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 1° 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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