Parallel Element-by-Element Spectral Multilevel
                        Techniques for Finite Elements

                         M. B. Davis and G. F. Carey
               ASE and EM Dept., University of Texas at Austin

                              December 19, 1994

                                   Abstract

A parallel element-by-element multilevel strategy is developed and applied to
two candidate PDE systems.  Spectral (p) finite elements are used to
discretize the problem and the multilevel solution strat-egy uses projections
between bases of different degree (level).  Hierarchic bases are particularly
well suited since the element matrices and vectors are nested and the
projections easily defined and performed.  The projection methods for p-m
ultilevel are particularly important and are developed and analyzed for
Lagrange and hierarchic bases.  The element-by-element (EBE) parallelization
is natural for the finite element method, and if basis degree is used to
specify the multigrid level, an EBE strategy is natural for the multilevel
technique as well.  Algorithm scalability and efficiency is analyzed and
tested.  Results are presented for two candidate nonlinear elliptic transport
problems:  the augmented drift-diffusion equations of semiconductor device mo
deling and the stream function-vorticity equations of incompressible fluid
dynamics.