Neumann Boundary Conditions with Raviart-Thomas Elements

Question

I am solving a system of PDEs on the unit square, where one of the unknowns, a vector field $\mathbf{q}$, must satisfy the boundary condition

$$ \mathbf{q} \cdot \mathbf{n} = 1 $$

on the "lower" or "southern" boundary. If I use a vector function space of Lagrange elements, enforcing this boundary condition is simple, since I only need to enforce that the y component be equal to -1 on that boundary. However, I want to implement this using Raviart-Thomas elements, and am right now enforcing the boundary conditions weakly. Is there a way to force the normal component to be equal to 1 explicitly, using these elements?

Comments

So do you find a solution about the boundary condition

q⋅n=1?

Answer 1

In mixed methods, Neumann boundary conditions become essential boundary condition. In other words, you need to impose them to function space "strongly".

from dolfin import * 

import numpy

u_0 = Expression('exp(x[1]) + 1') 

u_ex = Expression('exp(x[0])exp(x[1]) + x[0]x[1] + 1') 

sigma_ex = Expression(('exp(x[0])exp(x[1]) + x[1]', 'exp(x[0])exp(x[1]) + x[0]'))

f = Expression('2exp(x[0])exp(x[1])')

G = Expression(("exp(x[0])exp(x[1]) + x[1]", "exp(x[0])exp(x[1]) + x[0]"))

mesh = UnitSquareMesh(nx,nx)

Mh = FunctionSpace(mesh, "RT", 1) 

Vh = FunctionSpace(mesh, "DG", 0)

V = MixedFunctionSpace([Mh, Vh])

n = FacetNormal(mesh)

class Nbdy(SubDomain):

    def inside(self, x, on_boundary):

    return on_boundary and ((abs(x[1]-1.) < tol) or (abs(x[0]-1.) < tol) or (abs(x[1]) < tol))

bc = DirichletBC(V.sub(0), sigma_ex, Nbdy())

(tau, v) = TestFunctions(V)

(sigma, u) = TrialFunctions(V)

a = inner(sigma, tau)dx + udiv(tau)dx + div(sigma)vdx

L = fvdx + u_0inner(tau, n)*ds

su = Function(V)

solve(a == L, su, bc)

(sigma, u) = su.split()

u_error = errornorm(u_ex, u, norm_type = 'L2')

sigma_error = errornorm(sigma_ex, sigma, norm_type = 'L2')

Comments

I am confusing about your answer.

Would you please express your code more clear?
Thanks a lot!