How to solve simple 2D electrostatics problems using FEniCS?

Question

For example, one charged particle is situated in uniform electric field. How to calculate the potential function? I don't understand how to make boudary conditions of Laplace equation in this case.

Comments

Hi, you are more likely to get an answer if you state your problem in terms of equations. Is it $-\Delta \phi = \delta_0$ in the 2d domain where $\phi$ is the potential and $\delta$ the particle charge?

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I would write equation if I know how to write in TeX on this site.

Yes. But we can understand the particle as point charge, so \delta=0.

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http://fenicsproject.org/qa/49/how-does-one-use-mathjax-in-the-fenics-q

Answer 1

I found this:
http://fenicsproject.org/qa/2893/point-source-in-rhs-of-heat-eq

I think the syntax for RectangleMesh has changed. So use:
mesh = RectangleMesh(Point(-1,-1), Point(1, 1), 100, 100, "right/left")

For the uniform electric field, you can add Dirichlet boundary conditions on two of the boundaries with the desired values.

    from dolfin import *
mesh = RectangleMesh(Point(-1,-1), Point(1, 1), 100, 100, "right/left")
V = FunctionSpace(mesh, "CG", 1)
u = TrialFunction(V)
v = TestFunction(V)

class Left(SubDomain):
    def inside(self, x , on_boundary):
       return near(x[1], -1.0)

class Right(SubDomain):
    def inside(self, x , on_boundary):
       return near(x[1], 1.0)

bcs = [DirichletBC(V, Constant(0), Left()),
       DirichletBC(V, Constant(100), Right())]

a = inner(grad(u), grad(v))*dx
L = Constant(0)*v*dx
A, b = assemble_system(a, L, bcs)

delta = PointSource(V, Point(0., 0.,), 100)
delta.apply(b)

u = Function(V)
solve(A, u.vector(), b)

plot(u, interactive=True)