Adaptivity with moving grids

22 C. J. Budd, W. Huang and R. D. Russell
can be mapped to both circular and spherical domains. This is especially
useful for calculations in meteorology involving whole Earth models. However,
moving mesh methods are not ideal for mappings to and from nonconvex
regions, due to the inherent singularities associated with re-entrant
corners. See Dvinsky (1991) for a brief discussion of this point. The issue
of refining a mesh close to such a corner where the geometry of the solution
and the associated singularity is (of course) known apriorihas been extensively
covered in the literature of h-adaptive methods: see, for example,
Ainsworth and Oden (2000), Johnson (1987) and many other texts. This
approach can be very naturally coupled with a moving mesh approach by
using an h-adaptive method to construct a mesh in the computational domain,
which is refined close to the re-entrant corner. This mesh can then be
mapped, in a similar manner to that described earlier, to a moving mesh in
the physical domain. We will not pursue this further here as this article is
largely concerned with the construction of meshes adapted to the evolving
structures of time-dependent PDEs.
2.5. Equidistribution and monitor functions
Having considered the general aspects of the mesh geometry and the mesh
function F, we now consider the issues associated with calculating appropriate
functions F to give meshes with certain properties. There are several
general approaches to this, and we consider two closely related methods:
equidistribution-based and variational-based. Both methods generate
meshes determined by suitable monitor functions, which are typically determined
both by properties of the solution of the underlying partial differential
equation and by other considerations of the mesh regularity.
2.5.1. Equidistribution
At the heart of many r-adaptive methods is the concept of equidistribution,
introduced as a computational device by de Boor (1973). Equidistribution is
a widely used means of prescribing the optimum geometry of the mesh, but
many different strategies have been devised to move the mesh towards this
optimum state, leading to a variety of moving mesh methods. In a certain
sense, all meshes equidistribute some function, and to motivate equidistribution
we consider the fundamental Radon–Nikodym theorem from measure
theory. To do this we consider an invertible mesh mapping function F which
maps an arbitrary set A in Ω C to an image set B = F (A) inΩ P .Wecaninduce
a measure ν(B) onΩ P by ν(B) =|A|, where |A| is the usual Lebesgue
measure on Ω C . We then have the following.
Theorem 2.2. (Radon–Nikodym) If ν is a well-defined Borel measure
on Ω P , then there is a non-negative measurable function M :Ω P → R such