.. _demo_mixed_poisson:
Mixed formulation for Poisson equation
======================================
This demo is implemented in a single Python file,
:download:`demo_mixed-poisson.py`, which contains both the variational
forms and the solver.
This demo illustrates how to solve Poisson equation using a mixed
(two-field) formulation. In particular, it illustrates how to
* Use mixed and non-continuous finite element spaces
* Set essential boundary conditions for subspaces and H(div) spaces
* Define a (vector-valued) expression using additional geometry information
Equation and problem definition
-------------------------------
An alternative formulation of Poisson equation can be formulated by
introducing an additional (vector) variable, namely the (negative)
flux: :math:`\sigma = \nabla u`. The partial differential equations
then read
.. math::
\sigma - \nabla u &= 0 \quad {\rm in} \ \Omega, \\
\nabla \cdot \sigma &= - f \quad {\rm in} \ \Omega,
with boundary conditions
.. math::
u = u_0 \quad {\rm on} \ \Gamma_{D}, \\
\sigma \cdot n = g \quad {\rm on} \ \Gamma_{N}.
The same equations arise in connection with flow in porous media, and
are also referred to as Darcy flow.
After multiplying by test functions :math:`\tau` and :math:`v`,
integrating over the domain, and integrating the gradient term by
parts, one obtains the following variational formulation: find
:math:`\sigma \in \Sigma` and :math:`v \in V` satisfying
.. math::
\int_{\Omega} (\sigma \cdot \tau + \nabla \cdot \tau \ u) \ {\rm d} x
&= \int_{\Gamma} \tau \cdot n \ u \ {\rm d} s
\quad \forall \ \tau \in \Sigma, \\
\int_{\Omega} \nabla \cdot \sigma v \ {\rm d} x
&= - \int_{\Omega} f \ v \ {\rm d} x
\quad \forall \ v \in V.
Here :math:`n` denotes the outward pointing normal vector on the
boundary. Looking at the variational form, we see that the boundary
condition for the flux (:math:`\sigma \cdot n = g`) is now an
essential boundary condition (which should be enforced in the function
space), while the other boundary condition (:math:`u = u_0`) is a
natural boundary condition (which should be applied to the variational
form). Inserting the boundary conditions, this variational problem can
be phrased in the general form: find :math:`(\sigma, u) \in \Sigma_g
\times V` such that
.. math::
a((\sigma, u), (\tau, v)) = L((\tau, v))
\quad \forall \ (\tau, v) \in \Sigma_0 \times V
where the variational forms :math:`a` and :math:`L` are defined as
.. math::
a((\sigma, u), (\tau, v)) &=
\int_{\Omega} \sigma \cdot \tau + \nabla \cdot \tau \ u
+ \nabla \cdot \sigma \ v \ {\rm d} x \\
L((\tau, v)) &= - \int_{\Omega} f v \ {\rm d} x
+ \int_{\Gamma_D} u_0 \tau \cdot n \ {\rm d} s
and :math:`\Sigma_g = \{ \tau \in H({\rm div}) \text{ such that } \tau \cdot n|_{\Gamma_N} = g \}`
and :math:`V = L^2(\Omega)`.
To discretize the above formulation, two discrete function spaces
:math:`\Sigma_h \subset \Sigma` and :math:`V_h \subset V` are needed
to form a mixed function space :math:`\Sigma_h \times V_h`. A stable
choice of finite element spaces is to let :math:`\Sigma_h` be the
Brezzi-Douglas-Marini elements of polynomial order :math:`k` and let
:math:`V_h` be discontinuous elements of polynomial order :math:`k-1`.
We will use the same definitions of functions and boundaries as in the
demo for Poisson's equation. These are:
* :math:`\Omega = [0,1] \times [0,1]` (a unit square)
* :math:`\Gamma_{D} = \{(0, y) \cup (1, y) \in \partial \Omega\}`
* :math:`\Gamma_{N} = \{(x, 0) \cup (x, 1) \in \partial \Omega\}`
* :math:`u_0 = 0`
* :math:`g = \sin(5x)` (flux)
* :math:`f = 10\exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02)` (source term)
With the above input the solution for :math:`u` and :math:`\sigma` will look as
follows:
.. image:: mixed-poisson_u.png
:scale: 75 %
:align: center
.. image:: mixed-poisson_sigma.png
:scale: 75 %
:align: center
Implementation
--------------
This demo is implemented in the :download:`demo_mixed-poisson.py`
file.
First, the required modules are imported::
from dolfin import *
import matplotlib.pyplot as plt
Then, we need to create a :py:class:`Mesh ` covering
the unit square. In this example, we will let the mesh consist of 32 x
32 squares with each square divided into two triangles::
# Create mesh
mesh = UnitSquareMesh(32, 32)
.. index::
pair: FunctionSpace; Brezzi-Douglas-Marini
pair: FunctionSpace; Discontinous Lagrange
Next, we need to build the function space. ::
# Define finite elements spaces and build mixed space
BDM = FiniteElement("BDM", mesh.ufl_cell(), 1)
DG = FiniteElement("DG", mesh.ufl_cell(), 0)
W = FunctionSpace(mesh, BDM * DG)
The second argument to :py:class:`FunctionSpace
` specifies underlying
finite element, here mixed element obtained by ``*`` operator.
.. math::
W = \{ (\tau, v) \ \text{such that} \ \tau \in BDM, v \in DG \}.
Next, we need to specify the trial functions (the unknowns) and the
test functions on this space. This can be done as follows::
# Define trial and test functions
(sigma, u) = TrialFunctions(W)
(tau, v) = TestFunctions(W)
In order to define the variational form, it only remains to define the
source function :math:`f`. This is done just as for the :ref:`Poisson
demo `::
# Define source function
f = Expression("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)", degree=2)
We are now ready to define the variational forms a and L. Since,
:math:`u_0 = 0` in this example, the boundary term on the right-hand
side vanishes. ::
# Define variational form
a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx
It only remains to prescribe the boundary condition for the
flux. Essential boundary conditions are specified through the class
:py:class:`DirichletBC ` which takes three
arguments: the function space the boundary condition is supposed to be
applied to, the data for the boundary condition, and the relevant part
of the boundary.
We want to apply the boundary condition to the first subspace of the
mixed space. Subspaces of a mixed :py:class:`FunctionSpace
` can be accessed
by the method :py:func:`sub
`. In our case,
this reads ``W.sub(0)``. (Do *not* use the separate space ``BDM`` as
this would mess up the numbering.)
Next, we need to construct the data for the boundary condition. An
essential boundary condition is handled by replacing degrees of
freedom by the degrees of freedom evaluated at the given data. The
:math:`BDM` finite element spaces are vector-valued spaces and hence
the degrees of freedom act on vector-valued objects. The effect is
that the user is required to construct a :math:`G` such that :math:`G
\cdot n = g`. Such a :math:`G` can be constructed by letting :math:`G
= g n`. In particular, it can be created by subclassing the
:py:class:`Expression `
class. Overloading the ``eval_cell`` method (instead of the usual
``eval``) allows us to extract more geometry information such as the
facet normals. Since this is a vector-valued expression, we also need
to overload the ``value_shape`` method.
.. index::
single: Expression; (in Mixed Poisson demo)
::
# Define function G such that G \cdot n = g
class BoundarySource(UserExpression):
def __init__(self, mesh, **kwargs):
self.mesh = mesh
super().__init__(**kwargs)
def eval_cell(self, values, x, ufc_cell):
cell = Cell(self.mesh, ufc_cell.index)
n = cell.normal(ufc_cell.local_facet)
g = sin(5*x[0])
values[0] = g*n[0]
values[1] = g*n[1]
def value_shape(self):
return (2,)
G = BoundarySource(mesh, degree=2)
Specifying the relevant part of the boundary can be done as for the
Poisson demo (but now the top and bottom of the unit square is the
essential boundary): ::
# Define essential boundary
def boundary(x):
return x[1] < DOLFIN_EPS or x[1] > 1.0 - DOLFIN_EPS
Now, all the pieces are in place for the construction of the essential
boundary condition: ::
bc = DirichletBC(W.sub(0), G, boundary)
To compute the solution we use the bilinear and linear forms, and the
boundary condition, but we also need to create a :py:class:`Function
` to store the solution(s). The
(full) solution will be stored in the ``w``, which we initialise using
the :py:class:`FunctionSpace
` ``W``. The actual
computation is performed by calling :py:func:`solve
`. The separate components ``sigma`` and
``u`` of the solution can be extracted by calling the :py:func:`split
` function. Finally, we plot
the solutions to examine the result. ::
# Compute solution
w = Function(W)
solve(a == L, w, bc)
(sigma, u) = w.split()
# Plot sigma and u
plt.figure()
plot(sigma)
plt.figure()
plot(u)
plt.show()