Adaptivity with moving grids
Adaptivity with moving grids
Adaptivity with moving grids
You also want an ePaper? Increase the reach of your titles
YUMPU automatically turns print PDFs into web optimized ePapers that Google loves.
32 C. J. Budd, W. Huang and R. D. Russell<br />
dimensionless parameter θ. When θ =1/2, only the first term remains.<br />
Regarding well-posedness, it is noted that the first term of the functional<br />
is convex, and the existence, uniqueness, and the maximal principle for its<br />
minimizer are guaranteed; e.g., see Reshetnyak (1989). It is unclear if this<br />
result can apply to the whole functional.<br />
2.7.4. Mesh quality measures: alignment and equidistribution<br />
Mesh quality measures can also be developed based on the alignment and<br />
equidistribution conditions (2.23) and (2.22). Indeed, for a given matrixvalued<br />
monitor function M = M(x) and a coordinate transformation x =<br />
x(ξ) (or its inverse), we can use<br />
[ trace(J T ] n<br />
MJ) 2(n−1)<br />
Q ali =<br />
, (2.41)<br />
n det(J T MJ) 1 n<br />
Q eq = |J| √ det(M)|Ω C |<br />
∫Ω P<br />
√<br />
det(M)dy<br />
(2.42)<br />
to measure how closely the coordinate transformation (i.e., mesh) satisfies<br />
the alignment and equidistribution conditions (2.23) and (2.22), respectively.<br />
We note that Q ali is equivalent to<br />
[ trace(J<br />
ˆQ −1 M −1 J −T ] n<br />
) 2(n−1)<br />
ali =<br />
. (2.43)<br />
n det(J −1 M −1 J −T ) 1 n<br />
The quantity Q ali ranges from 1 to ∞, <strong>with</strong>Q ali ≡ 1 for the identity<br />
mapping, while Q eq takes values in (0, ∞), <strong>with</strong> max x Q eq = 1 implying an<br />
equidistributing mesh. Interestingly, Q ali reduces to an equivalence of Q geo<br />
when M = Id. In this sense, Q ali can be viewed as a geometric quality<br />
measure in the metric specified by M.<br />
2.8. Error control and associated monitor functions<br />
The measures for mesh quality and geometry described in Section 2.7 have<br />
largely been constructed in the absence of a clear application. For the<br />
majority of this article we are considering the effectiveness of a mesh for<br />
computing the solution of a partial differential equation. In this case we<br />
are expecting to impose some form of discretization of the system on the<br />
mesh. From this discretization we hope to solve a (typically rapidly evolving)<br />
partial differential equation. The mesh so constructed should attempt<br />
to minimize error (such as the truncation or the interpolation error) in some<br />
way. In this subsection we consider three forms of error, namely static truncation<br />
error, static interpolation error and dynamic errors, and in the first<br />
two cases look at monitor functions which lead to reduced errors. The difficulty<br />
in implementing such a procedures is, of course, that not only is