On the Use of Pade Forms for Postprocessing Polynomial Expansion

                  Jan S Hesthaven
         Division of Applied Mathematics
         Brown University, Providence, RI

                 Abstract

We shall discuss the use of Pade forms for post processing polynomial
expansions of functions of low regularity. In such cases, it is well
known that the polynomial representation looses its high-order
accuracy and may even loose pointwise convergence for discontinuous
functions, leading to the well known Gibbs phenomenon.

As we will discuss, postprocessing such data using rational functions,
or Pade forms, enables one to recover high accuracy almost everywhere
at very little computational cost and without knowing the location of
the point(s) of low regularity. This appears to hold true for both
data as well as polynomial expansions recovered from
solving partial differential equations using spectral methods although
the latter requires additional refinements,

An interesting extension of this involves multivariate Pade forms,
leading the way to truly multi-dimensional postprocessing techniques
on simplices.

Finally we shall discuss the use of the Pade postprocessing on top of
the celebrated Gegenbauer reconstruction method for overcoming the
Gibbs phenomenon. It is well known that this latter technique, the
only known way of recovering a spectrally accurate representation of
piecewise smooth data everywhere, is numerically ill conditioned. We
shall demonstrate that this issue can be completely overcome by
postprocessing the Gegenbauer expansion using the Pade forms.

This work was done in collaboration with Laura Lurati (IMA/Boeing) and
Sidi Kaber (Paris VI).