Adaptivity with moving grids

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42 C. J. Budd, W. Huang and R. D. Russell for a suitable constant 0
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
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