Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
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<strong>Adaptivity</strong> <strong>with</strong> <strong>moving</strong> <strong>grids</strong> 41<br />
det(I + α −1 |H(u)|) 2<br />
n+4 is large. Finally, the error bound reads as<br />
‖e h ‖ 2 L 2 (Ω P ) ≤ Cα2 N − 4 n . (2.68)<br />
From (2.67) it is not difficult to show that, for n ≤ 4, α is bounded as<br />
[ ∫<br />
] n+4<br />
1<br />
det(|H(u)|) 2<br />
2n<br />
n+4 dx ≤ α ≤<br />
2|Ω P | Ω P<br />
[ ∫<br />
] n+4<br />
1 ( ) 2n 2n<br />
trace(|H(u)|)<br />
n 2n<br />
n+4<br />
dx . (2.69)<br />
n+4 |ΩP | Ω P<br />
2.8.3. Dynamic error<br />
Usually when we apply a <strong>moving</strong> mesh method we are interested in solving a<br />
time-evolving PDE. This leads to additional dynamic errors (Li and Petzold<br />
1997, Li et al. 1998) such as oscillations around rapidly evolving fronts or<br />
miscalculations of the front speed. These depend significantly on the way<br />
in which the mesh is updated and coupled to the PDE. We consider these<br />
in more detail in the next section when we look at how the <strong>moving</strong> mesh<br />
equations are coupled to the underlying PDE.<br />
2.9. Monitor function smoothing and regularization<br />
Having considered the mesh quality, we now return to further considerations<br />
of the monitor function and of mesh smoothness. Recall that for onedimensional<br />
problems it is essential that the mesh should be quasi-uniform<br />
in order to have a low truncation error. Smoothing a mesh either directly<br />
or indirectly through smoothing/averaging aims to achieve this.<br />
2.9.1. The Dorfi and Drury method<br />
A direct approach to smoothing a one-dimensional mesh derived from an<br />
equidistribution principle is proposed in Dorfi and Drury (1987), and is<br />
often called the Dorfi and Drury method. In this method, if<br />
n i =(∆X i ) −1 ≡ ( ) −1,<br />
X i+1 − X i<br />
then a smoother mesh is given by computing ˆn i where<br />
ˆn i = n i − γ ( n i+1 − 2n i + n i−1<br />
)<br />
(2.70)<br />
for a suitable constant γ. A variant of this, considered in Li and Petzold<br />
(1997), is given by updating the mesh differences by<br />
∆ ˆX i =<br />
i+p<br />
∑<br />
j=i−p<br />
θ |i−j| ∆X i