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Mixed FE Methods on Polyhedral Meshes 37
To derive the required final discretization for the problem (4) we introduce
the new variable ˆp by the formula
ˆp = 1 ] [ē T
σ 2 ˜BH ū H − ē T F ≡ 1 [
]
σ 2 BHū 0 H + |E|f , (38)
where
f = − |E|ēT 1 F. (39)
Then, we get the system in terms of ū (k)
H
and ̂p k (we 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 − c k|E k |ˆp k = −|E k |f k ,
(40)
k = 1,n, complemented by the equations of continuity of normal fluxes on the
interfaces between neighboring polyhedral cells and by the equations for the
normal fluxes on ∂Ω. Here,
ĝ k =ḡ k − 1 σ 2 k
[ ] ˜B(k)
T
H ēk ē T k F k (41)
and the values of c k and f k are defined in (8). Recall that σ 2 k = c k|E k |.
Now, we return to the system (23) and consider the case when the coefficient
c ≡ 0inE k , i.e. Σ k is the zero matrix. In this case, the matrix
S h = B h M −1
h BT h (42)
in (26) is singular.
Let us consider the eigenvalue problem
S h ¯w = µD ¯w, (43)
where the subindex k staying for the number of the cell E = E k is again
omitted. This eigenvalue problem has one zero eigenvalue µ 1 =0andν − 1
positive eigenvalues µ 2 ≤ µ 3 ≤ ··· ≤ µ ν where ν is the dimension of S h .We
denote the system of D-orthonormal eigenvectors of problem (43) by
¯w 1 , ¯w 2 ,..., ¯w ν , (44)
where
¯w 1 = 1 ē. (45)
|E|
1/2
The spectral decomposition of the matrix S h with respect to eigenvalue
problem (43) is defined by the following formula:
S h = DWΛW T D, (46)