A FEniCS Tutorial - FEniCS Project

Figure 8: Hollow cylinder generated by mapping a rectangular mesh, stretched
toward the left side.
5.1 Working with Two Subdomains
Suppose we want to solve
∇·[k(x,y)∇u(x,y)] = 0, (88)
in a domain Ω consisting of two subdomains where k takes on a different value
in each subdomain. For simplicity, yet without loss of generality, we choose for
the current implementation the domain Ω = [0,1]×[0,1] and divide it into two
equal subdomains, as depicted in Figure 5.1,
Ω 0 = [0,1]×[0,1/2], Ω 1 = [0,1]×(1/2,1].
We define k(x,y) = k 0 in Ω 0 and k(x,y) = k 1 in Ω 1 , where k 0 > 0 and k 1 > 0
are given constants. As boundary conditions, we choose u = 0 at y = 0, u = 1
at y = 1, and ∂u/∂n = 0 at x = 0 and x = 1. One can show that the exact
solution is now given by
u(x,y) =
{ 2yk1
k 0+k 1
, y ≤ 1/2
(2y−1)k 0+k 1
k 0+k 1
, y ≥ 1/2
(89)
As long as the element boundaries coincide with the internal boundary y = 1/2,
this piecewise linear solution should be exactly recovered by Lagrange elements
of any degree. We use this property to verify the implementation.
Physically, the present problem may correspond to heat conduction, where
the heat conduction in Ω 1 is ten times more efficient than in Ω 0 . An alternative
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