MultiGrid Calculus

Author: Han de Bruijn (1999)
Latest revision: 2019 August

The well-known Newton-Raphson algorithm is used as a starting point for still another method for inverting tri-diagonal matrices. It is shown that this method is closely related to MultiGrid algorithms. The notion of Persistent Properties is developed. The quotient of the off-diagonal matrix coefficients proves to be an exponential function of the grid spacing. 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 Hyperbolic Connection in a safe domain. The safe domain is quite distinct from the dangerous one. It is shown that all safe solutions form a sampling of the analytical solutions of the second order linear ODE (Ordinary Differential Equation). But it is demanded that the discriminant of this ODE is positive or zero. The matrix coefficients can be expressed in the coefficients of the ODE and the grid spacing.

  1. Newton-Raphson
  2. $2 \times 2$ matrix
  3. MultiGrid
  4. Direct Solver
  5. Persistent Properties
  6. Quotient Function
  7. Evidence once more
  8. Some Stable Solutions
  9. Product Function
  10. Trigonometric Connection I
  11. Trigonometric Connection II
  12. The Hyperbolic Connection
  13. Governing Equation
  14. Upper and Lower case
  15. L.U. Decomposition
  16. All Possible Cases
  17. The Main Result
  18. P.P. Summary
  19. Incremental Jacobi method
  20. The End of 1-D MultiGrid?

Reference

S.V. Patankar, "Numerical Heat Transfer and Fluid Flow",
Hemisphere PublishingCompany U.S.A. 1980.