Hello,

I want to compute

where \Omega = [0,1]^2 with inflow boundary conditions u_{|\partial \Omega^{-}} = 1 for \partial \Omega^{-} = \{0\} \times [0,1] and \partial \Omega^{+} = \partial \Omega \setminus \partial \Omega^{-}.

By Greene’s identity

having used that the integral vanishes on \partial \Omega^{-}. Using either the left or right-hand side to numerically solve the problem should give the same result, but that is not the case in my code. Can someone point out what I am doing wrong?

```
from dolfin import *
import numpy as np
nx = ny = 20 ; degree = 1
mesh = UnitSquareMesh(nx,ny)
n = FacetNormal(mesh)
V = FunctionSpace(mesh, "CG", degree)
Vf = VectorFunctionSpace(mesh, "CG", degree=degree+1, dim=2)
f = Expression("cos(x[0] * M_PI) + sin(x[1] * M_PI)", degree=degree+1)
beta = Expression(('cos(x[0])', 'sin(2*x[1] - .2)'), degree=degree+1)
betaInterp = interpolate(beta, Vf)
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
class Outflow(SubDomain):
def inside(self, x, on_boundary):
tol = 1e-13
if x[0] > tol and on_boundary:
return True
class Inflow(SubDomain):
def inside(self, x, on_boundary):
tol = 1e-13
if near(x[0], 0, tol) and on_boundary:
return True
outflow = Outflow()
outflow.mark(boundaries, 1)
inflow = Inflow()
inflow.mark(boundaries, 2)
ds = Measure("ds", subdomain_data=boundaries)
u = TrialFunction(V) ; v = TestFunction(V)
a_v1 = dot(betaInterp, grad(u)) * v * dx
a_v2 = - u * dot(betaInterp, grad(v)) * dx \
- div(betaInterp) * u * v * dx \
+ dot(betaInterp, n) * u * v * ds(1)
f_rhs = f * v * dx
A_v1 = assemble(a_v1) ; A_v2 = assemble(a_v2) ; F = assemble(f_rhs)
bc = DirichletBC(V, 1.0, inflow)
bc.apply(A_v1, F)
bc.apply(A_v2)
u1 = Function(V) ; solve(A_v1, u1.vector(), F)
u2 = Function(V) ; solve(A_v2, u2.vector(), F)
print(errornorm(u1,u2)) #output is 18.217744114223798
print(np.where( np.abs(A_v1.array() - A_v2.array()) > 1e-14 )) # output is (array([ 2, 2, 231, 231]), array([ 0, 2, 210, 231]))
```

The last line shows where the two assembled matrices differ, which are only 4 entries.

Apologies if this question is basic and/or has already been asked in a different form, I couldn’t find any answer to it.

Thanks in advance.