Adaptivity with moving grids

6 C. J. Budd, W. Huang and R. D. Russell
and are now well established in many commercial codes. There is a significant
body of analysis supporting their use. In contrast, r-adaptive methods
are more recent and are less well understood. A significant criticism which
has often been made of them is that their implementation usually requires
the solution of auxiliary partial differential equations for the mesh, which
must be solved in parallel to the underlying partial differential equation.
This requires significant additional computational cost. Furthermore, the
equations to be solved to determine a suitable mesh can often be very stiff,
and thus expensive to solve. Furthermore, the methods get the best estimates
for a given N rather than errors necessarily lower than a specified
tolerance. However, r-adaptive methods do have significant advantages in
certain applications. Firstly, from a computational point of view, it is convenient
to work with the same number of mesh points and the same mesh
topology. This makes the linear algebra rather easier, as the matrices considered
have a constant sparsity structure, and there is no need for any form
of nested data structure to keep track of the node points (an issue which
always complicates the use of h-refinement methods). The discretization
strategy on the mesh is also easier, especially with a finite element method,
as the constancy in the mesh topology and connectivity implies that there is
no possibility of hanging nodes. There are further, structural advantages to
r-refinement methods. One of these is that the movement of the mesh points
may well correspond to natural structures of the PDE itself. An obvious example
is Lagrangian-based methods for fluid flow problems, in which mesh
points move with the fluid flow. A further such example is given by the use
of r-refinement methods for PDEs with natural scaling symmetries, in which
the mesh points automatically follow the motion of natural similarity variables
(and indeed the use of the r-refinement method becomes equivalent
to the use of an appropriate coordinate transformation). A third advantage
of r-refinement is that, under certain circumstances, the adaptive strategy
when coupled with the PDE can be regarded as one (large) dynamical system,
which may then be amenable to a combined analysis. One limitation
of having a fixed number of points means that it may never be possible to
resolve all of the fine structures of a PDE as it evolves (although it is surprising
what can be done with often a relatively small number of mesh points).
Also, all r-adaptive methods are, in principle, prone to mesh tangling, in
which lines connecting the mesh points can cross over during the evolution.
This generally leads to severe instabilities in the system and a failure of
the solution routine. Mesh tangling is often associated with mesh racing,
where some mesh points move very fast during the evolution, frequently
leading to a stiff set of equations to solve. The disadvantages of having to
solve an auxiliary set of partial differential equations are less severe than
they might originally appear to be. Firstly, the combined system of mesh
and underlying equations may be much smaller than the original system