prescribe values of the derivative of the solution, are
incorporated into the variational form. Essential (often termed
Dirichlet or strong) boundary conditions, which prescribe values
of the solution, become prescriptions on the function space. In
Firedrake, the former are naturally expressed as part of the
formulation of the variational problem, the latter are represented as
DirichletBC
objects and are applied when
solving the variational problem. Construction of such a strong
boundary condition requires a function space (to impose the boundary
condition in), a value and a subdomain to apply the boundary condition
over:
bc = DirichletBC(V, value, subdomain_id)
The subdomain_id
is an integer indicating which section of the
mesh the boundary condition should be applied to. The subdomain ids
for the various utility meshes are
described in their respective constructor documentation. For
externally generated meshes, Firedrake just uses whichever ids the
mesh generator provided. The value
may be either a scalar, or
more generally a UFL expression, for example a Function
or
Constant
, of the appropriate shape. You may also supply
an iterable of literal constants:
bc = DirichletBC(V, (1.0, 2.0), 1)
Strong boundary conditions are applied in the solve by passing a list of boundary condition objects:
solve(a == L, bcs=[bc])
See the next section for a more complete description of the interface Firedrake provides to solve PDEs. The details of how Firedrake applies strong boundary conditions are slightly involved and therefore have their own section in the manual.
Boundary conditions on interior facets¶
If you wish to apply strong boundary conditions to interior facets of your mesh, this is transparently supported. You should arrange that your mesh generator marks those facets on which you wish to apply boundary conditions, and just use the subdomain ids as usual.
Special subdomain ids¶
As well as integer subdomain ids that come from marked portions of the
mesh, Firedrake also supports the magic string "on_boundary"
to
apply a boundary condition to all exterior facets of the mesh.
Further, on :doc`:extruded meshes <extrudedmeshes>` the special
strings "top"
and "bottom"
can be used to apply a boundary
condition on respectively the top and bottom of the extruded domain.
Note
These special strings cannot be combined with integer ids, so if
you want to apply boundary data on an extruded mesh on (say) ids
1
and 2
as well as the top of the domain you would write
bcs = [DirichletBC(V, ..., (1, 2)), DirichletBC(V, ..., "top")]
Specifying conditions on components of a space¶
When solving a problem defined on either a
MixedFunctionSpace
or a rank1 FunctionSpace
, it is
common to want to specify boundary values for only some of the
components. In the former case, this is the only supported method of
setting boundary values, the latter also supports setting the value
for all components. In both cases, the syntax is the same. When
defining the DirichletBC
we must index the function space
used. For example, to specify that the third component of a
VectorFunctionSpace()
should take the boundary value 0, we write:
V = VectorFunctionSpace(mesh, ...)
bc = DirichletBC(V.sub(2), Constant(0), boundary_ids)
Note that when indexing a MixedFunctionSpace
in this
manner, one pulls out the indexed subspace, rather than a component.
For example, to specify the velocity values in a TaylorHood
discretisation we write:
V = VectorFunctionSpace(mesh, "CG", 2)
P = FunctionSpace(mesh, "CG", 1)
W = V*P
bcv = DirichletBC(W.sub(0), Constant((0, 0)), boundary_ids)
If we only wanted to specify a single component, we would have to index twice. For example, specifying that the xcomponent of the velocity is zero, using the same function space definitions:
bcv_x = DirichletBC(W.sub(0).sub(0), Constant(0), boundary_ids)
Boundary conditions in discontinuous spaces¶
Firedrake uses the topological association of nodes to facets to determine where to apply strong boundary conditions. For spaces where nodes are not topologically associated with the boundary facets, such as discontinuous Galerkin spaces, you should instead apply boundary conditions weakly.
Time dependent boundary conditions¶
Imposition of timedependent boundary conditions can by carried out by
modifying the value in the appropriate DirichletBC
object. Note that if you use a literal value to initialise the
boundary condition object within the timestepping loop, this will
necessitate a recompilation of code every time the boundary condition
changes. For this reason we either recommend using a
Constant
if the boundary condition is spatially uniform,
or a UFL expression if it has both space and
timedependence. For example, a purely timevarying boundary
condition might be implemented as:
c = Constant(sin(t))
bc = DirichletBC(V, c, 1)
while t < T:
solve(F == 0, bcs=[bc])
t += dt
c.assign(sin(t))
If the boundary condition instead has both space and time dependence we can write:
c = Constant(t)
e = sin(x[0]*c)
bc = DirichletBC(V, e, 1)
while t < T:
solve(F == 0, bcs=[bc])
t += dt
c.assign(t)
More complicated forms¶
UFL is a fullyfledged language for expressing variational problems, and hence has operators for all appropriate vector calculus operations along with special support for discontinuous galerkin methods in the form of symbolic expressions for facet averages and jumps. For an introduction to these concepts we refer the user to the UFL manual as well as the Firedrake tutorials which cover a wider variety of different problems.