6.2. Upstream Weighted Methods
165
FIGURE 6.7. Approximate expressions of the unknown concentration. (a) Linear Galerkin FEM or general MCBM;
(b)
Upstream
weighted
method.
and
c;
/
/
,
,
cj
~ , , ,
(a)
"
"
,
C. 'c
C,
,
m
"
~
cj
/ L'
/
"
___ "rn
.......
j
j
(b)
aki = xjYm - xmYj' b"i = Yj - Ym, Cki = x m - Xj'
akj = XmYi - xiYm, bkj = Ym - Yi, Ckj = Xi - X m ,
akm = XiYj - XjYi' bkm = Yi - Yj' Ckm = Xj - Xi·
(6.2.33)
Substituting the definition of C m , Eq. (6.2.31), into Eq. (6.2.32), we have
C(x, Y, t) = ~kiCi + ~kjCj + ~kkCk'
(6.2.34)
where
(x,y) E Aijm
~ki = rPki + WirPkm,
~kj = rPkj + WjrPkm,
~kk = w,. rPkm·
(6.2.35a)
(6.2.35b)
(6.2.35c)
As expressed in Eq. (6.2.34), lÄi' ~kj' and ~kk act as basis functions in the subelement Aijm and contain weighting coefficients, so they are called weighted
basis functions.
Similarly, for another sub-element Aimk with node i as its vertex, we have
C(x,y,t) = ~jiCi + fhjCj + ~jkCk'
(x, y) E Aimk,
where the weighted basis functions are
~ji = rPji + wirPjm,
fhj = wjf/ljm·
fhk = f/ljk + wkf/ljm,
(6.2.36)
(6.2.37a)
(6.2.37b)
(6.2.37c)