Adaptivity with moving grids

34 C. J. Budd, W. Huang and R. D. Russell
through equidistributing a suitable monitor function determined during the
computation.
It is important to note at this stage that the error in discretizing a differential
equation (or indeed in interpolating the solution to that equation or a
function in general) is a combination of the error that would occur on a uniform
mesh together with further errors that arise from the non-uniformity
of the mesh (variation in the size of the elements) and (in more than one
dimension) the mesh skewness. The latter errors have to be treated with
great care as they can easily dominate the truncation error on the uniform
mesh and make an adapted mesh worse than useless in solving the underlying
problem. However, in contrast, a common error in many numerical
analysis texts is to assume either that these errors add together to give a
larger error, or that, for example, the error due to the non-uniformity of
the mesh is always at a lower order than the error on the uniform mesh
and thus dominates the overall calculation. In fact, provided that the mesh
function is suitably smooth, for example if (in one dimension) the mesh is
quasi-uniform and obeys the condition (2.9), then the three errors can be
at the same order when expressed in terms of 1/N p . Inan‘optimal’mesh
it may be possible for the errors to cancel each other out to leading order.
However, such meshes are usually very hard to construct and require
a lot of aprioriinformation about the solution. Adaptive meshes generally
work by bounding the leading order error (regardless of the behaviour of
the underlying solution).
We start by looking at both optimal and adaptive non-uniform meshes on
which we can pose finite difference discretization of the Poisson equation:
− d2 u
= f(x). (2.46)
dx2 If the mesh is a function x(ξ) of the computational variable, then in the
computational domain we have
− 1 ( )
uξ
= f(x(ξ)), J = x ξ . (2.47)
J J
ξ
The equation (2.47) can then be discretized in the computational domain
for which we use the approximations
U j ≈ u(X j ), X j = x(j∆ ξ ), f j = f(X j ).
A natural centred difference approximation to (2.47) then takes the form
with
(
− 2
Uj+1 −U j
X j+1 −X j
− U j−U j−1
(∆ξ) 2 = f j , (2.48)
X j+1 − X j−1
X j −X j−1
)
∆ i = X i+1 − X i .