Adaptivity with moving grids

Adaptivity with moving grids 3
locally coarsen or refine this by the inclusion or deletion of mesh points. The
strategy for doing this is normally guided by some a posteriori estimate of
the solution error, and may consider problems in which the error is due
to the solution geometry (such as re-entrant corners) or high derivatives.
In p-refinement methods some finite element discretization of the PDE is
used with local polynomials of some particular order. This order is then increased
or decreased in accordance with the solution error. These methods
may be combined with h-refinement methods and with careful a posteriori
estimates to give hp methods (Ainsworth and Oden 2000). The principal
objective of the hp methods is to obtain solutions within prescribed error
bounds by such refinement procedures. There is not usually an upper bound
on the number of points used in the calculation. Such methods have now
been developed to a high degree of sophistication. However, they are necessarily
rather complex, need not take advantage of any dynamic properties
of the underlying solution, and the a posteriori error estimates rely heavily
on certain assumptions on the solution which may be hard to verify for
strongly nonlinear problems.
The r-refinement (relocation refinement) moving mesh methods which
will form the substance of this article are a more recent development than
hp methods. Whilst not as widely used as h- or p-adaptive methods,
r-adaptivity has been used with success in many applications including
computational fluid mechanics (Tang 2005), phase field models and crystal
growth (Mackenzie and Mekwi 2007a), and convective heat transfer
(Ceniceros and Hou 2001). It also has a natural application to problems
with a close coupling between spatial and temporal length scales,
such as in problems with symmetry, scaling invariance and self-similarity
(Barenblatt 1996, Budd and Williams 2006), where the mesh points become
the natural coordinates for an appropriately rescaled problem. Less is
known about the behaviour of r-adaptive methods than of the much more
extensively developed hp methods, and (at least in higher dimensions) they
have yet to become part of established large numerical codes. In particular,
as we shall see in this article, many outstanding open questions remain on
their convergence, the nature of the meshes that they generate and the error
estimates that can be obtained when using them to solve PDEs with rapidly
evolving structures. As a consequence, much of the analysis of such methods
has been for one-dimensional problems, and the one-dimensional PDE
solver MOVCOL (Huang and Russell 1996, Russell, Williams and Xu 2007)
and the celebrated continuation code AUTO (for solving two-point boundary
value problems amongst others) both make use of r-adaptive methods.
However, r-refinement methods show great potential for solving a much
greater range of problems, as we hope to demonstrate in this article.
The r-refinement methods start with a uniform mesh and then move the
mesh points, keeping the mesh topology and number of mesh points fixed