OPERATOR DEPENDENT GRIDS

                       Daniele Funaro
                  Department of Mathematics
                 University of Modena (Italy)
                       funaro@unimo.it


       I will try to introduce, with a series of examples, an idea
  that I developed  in the  last few years. Everything  started at
  the time  I was  looking  for finite-difference  preconditioning
  matrices  for spectral  discretization  of  convection-diffusion
  equations. The interest then shifted to stabilization techniques
  for the approximation of boundary layers, needed when the degrees
  of freedom are not sufficient for the  resolution of sharp fronts.
  I found out (see the refereces listed below) that the collocation
  at a  special set of  grid  points,  related to the  differential
  operator to be discretized, provided a wonderful treatment of such
  stiff problems,  giving also  suggestions  for  the choice of the
  preconditioners.   After  experiencing with  many other  kinds of
  equations,  I realized that the trick of adapting the grid to the
  operator can be applied in a lot of circumstances.   Therefore, I
  would like to  propose the theory  in a general framework, giving
  also some applications concerning finite-differences approximations
  of  advection diffusion equations.


                             REFERENCES

 [1]  Funaro D. (1993), A new scheme for the approximation of
     advection diffusion equations by collocation, SIAM J. Numer.
     Anal., 30, n.6,  pp.1664-1676.

 [2]  Funaro D., Russo A. (1993), Approximation of advection-diffusion
     problems by a modified Legendre grid, in Finite Elements in
     Fluids, New Trends and Applications (K.Morgan, E.Onate, J.Periaux,
     J.Peraire, O.C.Zienkiewicz Eds.), Pineridge Press, pp.1311-1318.

 [3]  Funaro D. (1997), Spectral Elements for Transport-Dominated
     Equations,  LNCSE, n.1, Springer, Heidelberg.

 [4]  Funaro D. (1997), Some remarks about the collocation method on a
     modified Legendre grid,  Computers  Math. Applic., Vol.33, n.1/2,
     pp.95-103.

 [5]  Funaro D. (1997), Improving the performances of implicit schemes
     for hyperbolic equations, J. of Scientific Computing, v.12, n.4.