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