Adaptivity with moving grids

Adaptivity with moving grids 11
As described in the Introduction, in an r-adaptive procedure a fixed numberofmeshpointsaremoved
in response to some user-designed condition.
Any r-adaptive method has two main features, a description of the optimal
geometry of the mesh (which is related both to intrinsic properties of the
mesh regularity and to the structure of the underlying solution of the PDE)
and a strategy for evolving the mesh towards this optimal geometry. Optimal
mesh geometries are typically expressed in terms of equidistribution
measures (related to the solution of the underlying PDE by monitor functions)
or using variational principles, and we review these here. Movement
strategies are generally methods for determining either the location of the
mesh points or the velocity of the mesh points. We discuss both briefly in
this section, and then in more detail in Sections 3 and 4.
Essential to mesh adaptation is the ability to control the shape, size and
orientation of mesh elements, and hence to control the error of the solution
of the underlying PDE. This is done in three steps. Firstly an estimate
of the solution error and/or mesh quality is made. Secondly the mesh is
aligned and moved according to this estimate. Thirdly the solution of the
underlying PDE is advanced on the new mesh. Typically this can be in
response to some structure of the solution of a PDE which is evolving in the
space supporting the mesh; however, there are more general circumstances
(such as in image processing) where we might wish to evolve a mesh in a
manner independent of any PDE.
In this section we will study the geometry of the meshes that arise from
various adaptive strategies, considering such aspects as local element size,
skewness and orientation as well as considering both isotropic and anisotropic
meshes, and looking at solution error estimation and control.
2.1. Mesh-mapping functions
To describe an r-adaptive mesh we consider a fixed computational domain
Ω C ⊂ R n , in which most of the computations associated with the PDE will
be made. The domain Ω C will have the usual Lebesgue measure and may
have a non-trivial topology. We now consider there to be a fixed mesh τ C
on the computational domain. This can either be uniform or non-uniform,
depending on the nature of the underlying problem, and in the simplest
case will be a uniform set of logical rectangles. It can also be triangular,
and usually takes this form when a finite element or finite volume method
is used to discretize the PDE in the physical domain. Alternatively, if
a finite difference or a spectral method is used to discretize the PDE in
the computational domain then a regular rectangular mesh may be more
appropriate. Note that we have a lot of apriorifreedom in the choice
of the computational domain Ω C , and hence when Ω C is simply connected
it is often convenient to consider it to be a logically rectangular domain,