Partial Differential Equations - Modelling and ... - ResearchGate

40 Yu.A. Kuznetsov
The matrix M 0 in (67) is obtained by the assembling of matrices M 0,(k)
H
defined in (36) if the coefficient c is a positive function in E k or in (58) if c ≡ 0
in E k , k = 1,n. Respectively, the components ˆp k of the vector ¯p H in (67) are
defined either in (9) if the coefficient c is a positive function in E k or in (65)
if c ≡ 0inE k , k = 1,n.
The elimination of ū H (condensation of the system (67)) results in the algebraic
system in terms of vector ¯p H and the interface and boundary Lagrange
multiplier vector ¯λ:
]
A =¯q. (68)
[¯pH¯λ
Here,
where
A k =
[ ]
ck |E k | 0
+
0 0
A =
n∑
N k A k Nk T , (69)
k=1
[
B
0,(k)
H
C k
] [
M 0,(k)
H
] −1
[ (
B 0,(k)
H
) T
C
T
k
]
(70)
are symmetric and positive definite matrices, and N k are the underlying assembling
matrices, k = 1,n. The formula for the vector ¯q in (68) can be easily
derived.
Remark 1. If the function f is constant in E ≡ E k then the vector F in (23)
is defined by the formula
F = −f E Dē, (71)
where f E is the value of f in E, and belongs to the null space of the matrix
S + in (51). To this end, instead of (54) we have
ū h = R 1 ū H , (72)
and ḡ in (57) is the zero vector. Simple analysis shows that the resulting
discretization (66) is equivalent to the “div-const” discretization proposed in
[KR03] (see also [KLS04, KR05]).
Remark 2. The previous remark is concerned the case when c ≡ 0inE ≡ E k ,
1 ≤ k ≤ n. Consider the case when c is a positive function in E, the diffusion
equation is discretized by the method (16)–(18) and the value n k for this cell is
equal to one. Under the assumptions made, the equation (index k is omitted)
B H ū H + B h ū h − ˜Σ ¯p = F 1 , (73)
where the matrix ˜Σ and the vector F 1 are defined in (20) and (21), respectively,
is the underlying counterpart of the third equation in (23). Similar to
(50), we can consider the following formula for the solution subvector ¯p:
¯p = S + [
h ˜BH ū H − ˜Σ
]
¯p − F 1 + αē (74)