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