"""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
"""
from dolfin import *
# Create mesh and define function space
mesh = UnitSquare(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
v = TestFunction(V)
u = TrialFunction(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(v), grad(u))*dx
L = v*f*dx - v*g*ds
U = Function(V)
# Assemble linear system
A = assemble(a)
b = assemble(L)
x = U.vector()
# Apply boundary conditions
bc.apply(A, b)
# Solve linear system
solver = KrylovSolver()
solver.solve(A, x, b)
# Save solution in VTK format
file = File("poisson.pvd")
file << U
# Plot solution
plot(U, interactive=True)