CENTER FOR COMPUTATIONAL MATHEMATICS COLLOQUIUM

                  UNIVERSITY OF COLORADO AT DENVER



TITLE:   Spectral Elements for Advection-Diffusion Equations
 

SPEAKER: Daniele Funaro, Dipartimento di Matematica, Universita di Modena,
         Modena, Italy 
         

DATE:    Monday, June 30, 1997


PLACE:   Math Conference Room 626 
         UCD Building, 1250 14th St., Denver


TIME:    noon (Refreshments served at 11:45 am) 



ABSTRACT

We would like to describe a way to approximate boundary-value problems 
by spectral elements, which is particularly suited for advection-diffusion
equations, where the second-order diffusive terms are largely dominated by the
first-order transport terms.

First of all, we need a good solver for the case of the single domain (i.e. the
usual square in 2-D or the cube in 3-D). We collocate  the equation at a set
of points obtained by shifting upwind (with respect to the direction of the flux 
individuated by the advective terms) the standard Legendre nodes. This procedure 
is interesting for two reasons: it provides stabilized approximated solutions
(not affected by spurious oscillations) and allows the construction of
very effective finite-differences preconditioners, giving a fast solution
of the systems by an iterative solver, with a number of iterations not depending 
on the polynomial degree and the size of the advective terms.

The second issue concerns the implementation of the spectral elements. An
iterative domain-decomposition algorithm reduces the global solution to a
sequence of subproblems in each single domain. The technique we adopted
is standard. In practice, we impose Dirichlet boundary conditions at the 
interfaces, and update the interface values according to the error of the 
normal derivatives of the multidomain solution across the interface sides. 
A speed up of the convergence is obtained by a suitable tridiagonal
preconditioner applied  to the values at the nodes of each interface side.
This is the crucial part of the implementation. We present a preconditioning 
matrix  capable to handle advection dominated equations, resulting in a 
convergence behavior which seems to be independent, as pointed out by numerical 
experiments, of both the polynomial degree and the size of the advective terms. 
To this purpose a special set of grid points has to be taken on each interface 
side. These nodes are located upwind with respect to the direction of the 
tangential component of the flux along the interface.