Adaptivity with moving grids

Adaptivity with moving grids 7
defined on a uniform mesh for the same level of accuracy. Indeed, in Section
5 we will give an example of the solution of focusing behaviour in the
one-dimensional nonlinear Schrödinger (Sulem and Sulem 1999) equation
(which can be written as four real first-order PDEs), where a discretization
of the PDE on a set of N = 81 moving mesh points is able to resolve singular
structures with a length scale of 10 −5 , and outperforms discretizations
on uniform meshes with 10 5 mesh points. Hence the additional 81 auxiliary
equations for the mesh gives a method which outperforms one with 10 5
equations, giving a very significant cost reduction. Secondly, although the
equations describing the mesh are indeed stiff in general, we do not need
(in general) to solve them exactly. After all, it is the underlying solution of
the PDE that we are interested in, and not the mesh on which it is solved.
Thus a quite rough approximation to the solution of the moving mesh equations
will often deliver a mesh more than adequate for the task of resolving
the structures of the underlying PDE. Indeed, we will argue in Section 3
(and demonstrate by example in Section 5) that a relaxed version of the
moving mesh equations can be solved using a simple explicit method, will
deliver similar performance for much stiffer equations than the meshes obtained
by more computationally expensive methods. Indeed, they may well
be stable, more regular, and deliver a better mesh quality than solving the
exact equations for the mesh. Finally, one of the main applications of hp
methods is to solve otherwise regular PDEs on irregular domains, typically
with re-entrant corners, that introduce significant errors due to a lack of
solution regularity at the corner. The r-adaptive methods as described in
this paper are not really the right tool for this job (though see the results
in Touringy (1998)). However, a combination of h and r methods may well
prove optimal in this case, where the h method is used to mesh around the
corner and the r method to follow any evolving solution structure. Future
attempts to combine these two types of adaptive refinement in a general
context should prove to be most interesting.
1.3. Computation on moving meshes
The problem of computing solutions of PDEs using a moving mesh method
separates into three related problems.
(1) As described, we need some monitor function to guide the mesh evolution,
which is typically constrained either to equidistribute this function,
or to relax towards an equidistributed state. In practice, whatever
the choice of monitor function, some spatial (and temporal) smoothing
is usually employed.
(2) Having determined the monitor function, we must determine a mesh
which equidistributes this in some way. The equidistribution problem
itself is a nonlinear algebraic problem, and a variety of techniques have