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§2.3 spatial semidiscretization: a cell-node mfd method 15
⊗
(i, j)
(i, j + 1)
⊗
•
( )
1 1 i+ 2 , j+ 2
⊗
⊗
(i + 1, j)
(i + 1, j + 1)
Figure 2.2: Stencil for the discrete divergence operator located at the (i+1/2, j+1/2)-
cell center. Symbols: ⊗ : W X
·,· , W Y
·,·; • : (DW h )i+1/2,j+1/2.
sides of the cell:
∂Ω i+1/2,j+1/2
∫
⟨w, n⟩ ds =
ℓ β
i+1/2,j
∫
+
∫
⟨w, n⟩ ds +
ℓα ℓ β
i+1/2,j+1
∫
⟨w, n⟩ ds +
i+1,j+1/2
ℓ α
i,j+1/2
⟨w, n⟩ ds
⟨w, n⟩ ds.
(2.13)
Since each side of the cell is a straight line, the outward unit vector normal to
, we have:
that side is constant. For instance, for the side ℓ β
i+1/2,j
n i+1/2,j =
(
yi+1,j − yi,j
|ℓ β
i+1/2,j |
, −(xi+1,j − xi,j)
|ℓ β
i+1/2,j |
) T
.
This normal vector is involved in the first integral on the right-hand side of (2.13).
Thus, we can approximate the value of such an integral by the trapezoidal rule
as: ∫
ℓ β
⟨w, n⟩ ds ≈
i+1/2,j
W X i,j + W X i+1,j
(yi+1,j − yi,j)
2
− W Y i,j + W Y i+1,j
2
(xi+1,j − xi,j),
(2.14)
where we have introduced the nodal values of the semidiscrete vector W h ∈ Hv.
The integrals over the other sides can be obtained analogously. As a result, the
discrete divergence D at a mesh cell can be expressed as:
(DW h ) i+1/2,j+1/2 =
(W X i+1,j+1 − W X i,j )(yi,j+1 − yi+1,j) − (W X i,j+1 − W X i+1,j )(yi+1,j+1 − yi,j)
2 |Ω i+1/2,j+1/2|
− (W Y i+1,j+1 − W Y i,j )(xi,j+1 − xi+1,j) − (W Y i,j+1 − W Y i+1,j )(xi+1,j+1 − xi,j)
,
2 |Ωi+1/2,j+1/2| (2.15)