"""This demo program solves Poisson's equation

    - div grad u(x, y) = f(x, y)

on the unit square with source f given by

    f(x, y) = 10*exp(-((x - 0.5)^2 + (y - 0.5)^2) / 0.02)

and boundary conditions given by

    u(x, y) = 0        for x = 0 or x = 1
du/dn(x, y) = -sin(5*x) for y = 0 or y = 1

This demo is identical to the Poisson demo with the only difference that
the source and flux term is expressed using SpatialCoordinates in the
variational formulation.
"""

# Copyright (C) 2010 Kristian B. Oelgaard
#
# This file is part of DOLFIN.
#
# DOLFIN is free software: you can redistribute it and/or modify
# it under the terms of the GNU Lesser General Public License as published by
# the Free Software Foundation, either version 3 of the License, or
# (at your option) any later version.
#
# DOLFIN is distributed in the hope that it will be useful,
# but WITHOUT ANY WARRANTY; without even the implied warranty of
# MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
# GNU Lesser General Public License for more details.
#
# You should have received a copy of the GNU Lesser General Public License
# along with DOLFIN. If not, see <http://www.gnu.org/licenses/>.
#
# Modified by Anders Logg 2011
#
# First added:  2010-03-05
# Last changed: 2012-11-12

from dolfin import *
import matplotlib.pyplot as plt


# Form compiler options
parameters["form_compiler"]["optimize"]     = True
parameters["form_compiler"]["cpp_optimize"] = True

# Create mesh and define function space
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "CG", 1)

# Define Dirichlet boundary (x = 0 or x = 1)
def boundary(x):
    return x[0] < DOLFIN_EPS or x[0] > 1.0 - DOLFIN_EPS

# Define boundary condition
u0 = Constant(0.0)
bc = DirichletBC(V, u0, boundary)

# Define variational problem
x = SpatialCoordinate(mesh)
u = TrialFunction(V)
v = TestFunction(V)
d_x = x[0] - 0.5
d_y = x[1] - 0.5
f = 10.0*exp(-(d_x*d_x + d_y*d_y) / 0.02)
g = -sin(5.0*x[0])
a = inner(grad(u), grad(v))*dx
L = f*v*dx + g*v*ds

# Compute solution
u = Function(V)
solve(a == L, u, bc)

# Save solution in VTK format
file = File("spatial-coordinates.pvd")
file << u

# Plot solution
plot(u)
plt.show()