Partial Differential Equations - Modelling and ... - ResearchGate

Mixed FE Methods on Polyhedral Meshes 31
∫
∫
[− div u − cp]dx=− f dx, k = 1,n. (5)
E k E k
The latter equality results in the discrete equation
∑s k
− u k,i |Γ k,i |−c k |E k |ˆp k = −|E k |f k , (6)
i=1
where
u k,i = 1 ∫
u · n k ds (7)
|Γ k,i | Γ k,i
is the mean value of the normal flux u · n k on Γ k,i ,
c k = 1
|E k |
∫
c dx and f k = 1 ∫
f dx (8)
E k
|E k | E k
are the mean values of c and f in E k , respectively,
∫
cp dx
ˆp k = ∫ E k
c dx
E k
(9)
is the c-weighted mean value of p in E k . Here, |Γ k,i | and |E k | denote the length
of Γ k,i and the area of E k , respectively, i = 1,s k ,andn k is the outward unit
normal to ∂E k , k = 1,n.
The equation (6) can be written in the matrix form by
B 0,(k)
H ū(k) − c k |E k |ˆp k = −|E k |f k , (10)
where
B 0,(k)
H
= − [ |Γ k,1 | ··· |Γ k,sk | ] ∈ R 1×s k
(11)
and ū (k) = [ ] T
u k,1 ,...,u k,sk ∈ R
s k
, k = 1,n. The matrix B 0,(k)
H
will be used
later to derive the final discretization for the problem (4).
The formula (9) assumes that the coefficient c is not equal identically to
zero in E k . In the case c ≡ 0inE k the discrete equation (6) is replaced by
the equation
−
s k
∑
i=1
and (10) is replaced by the equation
u k,i |Γ k,i | = −|E k |f k , (12)
B 0,(k)
H ū(k) = −|E k |f k . (13)