Adaptivity with moving grids

Adaptivity with moving grids 33
it difficult to measure (or indeed to precisely define) the error during the
calculation and adapting the mesh accordingly, but also some of the ‘best’
adaptive meshes for solving time-dependent problems lead to very stiff differential
equations, thus significantly increasing the computational cost of
the process and in some cases making the whole calculation significantly
unstable. However, it is intuitively reasonable that for solutions with small
length scales over part of the domain, and larger length scales elsewhere, we
might expect to gain significant efficiency by using a smaller mesh in the
region of high variation. Exactly this observation motivated the important
early work of Dorfi and Drury (1987).
2.8.1. Static truncation error and ‘optimal’ meshes
The most natural reason for using an adaptive mesh in the context of solving
a PDE is to control the overall error in any discretization. It is a
surprisingly difficult problem to obtain such an estimate in the context of a
(non-uniform) moving mesh. Typically there are contributions to the local
truncation error from the local mesh scale, the variation of the mesh from
one element to the next, and also the effects of the mesh motion, which
all need to be taken into consideration. It is also difficult to then extrapolate
from a local to a global error estimate in such cases. Accordingly, we
will confine ourselves to giving a flavour of this analysis by looking at some
simple one-dimensional problems for which we can perform the technical
calculations needed to analyse the error. Following this we will return to
more general ideas shortly. Accordingly, as examples of two steady-state
problems for which adaptivity may be required, we may wish to solve the
Poisson equation
−∆u = f(x,y,z,...) (2.44)
for a potentially singular right-hand side f. Alternatively we may seek to
solve the singular diffusion equation
−ɛu ′′ − c(x)u ′ = f(x), ɛ ≪ 1. (2.45)
An ideal mesh with N points, used to compute the function u, isone
which leads to low errors – ideally, in the case of (2.45), to errors which are
ɛ-independent and depend only upon N. Such a mesh should also keep computational
costs low. Essentially we can consider two types of non-uniform
mesh for the computation. One type is an optimal, fitted or a priori mesh,
which is prescribed in advance of the calculation and gives best possible errors
for that computation in some appropriate norm. Important examples
of this class are the Shishkin and Bakhvalov meshes for the singularly perturbed
problems (2.45) and optimal meshes for Poisson-type problems. In
Babuška and Rheinboldt (1979), a general analysis of such meshes is made
in the context of finite element calculations in one dimension. An adaptive
mesh, on the other hand, attempts to approximate an optimal mesh