Adaptivity with moving grids

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