Coupled one-dimensional Poisson equations with Dirichlet boundary conditions
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

The power of defining the equations in classes easily let you combine multiple equations. Let us now solve the following system

.. math::

   \begin{aligned}
   -\nabla^2 u&=w \\
   -\nabla^2 w&=-10u \\
   \end{aligned}

subject to :math:`u(-1)=u(1)=0` and :math:`w(-1)=-w(1)=1`. The code just create two instances of the ``PoissonEquation`` from the previous file :download:`poisson.py` with different names, combines both and couple both equations via the source terms:

.. code:: python

   from pyoomph import *
   from poisson import PoissonEquation  # Load the Poisson equation for the previous class


   class CoupledPoissonProblem(Problem):
       def define_problem(self):
           mesh = LineMesh(minimum=-1, size=2, N=100)
           self.add_mesh(mesh)

           u, w = var(["u", "w"])  # Bind the variables to use them mutually as sources
           # Create two instances of Poisson equations with different names and coupled sources
           equations = PoissonEquation(name="u", source=w) + PoissonEquation(name="w", source=-10 * u)
           equations += DirichletBC(u=0, w=1) @ "left"  # Dirichlet conditions u=0, w=1 on the left boundary
           equations += DirichletBC(u=0, w=-1) @ "right"  # and u=0, w=-1 on the right boundary
           equations += TextFileOutput()
           self.add_equations(equations @ "domain")


   if __name__ == "__main__":
       with CoupledPoissonProblem() as problem:
           problem.solve()  # Solve the problem
           problem.output()  # Write output

Also note how :py:class:`~pyoomph.meshes.bcs.DirichletBC` takes multiple keyword arguments to set multiple boundary values.

..  figure:: coupled_poisson.*
	:name: figspatialcoupledpoisson
	:align: center
	:alt: Coupled Poisson equations.
	:class: with-shadow
	:width: 50%
	
	Coupled Poisson equations with Dirchlet boundaries.


.. only:: html

	.. container:: downloadbutton

		:download:`Download this example <poisson_coupled.py>`
		
		:download:`Download all examples <../../tutorial_example_scripts.zip>`