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

24 Convection dominated problems
2.2.5 Galerkin least square approximation (GLS) in one
dimension
In the preceding sections we have shown that several, apparently di€erent,
approaches have resulted in identical (or almost identical) approximations. Here
yet another procedure is presented which again will produce similar results. In this
a combination of the standard Galerkin and least square approximations is made. 15;16
If Eq. (2.11) is rewritten as
with
L ‡ Q ˆ 0 ^ ˆ N ~ f …2:36a†
L ˆ U d d
ÿ
dx dx
k d
dx
the standard Galerkin approximation gives for the kth equation
… L
0
N kL…N† ~ f dx ‡
… L
0
…2:36b†
N kQ dx ˆ 0 …2:37†
with boundary conditions omitted for clarity.
Similarly, a least square residual minimization (see Chapter 3 of Volume 1, Sec.
3.14.2) results in
or
R ˆ L ^ ‡ Q and
… L
0
1
2
U dN k
dx
d
d ~ k
ÿ d
dx
… L
0
R 2 dx ˆ
… L
0
d…L ^ †
d ~ …L
k
^ ‡ Q† dx ˆ 0 …2:38†
k d
dx N k …L ^ ‡ Q† ˆ0 …2:39†
If the ®nal approximation is written as a linear combination of Eqs (2.37) and
(2.39), we have
… L
0
N k ‡ U dN k
dx
ÿ d
dx
k d
dx N k …L ^ ‡ Q† dx ˆ 0 …2:40†
This is of course, the same as the Petrov±Galerkin approximation with an undetermined
parameter . If the second-order term is omitted (as could be done assuming
linear N k and a curtailment as in Fig. 2.3) and further if we take
ˆ
j jh
2jUj
…2:41†
the approximation is identical to that of the Petrov±Galerkin method with the
weighting given by Eqs (2.21) and (2.22).
Once again we see that a Petrov±Galerkin form written as
… L
0
j j Uh dNk Nk ‡
2 jUj dx
d ^ d
U ÿ
dx dx
k d ^
dx
‡ Q dx ˆ 0 …2:42†