In the last few years, a new area of numerical methods - adaptive finite element methods (FEM's) - culminated in a very complex but advantageous technique for three-dimensional anisotropic adaptivity. This technique uses a very different approach from ``traditional'' FEM. These differences are so profound that, for some users or developers, the methodology is completely foreign. This document explains some of the more important issues which distinguish these versions of the finite element method. We will discuss some new ideas related to adaptive techniques and present new problems which appear as a result of the complexity of this method.
The aim of adaptive methods is to optimize the computational
process-to obtain the
best results for the least effort. The central parameters are the
conventional mesh
parameters that govern local accuracy: the mesh size 
, the order of
approximation
(e.g. the spectral order) and the location of gridpoints.
In - FEM, one can control both the local mesh size and spectral
order of approximation
simultaneously. A very complex issue involves combining these two
kinds of refinements
to achieve optimal improvement in the accuracy. In general, a strict
mathematical
solution is not known. However, there exists heuristic knowledge on
the use of -
FEM's for many classes of problems. The most recent advances in
adaptive FEM's allow
utilization of two- and three-dimensional - techniques for an
anisotropic refinement
strategy. In most cases, this allows one to obtain a solution to a
specified accuracy with a
significant savings in the computational load (relative to older,
isotropic algorithms).
The success of such a complex adaptive scheme depends upon several
properties of the
adaptive process: the data structure, the adaptive strategy, the
technique for a
posteriori error estimation, and the actual problem-solver
algorithms. The potential
payoff of a successful - adaptive strategy is substantial:
exponential rates of
convergence may be attained, meaning that complex problem features can
be resolved
using orders-of-magnitude fewer unknowns than those required by
conventional methods.
It is difficult (if not impossible) to separate adaptive FE methods
from their mathematical
formulation. In this discussion, however, we attempt to limit
ourselves to algorithmic
issues relating to the successful implementation of the - adaptive
techniques, at the
same time trying not to cloud the issue with too many small details
related to computer
programming, database implementation in a particular computer language, and
particular computer hardware.
The basic ideas relating to - adaptivity are presented using the
two-dimensional version of the method. Most of the extensions to
three-dimensional
problems are straightforward and based on identical concepts. Some special
cases related to 3D extensions will be presented in last section of
this appendix.
- adaptivity is the culmination of several years of research, and we
discuss these various stages of its design in order to separate basic
problems and their solutions.
Where appropriate, alternate approaches are presented, along with
discussion of their
relative drawbacks and advantages. Our focus, however, is one particular
implementation of - adaptivity, which is used in the ProPHLEX
Finite Element Library and several FEM codes built upon it, including
the commercial
CFD code, MSC/CFD , PHLEXsolids, designed
for solution of solid mechanics problems, and PHLEXstress: free DEMO
code available
from http://www.comco.com/main/TOC.html. This approach uses
hexahedral elements, and
allows for full anisotropic - adaptation. That is, each element
may be refined (split in
two) in any of its natural directions, and -enriched in several
different ways, again in
an arbitrary direction.
We will not consider here adaptivity obtained through remeshing of the domain
(i.e. building completely new, improved grid of finite elements based on some
error estimate from a previous, more coarse solution). We assume,
instead, that the
error estimate from the existing solution controls refinements and enrichments
(with possible unrefinements and un-enrichments) locally within a single
element (and sometimes including a few of its neighbors). The problem
of defining a reliable
and inexpensive error estimation based on an existing coarse solution, and the
choice of a reasonable adaptation strategy are also beyond the scope of this
presentation. We will only discuss what information should be provided by the
error estimator to fully utilize various adaptation possibilities existing
in modern - technology.