TripleGrid Calculus

Author: Han de Bruijn
Date: 2006 Nov / 2019 Aug

This document is meant as an extension to a previously published one, called MultiGrid Calculus. Upon retrospective, DoubleGrid Calculus would have been a better name for the latter. The reason being that "Multigrid" is much of a reserved word within the world of Numerical Analysis. Therefore its use in a pure mathematics context will likely give rise to confusion. (But maybe that's just intended?) Anyway, quite unexpectedly, it has been 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 these intervals larger or smaller, not with a factor two, but with a factor three. Ah, and now you could think that the next step is a MultiGrid Calculus employing a factor four or maybe five. 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. By the way, the DoubleGrid document has been available all the time as Multigrid Calculus.

  1. Elimination
  2. Quotient Calculus
  3. DoubleGrid Product
  4. TripleGrid Product
  5. Stationary Points
  6. Grid Refinement
  7. Via the Cubic
  8. TripleGrid Refinement
  9. Chebyshev and stuff

Disclaimers

Anything free comes without referee :-(
My English may be better than your Dutch.