Partial Differential Equations - Modelling and ... - ResearchGate

32 Yu.A. Kuznetsov
3 Mixed Finite Element Method
Let ∂ 0 E k,s be the part of the boundary ∂E k,s of a
⋂
polyhedral subcell E k,s
belonging to the interior of E k , i.e. ∂ 0 E k,s = ∂E k,s Ek , s = 1,n k , k = 1,n.
On ∂ 0 E k,s we define a set of t k,s non-overlapping flat polygons γ k,s,j which
satisfies the following two conditions:
1. ∂ 0 E k,s = ⋃ t k,s
j=1 ¯γ k,s,j,
2. each γ k,s,j belongs to ∂ 0 E k,s ′ for some s ′ ≠ s, s ′ ≤ n k ,
where t k,s is a positive integer, s = 1,n k , k = 1,n.
Examples of the partitionings of ∂ 0 E k,s into polygons γ k,s,j are given in
Figures 3 and 4. In Figure 3, the interface ∂ 0 E k,1 = ∂ 0 E k,2 between E k,1
and E k,2 consists of γ k,1 and γ k,2 . In Figure 4, ∂ 0 E k,1 consists of γ k,1,1 = γ 1 ,
γ k,1,2 = γ 2 ,andγ k,1,3 = γ 3 ,and∂ 0 E k,2 consists of γ k,2,1 = γ 3 , γ k,2,2 = γ 4 ,and
γ k,2,3 = γ 5 . Finally, ∂ 0 E k,3 consists of γ k,3,1 = γ 1 , γ k,3,2 = γ 2 , γ k,3,3 = γ 4 ,and
γ k,3,4 = γ 5 .
Let T h,k,s = {e k,s,i } be conforming tetrahedral partitionings of E k,s ,
s = 1,n k , k = 1,n. The conformity of a tetrahedral partitioning (tetrahedral
mesh) means that any two different intersecting closed tetrahedrons in
T h,k,s have either a common vertex, or a common edge, or a common face.
The boundaries ∂E k,s of E k,s are unions of polygons in {Γ k,i } and in
{γ k,s,j }, s = 1,n k , k = 1,n. We assume that each of the tetrahedral meshes
T h,k,s is also conforming with respect to the boundaries of polygons in {Γ k,i }
and in {γ k,s,j } belonging to ∂E k,s , i.e. these boundaries belong to the union of
E
k, 1
γ
1
γ
2
E
k, 3
γ
3
γ
4
E
k, 2
γ
5
Fig. 4. An example of partitionings ∂ 0E k,s into segments γ k,j,s , j = 1,t k , s = 1, 3