Adaptivity with moving grids

14 C. J. Budd, W. Huang and R. D. Russell
2.2. Static mesh properties: skewness, regularity and smoothness
We consider first some immediate properties of the mesh τ P which are related
to the function F. Broadly speaking these divide into local and global
properties. The global properties relate to the isotropy of the mesh, orthogonality
issues and the behaviour close to boundaries. We discuss these
presently.
The local properties relate to the size and shape of the elements of τ P .
If τ C is divided into logical regular rectangular or triangular elements τC e ,
then these are mapped to elements τP e in τ P . These elements will then be
distorted rectangles/triangles, possibly with small angles at the vertices.
Locally we can characterize each such element by the size h e of the largest
side and (in two dimensions) the radius ρ e of the largest inscribed circle. If a
partial differential equation is discretized over τ P using (say) a finite element
method in two dimensions, then the error in the solution has contributions
from the size and the shape of the elements, as well as from the derivatives
of the solution itself. For example, if the solution u is interpolated over
τp e by a piecewise linear interpolant Π(u), then the following apriorierror
estimates are standard (Johnson 1987):
max |Π(u) − u| ≤2h 2 τp
e e max
∂ 2 ∣
u ∣∣∣
∣ ,
∂x i ∂x j max
∂ ( ) ∣ ∣ Π(u) − u ∣∣ ≤ 6 h2 e
max
∂ 2 ∣
u ∣∣∣
∂x i ρ e
∣ .
∂x i ∂x j
τ e p
(2.2)
An adapted mesh will usually aim to control the size and the shape of each
element so that, for any particular solution u, the overall error is controlled.
Thus, for example, if the second derivatives of u are large over τ e p , then the
error (2.2) can be controlled locally by taking h e to be small, and ensuring
that h e /ρ e remains bounded. We discuss these issues in detail later in this
section. (See Cao (2005, 2007a) and Chen, Sun and Xu (2007) for a very
complete analysis of this problem.)
Both the size and the shape of the mesh elements can be described in
terms of the local properties of F , and in particular its Jacobian J, given by
The following is immediate.
J = ∂F
∂ξ . (2.3)
Condition 2.1. For the map to be locally well-posed we require that J
should be both bounded and invertible at all points in Ω C .