Adaptivity with moving grids

Adaptivity with moving grids 5
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Figure 1.1. A logically rectangular mesh, moved to
concentrate points around a ring in an evolving singular
solution of the nonlinear Schrödinger equation. Note the
good radial symmetry of the adapted mesh around the ring.
and Khoo 2007), shear layer calculations (Tang 2005), gas dynamics (Li
and Petzold 1997, Li, Petzold and Ren 1998), hyperbolic conservation laws
with high Mach number (Li and Petzold 1997, Tang 2005, Stockie, Mackenzie
and Russell 2000, Tang and Tang 2003), problems with high vorticity
(Ceniceros and Hou 2001), magneto-hydrodynamics (Tan 2007) and meteorological
problems (Budd and Piggott 2005). More details of such applications
are given in Section 5. In Figure 1.1 we give an example of an
r-adaptive mesh which has evolved to capture the structure of a singular
solution of the nonlinear Schrödinger equation which has its support concentrated
around a ring.
All r-adaptive methods have intimate connections with the geometry
of mapping one domain to another. They thus have intimate links with
problems in differential geometry such as optimal transport (Brenier 1991,
Gangbo and McCann 1996), mean curvature flows (Huang 2007) and harmonic
mappings (Dvinsky 1991). A natural application of such ideas arises
in image processing, and r-adaptivity has close connections with such image
processing procedures as image segmentation and image de-noising
(Sapiro 2003).
There are advantages and disadvantages to each of the strategies outlined
above. As discussed earlier, the hp methods have been in use for a long time