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54 E.J. Dean and R. Glowinski
Fig. 3. A uniform triangulation of Ω =(0, 1) 2 (h =1/8)
Remark 7. Suppose that Ω =(0, 1) 2 and that triangulation T h is like the one
shown in Figure 3.
Suppose that h = 1
I+1
, I being a positive integer greater than 1. In this
particular case, the sets Σ h and Σ 0h are given by
{
Σh = {P ij | P ij = {ih, jh}, 0 ≤ i, j ≤ I +1},
(40)
Σ 0h = {P ij | P ij = {ih, jh}, 1 ≤ i, j ≤ I},
implying that N h =(I +2) 2 and N 0h = I 2 . It follows then from the relations
(37) and (38) that (with obvious notation):
and
Dh11(ϕ)(P 2 ij )= ϕ i+1,j + ϕ i−1,j − 2ϕ ij
h 2 , 1 ≤ i, j ≤ I, (41)
Dh22(ϕ)(P 2 ij )= ϕ i,j+1 + ϕ i,j−1 − 2ϕ ij
h 2 , 1 ≤ i, j ≤ I, (42)
Dh12(ϕ)(P 2 ij )= (ϕ i+1,j+1 + ϕ i−1,j−1 +2ϕ ij )
2h 2
− (ϕ i+1,j + ϕ i−1,j + ϕ i,j+1 + ϕ i,j−1 ) /(2h 2 ), 1 ≤ i, j ≤ I. (43)
The finite difference formulas (41)–(43) are exact for the polynomials of degree
≤ 2. Also, as expected,
Dh11(ϕ)(P 2 ij )+Dh22(ϕ)(P 2 ij )= ϕ i+1,j + ϕ i−1,j + ϕ i,j+1 + ϕ i,j−1 − 4ϕ ij
h 2 ;
(44)
we have recovered, thus, the well-known 5-point discretization formula for the
finite difference approximation of the Laplace operator.
6.3 On the Least-squares Formulation of (E-MA-D) h
Inspired by Sections 3 to 5, we will discuss now the solution of (E-MA-D) h by
a discrete variant of the solution methods discussed there. The first step in