Adaptivity with moving grids

Adaptivity with moving grids 15
2.2.1. Mesh scaling
The local scaling factor 1/ρ of the transformation (also called the adaptation
factor) is given by
Λ ≡ 1/ρ =det(J) ≡|J|.
Assuming that J has a full set of singular values λ 1 ,...,λ n , the local
stretching is given by the determinant
Λ=|λ 1 ||λ 2 |···|λ n |. (2.4)
The adaptation factor controls the (possibly higher-dimensional) area |τ e P |
of the element τ e P so that ρ|τ e C| = |τ e P |. (2.5)
The area |τ e p | implicitly enters into the expression (2.2). Indeed, in socalled
shape-regular two-dimensional meshes there exist constants α and β
such that, for all elements τ e p ∈ τ p ,wehave
α|τ e p |≤h 2 e ≤ β|τ e p |.
Accordingly, many moving mesh methods (such as those based on equidistribution
or variational methods) aim to control the adaptation factor. Scaleinvariant
methods relate the adaptation factor to local length scales of the
underlying PDE. It is easily possible for the adaptation factor to vary over
many orders of magnitude, particularly when the adaptive method is being
used to compute singular structures in the underlying PDE in which the
solution u and/or its derivatives vary over similar orders of magnitude.
2.2.2. Mesh skewness
In the case of one-dimensional meshes, control of the adaptation factor for
each element completely describes the mesh. In higher dimensions many
more mesh properties are important, such as the local rotation or the skewness
of the mesh. A special class of irrotational meshes control the local
element rotation by requiring that J is symmetric, so that
or equivalently that
J T = J,
∇ ξ × F = 0.
This is by no means true of all such mappings, but it can be shown (Delzanno
et al. 2008, Brenier 1991) that meshes in an averaged sense closest to uniform
meshes have this property.
The shape of the element τP e , in particular the existence of any small
angles, is also important in the error estimate (2.2). A measure of this is
the local mesh skewness. A measure for the local skewness s of the mesh is