Page 315 - Numerical Methods for Chemical Engineering

304 6 Boundary value problems

Weighted-residual methods and the Galerkin formulation of FEM

We now examine how the FEM solves transport-type PDEs such as (6.9) using an unstruc-
tured grid. First, we collect in the PDE all nonzero terms and bring them to one side of the
equation to define a residual function that is zero everywhere and at all times for the true
solution,
∂ϕ 2
R(t,r) = + ∇ · (ϕv) − ∇ ϕ − s(r, t,ϕ) = 0 (6.194)
∂t
We can multiply (6.194) by any weight function w(r) and integrate over the domain to
obtain a weighted residual that similarly must be zero at all times for the true solution,

∂ϕ 2
' '
w(r)R(t,r)dr = w(r) + ∇ · (ϕv) − ∇ ϕ − s(r, t,ϕ) dr = 0 (6.195)
∂t

Let us consider the time-independent problem
' '
2
w(r)R(r)dr = w(r)[∇ · (ϕv) − ∇ ϕ − s(r,ϕ)]dr = 0 (6.196)

Equation (6.196) forms the basis of a numerical method if we parameterize a trial form
N
for ϕ(r) by some vector ϕ ∈ of coefficients ϕ p . In FEM, these coefficients are the
field values at each node p. We then choose N weight functions w p (r) to generate a set of
algebraic equations
'
f p (ϕ) = w p (r)R(r)dr = 0 (6.197)

What weight functions should we use? One choice originates from seeking to minimize the
integral
'
2 2
R = |R(r)| dr (6.198)
2
minimize {ϕ p }

for which the first-order optimality conditions yield
2 ' '
∂ R 2 ∂ 2 ∂ R
0 = = |R(r)| dr = 2 R(r) dr (6.199)
∂ϕ p ∂ϕ p ∂ϕ p

Thus, the least-squares weight functions are w p = ∂ R/∂ϕ p . These may be difficult to
compute and thus simpler methods are often favored in practice.
In the collocation method, we define the weight functions to be Dirac delta functions
centered on each node, so that the residual must be zero at each node. Unfortunately, even
though the residual is zero at each node, it may be large between nodes, especially with
strong convection. In simple geometries, accuracy is improved when the nodes are placed
at the zeros of orthogonal polynomials (see Chapter 4). This orthogonal collocation method
is discussed in Villadsen & Michelsen (1978).
Here, we use the Galerkin method, which is based upon writing the trial form of the
solution as a linear combination of each nodal value multiplied by a corresponding global