- Because one-dimensional space is a subspace of
**all**spaces of higher dimension. So how can you ever understand the latter when you don't even understand the former ? - Discarding the details of an 1-D theory is a manifestation of disdainful behaviour, which has become typical for people working on "demanding applications". Those who find being impressive more important than taking all steps which are necessary, in order to arrive at a thourough understanding of the matter.
- Simple & Straightforward and yet Satisfactory. This remembers to the
good old days of Mathematics. Maybe this report will take you back in time.
However, if you are not interested in Mathematics
**as such**, why then are you reading these pages anyway ? Please find your pleasure elsewhere ! - Apart from personal
emotion,
I find this 1-D MultiGrid Calculus among the best I have ever
done. So if you would have expected more, then I have to apologize for my
shortcomings.

The judgement of a (supposed) authority in the field:*I'm sorry, but I don't see anything in it.*

before you are going to read the real thing.

Generalizations of my (mis)conception of MG were gradually coming up but subsequently have died out:

- 2-D Full MultiGrid implementation

available as a Windows (Delphi) executable

(source code included, as usual)

Thus MG offers no advantage with such 3-D FEA.

The notion of Persistent Properties is developed. The behaviour of a product of the matrix coefficients can be understood in full detail, with help of a Connection to Trigonometry in a "dangerous" domain and a Connection to Hyperbolic functions in a "safe" domain. It turns out that the safe domain is quite distinct from the dangerous one. It is shown that safe solutions can only be found for a certain subset of all second order ODE's (Ordinary Differential Equations), namely those ODE's where the discriminant of the characteristic polynomial is positive (or zero).

The quotient of the off-diagonal matrix coefficients is an exponential function of the grid spacing. All matrix coefficients can be expressed in the coefficients of the accompanying ODE and the grid spacing.

My original intention has been to prove that time indeed has an "arrow". But the above work only explains why time can not be structured like one-dimensional space, with interactions in backward and forward direction. Thus resulting in stable and non-oscillatory matrices, which is contradictory to the nature of time-like events. Multigrid Calculus is incompatible with the time domain.

a = (1 - ε) / 2 b = (1 + ε) / 2 where ε = tanh(P.dx/2)

Here P = Péchlet number, dx = spacing of uniform mesh. A little algebra reveals that this is equivalent with:

a = 1/(e^{P.dx} + 1)
b = 1/(1 + e^{- P.dx}) or
b = e^{P.dx} / (e^{P.dx} + 1)

On the other hand, the following formulas are found in Patankar's *Numerical
Heat Transfer and Fluid Flow*, section 5.2-4, the so-called *Exponential Scheme*:

a_{e} = F_{e}
/ [ exp(P_{e}) - 1 ]
a_{w} = F_{w}
exp(P_{w}) / [ exp(P_{w}) - 1 ]
a_{p} = a_{e} +
a_{w} + ( F_{e} - F_{w} )

It can be demonstrated quite easily that Patankar's *exponential scheme*
is compatible with the one from *Multigrid Calculus*.
For uniform meshes, we can drop the suffixes e and w and normalize the flux F
to unity = 1 , resulting in:

a = 1 / [ exp(P) - 1 ] b = exp(P) / [ exp(P) - 1 ] a + b = [ exp(P) + 1 ] / [ exp(P) - 1 ]

Now multiply both a and b with [ exp(P) - 1 ] / [ exp(P) + 1 ] and we are done.

Anyway, quite unexpectedly, I have discovered that there exists another
kind of Multigrid Calculus, which is distinct from DoubleGrid. It also
works with coarsening and refinement of grids, but does not double or
halve the intervals. Instead, it makes the intervals larger or smaller,
not with a factor two, but with a factor **three**. Ah, and now you would
think that the next step is a MultiGrid Calculus employing a factor 4
or maybe 5 . But this is not so. The factor four being already covered
by a double doubling in the first place. Furthermore, it can be proved
that factors five or higher are not an option, except as a powers of 2
and 3 . Thus all possibilities for MultiGrid are exhausted with DoubleGrid
and TripleGrid.