Adaptivity with moving grids

40 C. J. Budd, W. Huang and R. D. Russell
≥ n 2 |Ω C | − 4 n
[∫
] n+4
|J|det(I + α −1 |H(u)|) 2
n
n+4 dξ
Ω C
[∫
(2.61)
] n+4
= n 2 |Ω C | − 4 n det(I + α −1 |H(u)|) 2
n
n+4 dx . (2.62)
Ω P
We note that equality in (2.61) holds when the mesh satisfies
|J|det(I + α −1 |H(u)|) 2
n+4 =
1
|Ω C |
∫
Ω P
det(I + α −1 |H(u)|) 2
n+4 dy.
Comparing this with the equidistribution condition (2.22), we have
√
det(M) = det(I + α −1 |H(u)|) 2
n+4 .
From this and (2.60), the optimal matrix-valued monitor function to minimize
the interpolation error is given by
M =det(I + α −1 |H(u)|) − 1
n+4 [I + α −1 |H(u)|]. (2.63)
Inserting (2.62) into (2.57), the interpolation error bound for a mesh satisfying
(2.23) and (2.22) with optimal M given in (2.63) is then
[∫
] n+4
‖e h ‖ 2 L 2 (Ω P ) ≤ Cα2 N − 4 n det(I +α −1 |H(u)|) 2
n
n+4 dx + h.o.t. (2.64)
Ω P
We now discuss how to choose α. We first notice that conditions (2.57)
and (2.62) are invariant under scaling transformations of M of the form
M → cM, for any positive constant c. Thus, if |H(u)| is strictly positive
definite on Ω P ,wecantakeα → 0 in (2.63) and (2.64). This gives
M =det(|H(u)|) − 1
n+4 |H(u)|, (2.65)
[∫
] n+4
‖e h ‖ 2 L 2 (Ω P ) ≤ CN− 4 n det(|H(u)|) 2
n
n+4 dx + h.o.t. (2.66)
Ω P
When |H(u)| vanishes locally, the monitor function cannot be defined by
(2.65) since the right-hand side is not positive definite. In this case, a positive
α should be used. Huang (2001b) suggests that α be defined implicitly
via
∫
det(I + α −1 |H(u)|) 2
n+4 dx =2|ΩP |. (2.67)
Ω P
It is easy to show that (2.67) has a unique solution for α. A simple iteration
method such as the bisection method can be used for solving this equation.
Moreover, when α is defined in this way, M is invariant for scaling transformation
of |H(u)|. Furthermore, it is shown in Huang (2001b) that about
fifty per cent of the mesh points are then concentrated in regions where