# Mixed formulation for the Poisson equation
# ==========================================
#
# We're considering the Poisson equation :math:`\nabla^2 u = -f` using a mixed
# formulation on two coupled fields. We start by introducing the negative flux
# :math:`\sigma = \nabla u` as an auxiliary vector-valued variable. This leaves
# us with the PDE on a unit square :math:`\Omega = [0,1] \times [0,1]` with
# boundary :math:`\Gamma`
#
# .. math::
#
# \sigma - \nabla u = 0 \ \textrm{on}\ \Omega
#
# \nabla \cdot \sigma = -f \ \textrm{on}\ \Omega
#
# u = u_0 \ \textrm{on}\ \Gamma_D
#
# \sigma \cdot n = g \ \textrm{on}\ \Gamma_N
#
# for some known function :math:`f`. The solution to this equation will be some
# functions :math:`u\in V` and :math:`\sigma\in \Sigma` for some suitable
# function space :math:`V` and :math:`\Sigma` that satisfy these equations. We
# multiply by arbitrary test functions :math:`\tau \in \Sigma` and :math:`\nu \in
# V`, integrate over the domain and then integrate by parts to obtain a
# weak formulation of the variational problem: find :math:`\sigma\in \Sigma` and
# :math:`\nu\in V` such that:
#
# .. 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.
#
# The flux boundary condition :math:`\sigma \cdot n = g` becomes an *essential*
# boundary condition to be enforced on the function space, while the boundary
# condition :math:`u = u_0` turn into a *natural* boundary condition which
# enters into the variational form, such that the variational problem can be
# written as: 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
#
# with the variational forms :math:`a` and :math:`L` 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
#
# The essential boundary condition is reflected in function spaces
# :math:`\Sigma_g = \{ \tau \in H({\rm div}) \text{ such that } \tau \cdot
# n|_{\Gamma_N} = g \}` and :math:`V = L^2(\Omega)`.
#
# We need to choose a stable combination of discrete function spaces
# :math:`\Sigma_h \subset \Sigma` and :math:`V_h \subset V` to form a mixed
# function space :math:`\Sigma_h \times V_h`. One such choice is
# Brezzi-Douglas-Marini elements of polynomial order :math:`k` for
# :math:`\Sigma_h` and discontinuous elements of polynomial order :math:`k-1`
# for :math:`V_h`.
#
# For the remaining functions and boundaries we choose:
#
# .. math::
#
# \Gamma_{D} = \{(0, y) \cup (1, y) \in \partial \Omega\},
# \Gamma_{N} = \{(x, 0) \cup (x, 1) \in \partial \Omega\}
#
# u_0 = 0,
# g = \sin(5x)
#
# f = 10~e^{-\frac{(x - 0.5)^2 + (y - 0.5)^2}{0.02}}
#
# To produce a numerical solution to this PDE in Firedrake we procede as
# follows:
#
# The mesh is chosen as a :math:`32\times32` element unit square. ::
from firedrake import *
mesh = UnitSquareMesh(32, 32)
# As argued above, a stable choice of function spaces for our problem is the
# combination of order :math:`k` Brezzi-Douglas-Marini (BDM) elements and order
# :math:`k - 1` discontinuous Galerkin elements (DG). We use :math:`k = 1` and
# combine the BDM and DG spaces into a mixed function space ``W``. ::
BDM = FunctionSpace(mesh, "BDM", 1)
DG = FunctionSpace(mesh, "DG", 0)
W = BDM * DG
# We obtain test and trial functions on the subspaces of the mixed function
# spaces as follows: ::
sigma, u = TrialFunctions(W)
tau, v = TestFunctions(W)
# Next we declare our source function ``f`` over the DG space and initialise it
# with our chosen right hand side function value. ::
x, y = SpatialCoordinate(mesh)
f = Function(DG).interpolate(
10*exp(-(pow(x - 0.5, 2) + pow(y - 0.5, 2)) / 0.02))
# After dropping the vanishing boundary term on the right hand side, the
# bilinear and linear forms of the variational problem are defined as: ::
a = (dot(sigma, tau) + div(tau)*u + div(sigma)*v)*dx
L = - f*v*dx
# The strongly enforced boundary conditions on the BDM space on the top and
# bottom of the domain are declared as: ::
bc0 = DirichletBC(W.sub(0), as_vector([0.0, -sin(5*x)]), 3)
bc1 = DirichletBC(W.sub(0), as_vector([0.0, sin(5*x)]), 4)
# Note that it is necessary to apply these boundary conditions to the first
# subspace of the mixed function space using ``W.sub(0)``. This way the
# association with the mixed space is preserved. Declaring it on the BDM space
# directly is *not* the same and would in fact cause the application of the
# boundary condition during the later solve to fail.
#
# Now we're ready to solve the variational problem. We define `w` to be a function
# to hold the solution on the mixed space. ::
w = Function(W)
# Then we solve the linear variational problem ``a == L`` for ``w`` under the
# given boundary conditions ``bc0`` and ``bc1``. Afterwards we extract the
# components ``sigma`` and ``u`` on each of the subspaces with ``split``. ::
solve(a == L, w, bcs=[bc0, bc1])
sigma, u = w.split()
# Lastly we write the component of the solution corresponding to the primal
# variable on the DG space to a file in VTK format for later inspection with a
# visualisation tool such as `ParaView `__ ::
File("poisson_mixed.pvd").write(u)
# We could use the built in plot function of firedrake by calling
# :func:`plot ` to plot a surface graph. Before that,
# matplotlib.pyplot should be installed and imported::
try:
import matplotlib.pyplot as plt
except:
warning("Matplotlib not imported")
try:
fig, axes = plt.subplots()
colors = tripcolor(u, axes=axes)
fig.colorbar(colors)
except Exception as e:
warning("Cannot plot figure. Error msg '%s'" % e)
# Don't forget to show the image::
try:
plt.show()
except Exception as e:
warning("Cannot show figure. Error msg '%s'" % e)
# This demo is based on the corresponding `DOLFIN mixed Poisson demo
# `__
# and can be found as a script in `poisson_mixed.py `__.