Adaptivity with moving grids

Adaptivity with moving grids 39
conditions (2.23) and (2.22) are concerned. Then the optimal monitor function
is obtained by minimizing the bound among all possible matrix-valued
functions M. Consider a simple situation where a function u ∈ H 2 (Ω P )is
interpolated by piecewise linear polynomials on a simplicial mesh (of N elements)
and the error is measured in L 2 -norm. Then an anisotropic asymptotic
bound (as N →∞) can be obtained from the interpolation theory of
Sobolev spaces (Huang and Sun 2003), namely
∫
‖e h ‖ 2 L 2 (Ω P ) ≤ Cα2 N − ( ( 4
n trace J T [I+α −1 |H(u)|]J )) 2 dx + h.o.t., (2.57)
Ω P
where H(u) denotes the Hessian of the function u, |H(u)| = √ H(u) T H(u),
and α>0 is an arbitrary number which serves as a regularization parameter,
whose value will be determined later. Note that a rigorous bound can be
obtained. But in this situation the derivation has to be associated with a
discrete form; e.g., see Huang (2007). Since the procedure is the same for
both, for simplicity we use the non-rigorous continuous form. Noticing that
a mesh satisfying (2.23) and (2.22) (and a proper boundary correspondence)
is a function of M, we can regard the integral on the right-hand side of (2.57)
as a function of M, namely
∫
( (
B(M) = trace J T [I + α −1 |H(u)|]J )) 2 dx. (2.58)
Ω P
In the following analysis, we consider only meshes satisfying (2.22) and
(2.23) and derive the optimal monitor function by minimizing B(M) among
all possible matrix-valued functions M. First we notice that (2.23) is mathematically
equivalent to
1
n trace(J T MJ) =det(J T MJ) 1 n . (2.59)
A direct comparison of (2.59) suggests that M can be chosen in the form
M = θ(x)[I + α −1 |H(u)|], (2.60)
where θ = θ(x) is a scalar function. For matrix-valued monitor functions in
this form, (2.59) reduces to
1
n trace(J T [I + α −1 |H(u)|]J) =det(J T [I + α −1 |H(u)|]J) 1 n .
Inserting this into (2.58) and using Hölder’s inequality, we have
B(M) =n
∫Ω 2 |J| 4 n det(I + α −1 |H(u)|) 2 n dx
∫ P
= n 2 [
|J|det(I + α −1 |H(u)|) 2
n+4
Ω C
] n+4
n
dξ