


CAIMS Research Prize / CAIMS Prix de Recherche
 ROBERT D. RUSSELL, Dept. of Mathematics, Simon Fraser University, Burnaby, BC
V5A 1S6
A Review of Some Adaptive Methods for Solving Differential
Equations

Supercomputers notwithstanding, the solution of timedependent partial
differential equations requires the use of adaptive techniques for
which the computational mesh is suitably chosen to capture special
solution features. Given the wide array of available techniques, it
can be a daunting task for users to determine which one might work
best for their particular applications.
In this talk, we focus on two basic types of methods of moving the
mesh in time and discuss how they encompass many of the approaches
which have been used. The history of their development is a long one,
with marked practical and theoretical improvements in the methods made
by scientists and engineers from a diversity of fields (including many
"pure" areas of mathematical analysis). The first approach is based
upon minimizing a suitable variational form involving the mesh
transformation itself, while the second involves computing mesh
velocities directly. Interpreting them both as ways of finding a
coordinate transformation from physical to computational coordinates
provides insight into why each faces different difficulties for higher
dimensional problems. We discuss some recent theoretical developments
which help to both explain how these traditional difficulties may be
partly overcome. We also relate these adaptive mesh problems to some
other general problems in science and engineering. Finally, some
numerical examples are given which illustrate the practicability of
these new approaches.

