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Mixed FE Methods on Polyhedral Meshes 33
edges of tetrahedrons in T h,k,s , s = 1,n k , k = 1,n. We do not assume that the
tetrahedral meshes T h,k,s and T h,k′ ,s′ are conforming on the interfaces between
neighboring cells E k and E k ′ when k ′ ≠ k as well as on the interfaces between
neighboring subcells E k,s and E k,s ′ when k ′ = k.
Let T h be a tetrahedral partitioning of Ω such that its restrictions onto
E k,s coincide with the tetrahedral meshes T h,k,s , and let RT 0 (E k,s )bethe
lowest order Raviart–Thomas finite element spaces on T h,k,s , s = 1,n k , k =
1,n. We define the finite element spaces V h,k,s consisting of vector functions
w ∈ RT 0 (E k,s ) which have constant normal fluxes w · n k,s on each of the flat
polygons Γ k,i and γ k,j belonging to ∂E k,s ,wheren k,s are the outward unit
normals to ∂E k,s , s = 1,n k , k = 1,n. Then, we define the spaces V h,k on E k
assuming that the restrictions w k,s of any vector function w k ∈ V h,k onto
E k,s belong to the spaces V h,k,s , s = 1,n k , and the normal components of
w k are continuous through γ k,s,j , j = 1,t k . To satisfy the latter condition
we assume that on each polygon γ k,s,j belonging to ∂E k,s ∩ ∂E k,s ′, s ′ ≠ s,
the outward normal components of vector functions w k,s and w k,s ′ satisfy
the equalities w k,s · n k,s + w k,s ′ · n k,s ′ = 0 (we recall that n k,s + n k,s ′ =0),
j = 1,t k,s , k = 1,n.
Finally, we define the finite element space V h assuming that the restrictions
w k of any vector function w ∈ V h onto E k belong to the spaces V h,k
and the normal components of w are continuous on the interfaces ∂E k ∩ ∂E l
between E k and E l . To satisfy the latter condition we assume that on each
polygon Γ k,i belonging to ∂E k ∩∂E l the outward normal components of vector
functions w k and w l satisfy the condition w k · n k + w l · n l =0,1≤ i ≤ s k ,
l ≠ k, k, l = 1,n.
We define the finite element space Q h for the solution function p by setting
that functions in Q h are constant in each of the tetrahedrons in the partitionings
T h,k,s , s = 1,n k , k = 1,n. With the defined FE spaces V h and Q h ,
the mixed finite element discretization to (4) is as follows: Find u h ∈ V h ,
u h · n =0on∂Ω, andp h ∈ Q h , such that
∫
∫
(
a −1 )
u h · v dx − p h div v dx = 0,
Ω
Ω
∫
∫
∫
(14)
− div u h q dx − cp h q dx = − fqdx
Ω
Ω
for all v ∈ V h , v · n =0on∂Ω, andq ∈ Q h .
Finite element problem (14) results in the system of linear algebraic equations
Mū + B T ¯p + C T ¯λ =0,
Bū − Σ ¯p = F,
(15)
Cū =0.
Here, M ∈ Rˆn׈n is a symmetric positive definite matrix, Σ ∈ R N×N is either
a symmetric positive definite or a symmetric positive semidefinite matrix,
B ∈ R N׈n ,andC ∈ Rñ׈n ,whereˆn =dimV h , N is the total number of
Ω