Partial Differential Equations - Modelling and ... - ResearchGate

34 Yu.A. Kuznetsov
tetrahedrons in T h ,andñ is the total number of polygons Γ k,i , i = 1,s k ,
k = 1,n, belonging to ∂Ω. The components of the Lagrange multiplier vector
¯λ ∈ Rñ represent the mean values of the solution function p on the polygons
Γ k,i ⊂ ∂Ω, i = 1,s k , k = 1,n. The third matrix equation in (15) takes care of
the Neumann boundary condition on ∂Ω.
We also consider another discretization to (4): Find u h ∈ V h , u h · n =0
on ∂Ω, andp h ∈ Q h such that
∫
∫
(
a −1 )
u h · v dx − ˜p h div v dx = 0,
Ω
Ω
∫
∫
∫
(16)
− div u h q dx − c˜p h q dx = − ˜f h q dx
Ω
Ω
for all v ∈ V h , v · n =0on∂Ω, andq ∈ Q h . Here,
˜p h (x) = 1 ∫
p h (x ′ )dx ′ ,
|E k,s | E k,s
x ∈ E k,s , (17)
and
˜f h (x) = 1 ∫
f(x ′ )dx ′ ,
|E k,s | E k,s
x ∈ E k,s , (18)
where |E k,s | is the volume of E k,s , s = 1,n k , k = 1,n.
The finite element problem (16) results in the system of linear algebraic
equations
Mū + B T ¯p + C T ¯λ =0,
Bū − ˜Σ ¯p = F 1 ,
(19)
Cū =0,
where the matrices M, B, andC are the same as in the system (15). The
matrix ˜Σ ∈ R N×N is a block diagonal matrix with Ñ = ∑ n
k=1 n k diagonal
submatrices
˜Σ k,s = 1
|E k,s | c k,sD k,s ē k,s ē T k,sD k,s ∈ R N k,s×N k,s
(20)
and the vector F 1 ∈ R N consists of Ñ subvectors
F k,s = −f k,s D k,s ē k,s ∈ R N k,s
(21)
(one matrix ˜Σ k,s and one vector F k,s per subcell E k,s ), where c k,s is the
value of the coefficient c in E k,s , f k,s is the value of the function ˜f h in E k,s ,
ē k,s =(1,...,1) T ∈ R N k,s
,andN k,s is the total number of tetrahedrons in
T h,k,s , s = 1,n k , k = 1,n. Here, D k,s are diagonal N k,s × N k,s matrices with
the volumes of tetrahedrons {e k,s,i } in T h,k,s on the diagonals, s = 1,n k ,
k = 1,n.
In Section 4, we shall prove that the method (16)–(18) is equivalent to
the “div-const” mixed finite element method [KR03, KR05] on the polyhedral
mesh consisting of the polyhedral mesh cells E k,s , s = 1,n k , k = 1,n.
Ω