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