FULLY AUTOMATIC hp-ADAPTIVE FINITE ELEMENTS
         Integration of hp-Adaptivity with a multigrid solver

                         Leszek Demkowicz
      Texas Institute for Computational and Applied Mathematics

                   
                   Seminario di Matematica Applicata
         Dipartimento di Matematica "F. Casorati" - Pavia 
                     Pavia,  Tue, Feb. 26, 2002



   Most of commercial implementations of various versions of hp finite
element methods rely on a-priori information about corner and edge 
singularities, or boundary layers, and begin the solution process
with a generation of judiciously constructed initial meshes like the
geometrically graded meshes of Babuska, or Shiskin type meshes for
handling the boundary layers. Once the mesh is known, uniform p
refinements are made. In more sophisticated implementations,
adaptive p-refinements are used. If the nature of the singularity is
known a priori, these techniques are very effective and difficult to
beat. 
   The situation becomes less clear, if the regularity results are not 
at hand. An unoptimal initial mesh, followed by p-refinements, may lead
to meshes that deliver results worse than than standard h-adaptive
methods. Years ago we heard from Prof. Oleg Zienkiewicz that, for error
levels 1-5 percent (measured in energy norm relative to the norm of the
solution), the h-adaptive meshes of quadratic elements are the best,
and there is little need for any extra sophistication. Indeed, for
many practical problems, the quadratic elements offer probably the best 
balance between the complexity of coding (one d.o.f. per node, no need
for orientation of nodes) and the results they can deliver.
   In the presented work I will show our recent results on a fully
automatic hp-adaptive strategy that not only delivers predicted, optimal
exponential convergence rates, but remains fully competitive with 
h-adaptive quadratic elements in the preasymptotic range. The method
draws on our recent progress in the understanding of the idea of hp-
interpolation, and a full integration with two- and multi-grid solvers
for hp meshes. The method delivers a sequence of 'coarse' meshes, with
corresponding sequence of 'fine' meshes obtained from the coarse ones
through a global hp-refinement. Construction of a next coarse mesh
in the sequence is based on minimizing the hp-interpolation error of
the fine mesh solution. Numerical evidence suggests that the solution
on the fine mesh can be replaced with a couple of smoothing operations.