#! /usr/bin/env python
#
from fenics import *

def mitchell12 ( ):

#*****************************************************************************80
#
## mitchell12 sets up Mitchell test #12.
#
#  Multiple difficulties
#  -Laplace(u) = f on the upside down L shaped region
#  u = u0 on the boundary.
#  u0 = awful formula
#  f = -(u'')
#
#  Discussion:
#
#    Suggested parameter values:
#      omega = 3 * pi / 2
#      xw =  0.0
#      yw = -0.75
#      r0 =  0.75
#      alphaw = 200.0
#      xp = sqrt(5)/4
#      yp = -0.25
#      alphap = 1000
#      epsilon = 0.01
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 August 2014
#
#  Author:
#
#    John Burkardt
#
#  Reference:
#
#    William Mitchell,
#    A collection of 2D elliptic problems for testing adaptive
#    grid refinement algorithms,
#    Applied Mathematics and Computation,
#    Volume 220, 1 September 2013, pages 350-364.
#
  import matplotlib.pyplot as plt
#
#  Set parameters.
#
  omega = 3.0 * pi / 2.0
  xw =  0.0
  yw = -0.75
  r0 =  0.75
  alphaw = 200.0
  xp = sqrt ( 5.0 ) / 4.0
  yp = -0.25
  alphap = 1000
  epsilon = 0.01
#
#  Use a precomputed mesh.
#
  mesh = Mesh ( "mitchell12_mesh.xml" )
#
#  Plot the mesh.
#
  plot ( mesh, title = 'Mitchell Test 12 Mesh' )
  filename = 'mitchell12_mesh.png'
  plt.savefig ( filename )
  print ( '  Graphics saved as "%s"' % ( filename ) )
  plt.close ( )
#
#  Define the function space.
#
  V = FunctionSpace ( mesh, "Lagrange", 1 )
#
#  Define the exact solution.
#
  u0 = Expression ( "?" )
#
#  Apply the boundary condition to points which are on the boundary of the mesh.
#
  def u0_boundary ( x, on_boundary ):
    return on_boundary
#
#  The value to be used at Dirichlet boundary points is U0.
#
  bc = DirichletBC ( V, u0, u0_boundary )
#
#  Define the variational problem.
#
  u = TrialFunction ( V )
  v = TestFunction ( V )
#
#  The right hand side function is simply -Laplacian(u0).
#
  f = Expression ( "?" )
#
  a = inner ( nabla_grad ( u ), nabla_grad ( v ) ) * dx
  L = f * v * dx
#
#  Compute the solution.
#
  u = Function ( V )
  solve ( a == L, u, bc )
#
#  Plot the solution.
#
  plot ( u )

  return

def mitchell12_test ( ):

#*****************************************************************************80
#
## mitchell12_test tests mitchell12.
#
#  Licensing:
#
#    This code is distributed under the GNU LGPL license.
#
#  Modified:
#
#    22 October 2018
#
#  Author:
#
#    John Burkardt
#

#
#  Report level = only warnings or higher.
#
  level = 30
  set_log_level ( level )

  print ( '' )
  print ( 'mitchell12_test:' )
  print ( '  FENICS/Python version' )
  print ( '  Mitchell test problem #12.' )

  mitchell12 ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'mitchell12_test:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):

  mitchell12_test ( )