A FEniCS Tutorial - FEniCS Project

with exact solution u(x) = x 2 . Our aim is to formulate and solve this problem
in a 2D and a 3D domain as well. We may generalize the domain [0,1] to a
rectangle or box of any size in the y and z directions and pose homogeneous
Neumann conditions ∂u/∂n = 0 at all additional boundaries y = const and
z = const to ensure that u only varies with x. For example, let us choose a unit
hypercube as domain: Ω = [0,1] d , where d is the number of space dimensions.
The generalized d-dimensional Poisson problem then reads
∇ 2 u = 2 in Ω,
u = 0 on Γ 0 ,
u = 1 on Γ 1 ,
∂u
∂n
= 0 on ∂Ω\(Γ 0 ∪Γ 1 ),
(37)
where Γ 0 is the side of the hypercube where x = 0, and where Γ 1 is the side
where x = 1.
ImplementingaPDEforanydisnomorecomplicatedthansolvingaproblem
with a specific number of dimensions. The only non-trivial part of the code is
actually to define the mesh. We use the command line for the user-input to the
program. The first argument can be the degree of the polynomial in the finite
element basis functions. Thereafter, we supply the cell divisions in the various
spatial directions. The number of command-line arguments will then imply the
number of space dimensions. For example, writing 3 10 3 4 on the command
line means that we want to approximate u by piecewise polynomials of degree 3,
and that the domain is a three-dimensional cube with 10×3×4 divisions in the
x, y, and z directions, respectively. The Python code can be quite compact:
degree = int(sys.argv[1])
divisions = [int(arg) for arg in sys.argv[2:]]
d = len(divisions)
domain_type = [UnitInterval, UnitSquare, UnitCube]
mesh = domain_type[d-1](*divisions)
V = FunctionSpace(mesh, ’Lagrange’, degree)
First note that although sys.argv[2:] holds the divisions of the mesh, all
elements of the list sys.argv[2:] are string objects, so we need to explicitly
convert each element to an integer. The construction domain_type[d-1] will
pick the right name of the object used to define the domain and generate the
mesh. Moreover, the argument *divisions sends all the component of the
list divisions as separate arguments. For example, in a 2D problem where
divisions has two elements, the statement
mesh = domain_type[d-1](*divisions)
is equivalent to
mesh = UnitSquare(divisions[0], divisions[1])
The next part of the program is to set up the boundary conditions. Since
the Neumann conditions have ∂u/∂n = 0 we can omit the boundary integral
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