Chapter 8 Configuring Fluidity - The Applied Modelling and ...

26 Numerical discretisation
3.2.1.1 Weak form
Development of the finite element method begins by writing the equations in weak form. The weak
form of the advection-diffusion equation is obtained by pre-multiplying it with a test function ϕ and
integrating over the domain Ω, such that
� � �
∂c
ϕ + u · ∇c − ∇ · µ · ∇c = 0. (3.4)
∂t
Ω
Integrating the advection and diffusion terms by parts yields
�
ϕ ∂c
�
− ∇ϕ · uc + ∇ϕ · µ · ∇c +
∂t
n · uc − n · µ · ∇c = 0. (3.5)
Ω
For simplicity sake let us first assume the boundaries are closed (u · n = 0) and we apply a homogeneous
Neumann boundary condition everywhere, such that ∂c/∂n = 0, which gives
�
Ω
∂Ω
ϕ ∂c
− ∇ϕ · uc + ∇ϕ · µ · ∇c = 0. (3.6)
∂t
Note that due to the Neumann boundary condition the boundary term has dropped out. The tracer
field c is now called a weak solution of the equations if (3.6) holds true for all ϕ in some space of test
functions V . The choice of a suitable test space V is dependent on the equation (for more details see
Elman et al. [2005]).
An important observation is that (3.5) only contains first derivatives of the field c, so that we can
now include solutions that do not have a continuous second derivative. For these solutions the
original equation (2.44c) would not be well-defined. These solutions are termed weak as they do not
have sufficient smoothness to be classical solutions to the problem. All that is required for the weak
solution is that the first derivatives of c can be integrated along with the test function. A more precise
definition of this space, the Sobolev space, can again be found Elman et al. [2005].
3.2.1.2 Finite element discretisation
Instead of looking for a solution in the entire function (Sobolev) space, in finite element methods
discretisation is performed by restricting the solution to a finite-dimensional subspace. Thus the
solution can be written as a linear combination of a finite number of functions, the trial functions ϕi
that form a basis of the trial space, defined such that
c(x) = �
ciϕi((x)).
i
The coefficients ci can be written in vector format. The dimension of this vector equals the dimension
of the trial space. In the sequel any function in this way represented as a vector will be denoted as c.
Since the set of trial functions is now much smaller (or rather finite as opposed to infinitedimensional),
we also need a much reduced set of test functions for the equation in weak form (3.6)
in order to find a unique solution. A common choice is, in fact, to choose the same test and trial space.
Finite element methods that make this choice are referred to as Galerkin methods — the discretisation
can be seen as a so called Galerkin projection of the weak equation to a finite subspace.
There are many possibilities for choosing the finite-dimensional trial and test spaces. A straightforward
choice is to restrict the functions to be polynomials of degree n � N within each element.
These are referred to as PN discretisations. As we generally need functions for which the first derivatives
are integrable, a further restriction is needed. If we allow the functions to be any polynomial
of degree n � N within the elements the function can be discontinuous in the boundary between
elements. Continuous Galerkin methods therefore restrict the test and trial functions to arbitrary