4.3 CONVERGENCE AND ERROR ANALYSIS 111
where I mnJ is the length of edge {m, n}, and h, h' are to be counted
algebraically, in the direction of the outward normal (thus, h < 0 if the
circumcenter C is outside T, as on Fig. 4.10, right part). This way, in the
case where ~t is uniform,
M m n =--~L
(h + h')/Imnl, negative indeed if
the circle condition is satisfied.
This quantity happens to be the flux of ~t grad Kn out of the Voronoi
cell of node m (Exercise 4.12: Prove this, under some precise assumption).
This coincidence is explained in inset: Although
the Voronoi cell and the barycentric box don't
coincide, the flux through their boundaries is the
same, because ~t VK n is divergence-free in the
region in between. But beware: Not only does
this argument break down when there is an obtuse
angle (cf. Exer. 4.12), but it doesn't extend to
dimension 3, where circumcenter and gravity
center of a face do not coincide.
Still, there is some seduction in a formula
m
such as (5), and it
has
a three-dimensional analogue. Look again at Fig.
4.9, middle. The formulas
,.-,_., ,.-,_., ,.-,_.,
Mmn = - [~
F
(area(F) ~(F)]/ I mnl, Mnn = - ~m~ N Mmn'
where F is an ad-hoc index for the small triangles of the dual cell
{m, n}", do provide negative exchange coefficients between n and m,
and hence a matrix with Stieltjes principal submatrices. This is a quite
interesting discretization method, but not the finite element one, and
M~M.
Exercise 4.13. Interpret this "finite volume" method in terms of fluxes
through Voronoi cells.
4.3 CONVERGENCE AND ERROR ANALYSIS
We now consider a family M of tetrahedral meshes of a bounded spatial
domain D. Does q0 m converge toward % in the sense that IIq0- q01I,
tends to zero, when rn... when
m what,
exactly? The difficulty is
mathematical, not semantic: We need some structure on the set M to
validly talk about convergence and limit.
1°The right concept is that
of filter
[Ca]. But it would be pure folly to smuggle that into
an elementary course.

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