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4 Hybridization and Condensation
Mixed FE Methods on Polyhedral Meshes 35
The underlying system of algebraic equations for the problem (14) can be
written in the macro-hybrid form as follows:
M k ū k + B T k ¯p k + C T k ¯λ k =0,
B k ū k − Σ k ¯p k = F k ,
(22)
k = 1,n, complemented by the continuity conditions for the normal fluxes on
the interfaces ∂E k ∩ ∂E l between neighboring cells E k and E l , k, l = 1,n,and
by the Neumann boundary condition for the normal fluxes on ∂Ω. The vector
¯λ k ∈ R s k
represents the mean values of the solution function p on polygons
Γ k,i , i = 1,s k , k = 1,n. The matrices Σ k are diagonal blocks of the matrix
Σ and the vectors F k are subvectors of the vector F in (15). The matrices
M and B in (15) can be defined by assembling of the matrices M k and B k in
(22), respectively.
We partition the components of the vector ū k in (22) into two groups. In
the first group, denoted by subindex H, we include the DOF assigned for the
polygons Γ k,i , i = 1,s k , on the boundary of E k , and to the second group,
denoted by subindex h, we include the rest of the DOF which are interior for
the cell E k , k = 1,n. Then, the equations (22) can be written in the equivalent
block form (the subindex k is omitted) as follows:
M H ū H + M Hh ū h + BH T ¯p + CT ¯λ =0,
M hH ū H + M h ū h + Bh T ¯p =0,
B H ū H + B h ū h − Σ ¯p = F.
(23)
At first, we consider the case when the coefficient c is a positive function
in E k , i.e. the matrix Σ k in (22) is symmetric and positive definite, 1 ≤ k ≤ n.
We eliminate the vectors ū h and ¯p from (23) in two steps. At the first step,
we eliminate the vector ū h and get the system
˜M H ū H + ˜B H T ¯p + CT ¯λ =0,
˜B H ū H − S h ¯p = F,
(24)
where
˜M H = M H − M Hh M −1
h M hH, ˜BH = B H − B h M −1
h M hH, (25)
and
S h = B h M −1
h BT h + Σ. (26)
It is obvious that the matrices ˜M H and S h are symmetric and positive
definite. Moreover, the dimension of the null space of the matrix B h M −1
h
BT h
equals to one, and the vector ē = ( 1,...,1 ) T
belongs to the null space of this
matrix (ē ∈ ker Bh T ).