"""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
"""
#
#  Location:
#
#    http://people.sc.fsu.edu/~jburkardt/examples/pbs/poisson.py
#
# Copyright (C) 2007-2011 Anders Logg
#
# 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/>.
#
# First added:  2007-08-16
# Last changed: 2012-11-12

# Begin demo

from dolfin import *

# Create mesh and define function space
mesh = UnitSquareMesh(32, 32)
V = FunctionSpace(mesh, "Lagrange", 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
u = TrialFunction(V)
v = TestFunction(V)
f = Expression("10*exp(-(pow(x[0] - 0.5, 2) + pow(x[1] - 0.5, 2)) / 0.02)")
g = Expression("sin(5*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("poisson.pvd")
file << u

# Plot solution
# plot(u, interactive=False)