Adaptivity with moving grids

Adaptivity with moving grids 43
where high resolution is required, and half where it is not. Such meshes
are called 50:50 meshes (see Budd et al. (2005), Huang (2001a), Huang
et al. (2002)) and a regularization of M to ensure that such meshes arise in
practice has been proposed by Beckett and Mackenzie (2000). To show how
such problems arise in a calculation in n dimensions, suppose that we have
a scalar monitor function M for which ∫ Ω P
M dx = θ. Consider now the
situation in which there are two subsets A and B of Ω P ,withΩ P = A ∪ B
so that the monitor function is designed to concentrate points in a small
region A (so that A may be the support of a singularity or of a front). The
preimage A ′ = F −1 (A) ⊂ Ω C represents those points in the computational
domain which are mapped to A, with a similar set B ′ . Suppose now that
|A ′ | and |B ′ | are the areas of these sets in Ω C , with respective areas |A| and
|B| in Ω P . These areas measure the proportion of mesh points allocated
to the corresponding sets A and B. Note that in most applications of an
adaptive method, where mesh points have to be concentrated into a small
region we would expect that
|A ′ | = O(1), |A| = o(1), |B ′ | = O(1), |B| ≈|Ω P |. (2.71)
Problems arise with mesh regularity and solution resolution away from
the set A if |B ′ |≪|A ′ |. It follows immediately from the equidistribution
principle that if θ = ∫ Ω P ≡A∪B
M dx then
∫
Λ= |A′ |
|B ′ | = ∫ A M dx
B M dx = θ
− 1.
∫B
M dx
Furthermore,
∫
|A ′ A
| =
M dx .
θ
It follows from the conditions on A ′ and A in (2.71) that over the set A we
have M ≫ θ. However, if the monitor function is so constructed such that
over the set B we have M ≪ θ and hence ∫ B
M dx ≪ θ (so that the integral
of M is concentrated in A), then Λ will be very large and the mesh will lose
regularity. Indeed, we will have |A ′ |≈1. Exactly such problems arose in
some of the blow-up calculations reported in Budd et al. (1996). In such
cases we must replace M by the regularized function introduced by Beckett
and Mackenzie (2000) and given by
ˆM = M +
θ
|Ω P | . (2.72)
Observe that over the set A we have ˆM ≈ M, andoverB we have ˆM ≈
θ/|Ω P |, and trivially
∫
ˆM dx =2θ.
Ω P