# Singularity in gradient caused by Dirichlet boundary condition

I am looking for a mathematical explanation for the singularity caused by a Dirichlet boundary condition partially imposed at a boundary.

For instance $$\nabla^2u=0 ~ \text{in}~\Omega$$ where $$\Omega$$ is a rectangle of 10 by 4. Dirichlet boundary condition $$u=0$$ is imposed at $$y=0 ~\text{and}~ y = 4 ~\text{and}~ x < 2$$ and a flux is applied at the boundary $$x=10$$

Plotting the flux, you can see the singularity where the Dirichlet boundary condition ends

I plotted this with the following FEniCS code

from fenics import *

mesh = RectangleMesh(Point(0, 0), Point(10, 4), 400, 160, diagonal='crossed')

V = FunctionSpace(mesh, 'CG', 1)
u, v = TrialFunction(V), TestFunction(V)

def MyDirichlet(x, on_boundary):
return on_boundary and (x[0] < 2.0) and (x[0] > 0.0)

class MyNeumann(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and near(x[0], 10.0)

subdomain = MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
subdomain.set_all(0)
myneumann = MyNeumann()
myneumann.mark(subdomain, 1)
ds = Measure('ds', domain=mesh, subdomain_data=subdomain)

bc = DirichletBC(V, Constant(0.0), MyDirichlet)

L = Constant(1.0)*v*ds(1)

u_sol = Function(V)

solve(a==L, u_sol, bcs=bc)
File("sing_example.pvd") << u_sol
Vflux = VectorFunctionSpace(mesh, 'CG', 1)


• Just out of curiosity, is there any particular reason to use this weird configuration of boundary conditions for Laplace equation? – Alone Programmer Jan 3 at 0:56
• No specific reason. – balborian Jan 3 at 3:10
• So, is this for research or right now just for learning and then moving on to solving a real problem in the future? – Alone Programmer Jan 3 at 3:37
• Both, the real problem is a convection diffusion problem where the advection field enters the right side of the rectangle and exists through the left side. The idea of placing the Dirichlet boundary conditions this way is to keep them away from the right side (the inlet). I have not thought of an alternative way to pose this real problem, but in any case I would like to know why this phenomenon happens. – balborian Jan 3 at 15:59
• As far as I know, you need to have at least some sort of boundary condition specified on every part of the boundary. Since you do not specify any conditions on $x=0$ or $(x>2)(x<10)$ I have no idea what Fenics will do. It probably depends on the details of solve. – knl Feb 1 at 21:15