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