A FEniCS Tutorial - FEniCS Project

3.1 A Diffusion Problem and Its Discretization
Our time-dependent model problem for teaching purposes is naturally the simplest
extension of the Poisson problem into the time domain, i.e., the diffusion
problem
∂u
∂t = ∇2 u+f in Ω, for t > 0, (64)
u = u 0 on ∂Ω, for t > 0, (65)
u = I at t = 0. (66)
Here, u varies with space and time, e.g., u = u(x,y,t) if the spatial domain Ω is
two-dimensional. The source function f and the boundary values u 0 may also
vary with space and time. The initial condition I is a function of space only.
A straightforward approach to solving time-dependent PDEs by the finite
element method is to first discretize the time derivative by a finite difference
approximation, which yields a recursive set of stationary problems, and then
turn each stationary problem into a variational formulation.
Let superscript k denote a quantity at time t k , where k is an integer counting
time levels. For example, u k means u at time level k. A finite difference
discretization in time first consists in sampling the PDE at some time level, say
k:
∂
∂t uk = ∇ 2 u k +f k . (67)
The time-derivative can be approximated by a finite difference. For simplicity
and stability reasons we choose a simple backward difference:
∂
∂t uk ≈ uk −u k−1
, (68)
∆t
where ∆t is the time discretization parameter. Inserting (68) in (67) yields
u k −u k−1
= ∇ 2 u k +f k . (69)
∆t
This is our time-discrete version of the diffusion PDE (64). Reordering (69) so
that u k appears on the left-hand side only, shows that (69) is a recursive set of
spatial(stationary)problemsforu k (assumingu k−1 isknownfromcomputations
at the previous time level):
u 0 = I, (70)
u k −∆t∇ 2 u k = u k−1 +∆tf k , k = 1,2,... (71)
Given I, we can solve for u 0 , u 1 , u 2 , and so on.
We use a finite element method to solve the equations (70) and (71). This
requires turning the equations into weak forms. As usual, we multiply by a
test function v ∈ ˆV and integrate second-derivatives by parts. Introducing the
symbol u for u k (which is natural in the program too), the resulting weak form
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