Partial Differential Equations - Modelling and ... - ResearchGate

is symmetric and positive definite,
Mixed FE Methods on Polyhedral Meshes 39
̂B T H = B T H + R T 1 B T h , (59)
and
ḡ = − ( M Hh + R T 1 M h
)
R2 F. (60)
Let us analyze the matrix ̂B T H
in (59):
̂B T H = B T H − [ M T hH + ˜B T HS + h B h]
M
−1
= ( BH T − MhHM T −1
h
BT h
h BT h =
)(
I − S
+
h S h)
=
= 1
|E| BT Hēē T D. (61)
To derive the latter formula we used the identity
I − S + h S h = 1 D (62)
|E|ēēT
and the fact that ē ∈ ker B T h .
Thus, the equation (57) is equivalent to the equation
M 0 Hū H + [ B 0 H] T
ˆp + C
T ¯λ =ḡ (63)
where the matrix B 0 H
is defined in (11), i.e.
B 0 H =ē T B H , (64)
and
ˆp = |E|ēT 1 D¯p ≡ 1 ∫
p h dx (65)
|E| E
is the mean value of p h in the polyhedral cell E.
Complementing the equation (63) in E ≡ E k by the equation (10) with
c k = 0, we get the system in terms of ū (k)
H
and ˆp k (we again return the index k):
M 0,(k)
H ū(k) H
+ [ B 0,(k) ] T
H ˆpk + Ck T ¯λ =ĝ k ,
B 0,(k)
H ū(k) H
= −|E k |f k ,
(66)
where M 0,(k)
H
= MH 0 and M H 0 is defined in (58). Recall that the equations (66)
are derived for the case c ≡ 0inE k ,1≤ k ≤ n.
Using the assembling procedure we get the system in terms of ū H ,¯p H ,and
the boundary Lagrange multipliers ¯λ:
M 0 ū H + [ BH] 0 T
¯pH + [ C 0]T ¯λ =ḡ 0 ,
BHūH 0 − Σ 0 ¯p H = F 0 ,
C 0 ū H =0.
(67)