Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

18 Convection dominated problems
changing the central ®nite di€erence form of the approximation to the governing
equation as given by Eq. (2.15) to
…ÿ2Pe ÿ 1† ~ i ÿ 1 ‡…2 ‡ 2Pe† ~ i ÿ ~ i ‡ 1 ‡ Qh2
ˆ 0 …2:20†
k
With this upwind di€erence approximation, realistic (though not always accurate)
solutions can be obtained through the whole range of Peclet numbers of the example
of Fig. 2.2 as shown there by curves labelled ˆ 1. However, now exact nodal solutions
are only obtained for pure convection …Pe ˆ1†, as shown in Fig. 2.2, in a similar
way as the Galerkin ®nite element form gives exact nodal answers for pure di€usion.
How can such upwind di€erencing be introduced into the ®nite element scheme and
generalized to more complex situations? This is the problem that we shall now
address, and indeed will show that again, as in self-adjoint equations, the ®nite
element solution can result in exact nodal values for the one-dimensional approximation
for all Peclet numbers.
2.2.2 Petrov±Galerkin methods for upwinding in one dimension
The ®rst possibility is that of the use of a Petrov±Galerkin type of weighting in which
Wi 6ˆ Ni. 6ÿ9 Such weightings were ®rst suggested by Zienkiewicz et al. 6 in 1975 and
used by Christie et al. 7 In particular, again for elements with linear shape functions
Ni, shown in Fig. 2.1, we shall take, as shown in Fig. 2.3, weighting functions
constructed so that
Wi ˆ Ni ‡ Wi …2:21†
where Wi is such that
…
Wi dx ˆ h
…2:22†
2
N i
W i *
or
h
e
Fig. 2.3 Petrov±Galerkin weight function W i ˆ N i ‡ W i . Continuous and discontinuous de®nitions.
i