A FEniCS Tutorial - FEniCS Project

can be conveniently written in the standard notation: a 0 (u,v) = L 0 (v) for (70)
and a(u,v) = L(v) for (71), where
∫
a 0 (u,v) = uvdx, (72)
Ω
∫
L 0 (v) = Ivdx, (73)
Ω
∫
a(u,v) = (uv +∆t∇u·∇v) dx, (74)
Ω
∫
(
L(v) = u k−1 +∆tf k) vdx. (75)
Ω
Thecontinuousvariationalproblemistofindu 0 ∈ V suchthata 0 (u 0 ,v) = L 0 (v)
holds for all v ∈ ˆV, and then find u k ∈ V such that a(u k ,v) = L(v) for all v ∈ ˆV,
k = 1,2,....
Approximate solutions inspacearefoundbyrestrictingthefunctional spaces
V and ˆV to finite-dimensional spaces, exactly as we have done in the Poisson
problems. We shall use the symbol u for the finite element approximation at
time t k . In case we need to distinguish this space-time discrete approximation
from the exact solution of the continuous diffusion problem, we use ue for the
latter. By u k−1 we mean, from now on, the finite element approximation of the
solution at time t k−1 .
Note that the forms a 0 and L 0 are identical to the forms met in Section 1.9,
except that the test and trial functions are now scalar fields and not vector
fields. Instead of solving (70) by a finite element method, i.e., projecting I
onto V via the problem a 0 (u,v) = L 0 (v), we could simply interpolate u 0 from
I. That is, if u 0 = ∑ N
j=1 U0 j φ j, we simply set U j = I(x j ,y j ), where (x j ,y j )
are the coordinates of node number j. We refer to these two strategies as
computing the initial condition by either projecting I or interpolating I. Both
operations areeasy to compute through one statement, usingeither theproject
or interpolate function.
3.2 Implementation
Our program needs to perform the time stepping explicitly, but can rely on
FEniCS to easily compute a 0 , L 0 , a, and L, and solve the linear systems for
the unknowns. We realize that a does not depend on time, which means that
its associated matrix also will be time independent. Therefore, it is wise to
explicitly create matrices and vectors as in Section 1.15. The matrix A arising
from a can be computed prior to the time stepping, so that we only need to
compute the right-hand side b, corresponding to L, in each pass in the time
loop. Let us express the solution procedure in algorithmic form, writing u for
the unknown spatial function at the new time level (u k ) and u 1 for the spatial
solution at one earlier time level (u k−1 ):
• define Dirichlet boundary condition (u 0 , Dirichlet boundary, etc.)
60