Adaptivity with moving grids

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