A FEniCS Tutorial - FEniCS Project

def Dirichlet_boundary(x, on_boundary):
if on_boundary:
if x[0] == 0 or x[0] == 1:
return True
else:
return False
else:
return False
A more compact implementation reads
def Dirichlet_boundary(x, on_boundary):
return on_boundary and (x[0] == 0 or x[0] == 1)
As pointed out already in Section 1.3, testing for an exact match of real numbers
is not good programming practice so we introduce a tolerance in the test:
def Dirichlet_boundary(x, on_boundary):
tol = 1E-14 # tolerance for coordinate comparisons
return on_boundary and \
(abs(x[0]) < tol or abs(x[0] - 1) < tol)
The second adjustment of our program concerns the definition of L, where
we have to add a boundary integral and a definition of the g function to be
integrated:
g = Expression(’-4*x[1]’)
L = f*v*dx - g*v*ds
The ds variable implies a boundary integral, while dx implies an integral over
the domain Ω. No more modifications are necessary.
The file dn1_p2D.py in the stationary/poisson directory implements this
problem. Running the program verifies the implementation: u equals the exact
solution at all the nodes, regardless of how many elements we use.
1.14 Multiple Dirichlet Conditions
The PDE problem from the previous section applies a function u 0 (x,y) for setting
Dirichlet conditions at two parts of the boundary. Having a single function
to set multiple Dirichlet conditions is seldom possible. The more general case
is to have m functions for setting Dirichlet conditions on m parts of the boundary.
The purpose of this section is to explain how such multiple conditions are
treated in FEniCS programs.
Let us return to the case from Section 1.13 and define two separate functions
for the two Dirichlet conditions:
−∇ 2 u = −6 in Ω,
u = u L on Γ 0 ,
u = u R on Γ 1 ,
− ∂u
∂n = g on Γ N .
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