Adaptivity with moving grids

Adaptivity with moving grids 13
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Ω C
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Ω P
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η
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y
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0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
ξ
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x
Figure 2.1. A typical map (x(ξ,η),y(ξ,η)) from a
computational domain Ω C to a physical domain Ω P .
2006), the boundary map is obtained automatically as part of the algorithm.
This is an attractive feature from the perspective of algorithmic complexity,
although it does lead to a reduction of control of the boundary points.
The great merit of this approach is that it transforms the problem of
finding (and describing) a mesh in Ω P (which in the case of h-adaptive
methods can require subtle data structures, including hierarchical trees)
into the much simpler problem of describing the function F. Much of this
article is devoted to deriving suitable equations for F and seeking effective
solution strategies for them. It goes without saying that it should not be
more difficult to find F than to solve the underlying PDE, and indeed that
many such functions F may give appropriate meshes on which to solve
the PDE. The properties of the mesh τ P then follow immediately from the
structure of the map F. This simple observation is a key to the success
of r-adaptive methods, as it allows the use of powerful mathematical tools
to describe, construct and control the mesh behaviour. These include the
application of methods from differential geometry (especially the theory
of optimal transport) to describe the static structure of τ P , and methods
from the theory of dynamical systems to describe its evolution. The latter
is especially appropriate when coupled to the partial differential equations
which are often solved to find F. The unity that r-adaptive meshes give
for both solving the underlying PDE, and finding the mesh, is a significant
advantage of r-adaptivity over other adaptive approaches.