Partial Differential Equations - Modelling and ... - ResearchGate

38 Yu.A. Kuznetsov
where
}
Λ = diag
{µ 1 ,µ 2 ,...,µ ν
(47)
and
W = [ ]
¯w 1 ¯w 2 ··· ¯w ν . (48)
Consider the second equation in (24) in the form
S h ¯p = ˜B H ū H − F. (49)
A solution vector ¯p of this system can be presented by the formula
¯p = S + [
h ˜BH ū H − F ] + αē (50)
with an arbitrary coefficient α ∈ R in the right-hand side and
S + h = WΛ+ W T . (51)
Here,
Λ + = diag { 0,µ −1 }
2 ,...,µ−1 ν
(52)
is a diagonal matrix.
Substituting vector ¯p in (50) to the second equation in (23), we get the
equation [
MhH + Bh T S + ˜B h H
]ūH + M h ū h = Bh T S + h
F. (53)
Thus,
ū h = R 1 ū H + R 2 F, (54)
where
R 1 = −M −1 [
h MhH + Bh T S + ˜B
]
h H (55)
and
R 2 = M −1
h BT h S + h . (56)
Now, we replace the first two equations in (23) by a single equation. To
derive this equation, we multiply the first two equations in (23) by the matrix
[
IH R1 T , ]
where I H is the identity s k × s k matrix, and then substitute the vector ū h
defined by formula (54) into the new equation. We get the resulting equation
in terms of vectors ū H ,¯p, and¯λ in the following form:
M 0 Hū H + ̂B T H ¯p + C T ¯λ =ḡ, (57)
where the matrix
MH 0 = [ ] [ ][ ]
I H R1
T M H M Hh IH
M hH M h R 1
(58)