Partial Differential Equations - Modelling and ... - ResearchGate

30 Yu.A. Kuznetsov
A mesh cell E with discontinuities either of the entries of the matrix a, or
the coefficient c, or both is said to be a mixed cell.
On the boundary ∂E k of a polyhedral cell E k we define a set of s k nonoverlapping
flat polygons Γ k,i , i = 1,s k , which satisfies the following three
conditions:
1. ∂E k = ⋃ s k
i=1
Γ k,i ;
2. each Γ k,i belongs to ∂E k,s for some s ≤ n k ;
3. each Γ k,i belongs either to ∂Ω or to ∂E k′ ,s ′ for some k′ ≠ k, s ′ ≤ n k ′,
k ′ ≤ n,
where s k is a positive integer, k = 1,n. A 2D example of the partitioning of
∂E k into Γ k,i , i = 1,s k , with s k = 8 is given in Figure 3.
The goal of this paper is to develop a mixed finite element method for the
diffusion problem (4) on the above described polyhedral meshes under special
conditions on the degrees of freedom (DOF) which can be used for discretization.
Namely, the final discretization can use only one DOF representing the
normal component of the solution flux vector function u in (4) on each Γ k,i ,
i = 1,s k , and only one DOF representing the solution function p in (4) in
each E k , k = 1,n.
To predict the final discretization scheme to be derived in Section 4, we
define the required discrete equation in E k for the second equation in (4) by
integrating this equation over the mesh cell E k :
Ω 3
Ω 2
Γ
k, 2
Γ
k, 3
E
k, 1
Γ
k, 4
γ
k, 2
Γ
k, 5
Γ
k, 1
γ
k, 1
E
k, 2
Γ
k, 6
Γ
k, 8
Γ
k, 7
Ω 1
Fig. 3. A 2D example of the partitionings ∂E k into Γ k,i , i = 1, 8, and ∂E k,1
⋂
∂Ek,2
into γ k,j , j =1, 2