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

def mitchell04 ( ):

#*****************************************************************************80
#
## mitchell04 sets up Mitchell test #4.
#
#  Peak
#  -Laplace(u) = f on the unit square.
#  u = u0 on the boundary.
#  u0 = exp(-alpha*(x-xc)^2+(y-yc)^2)
#  f = -(u0'')
#
#  Discussion:
#
#    Suggested parameter values:
#    * alpha = 1000,   (xc,yc) = (0.5,0.5)
#    * alpha = 100000, (xc,yc) = (0.51,0.117)
#
#  Modified:
#
#    24 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
#
  alpha = 1000
  xc = 0.5
  yc = 0.5
#
#  Divide the unit square into a 10x10 mesh of quadrilaterals, and
#  split each into triangles.
#
  mesh = UnitSquareMesh ( 10, 10 )
#
#  Plot the mesh.
#
  plot ( mesh, title = 'Mitchell Test 04 Mesh' )
  filename = 'mitchell04_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.
#  Frap-dapping FENICS is failing to compile this expression.
#
  u0 = Expression ( "exp ( - alpha * ( pow ( x[0] - xc, 2 ) + pow ( x[1] - yc, 2 ) ) )", \
                  alpha = alpha, \
                  xc = xc, \
                  yc = yc, degree = 10 )

#  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 )
#
#  Define the right hand side, which is simply -(u0''):
#
  f = Expression ( "4.0 * alpha * ( 1.0 - alpha "\
                 "* ( pow ( x[0] - xc, 2 ) + pow ( x[1] - yc, 2 ) ) )"\
                 "* exp ( - alpha * ( pow ( x[0] - xc, 2 ) + pow ( x[1] - yc, 2 ) ) )", \
                 alpha = alpha, \
                 xc = xc, \
                 yc = yc, degree = 10 )

  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 mitchell04_test ( ):

#*****************************************************************************80
#
## mitchell04_test tests mitchell04.
#
#  Modified:
#
#    22 October 2018
#
#  Author:
#
#    John Burkardt
#

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

  print ( '' )
  print ( 'mitchell04_test:' )
  print ( '  FENICS/Python version' )
  print ( '  Mitchell test problem #4.' )

  mitchell04 ( )
#
#  Terminate.
#
  print ( '' )
  print ( 'mitchell04_test:' )
  print ( '  Normal end of execution.' )
  return

if ( __name__ == '__main__' ):

  mitchell04_test ( )