Adaptivity with moving grids

Adaptivity with moving grids 17
It is shown by Huang (2005a) that measures s defined in (2.6) and Q geo are
mathematically equivalent.
Some other quality measures can be found in Liseikin (1999, Chapter 3),
Knupp (2001) and Shewchuk (2002). Again, a good adaptive method aims
to control some or all of these measures of skewness, either explicitly or implicitly
throughout the calculation, and we discuss this later in this section.
In the case of scale-invariant meshes we shall show that, whilst the adaptation
factor changes a great deal, the skewness hardly varies. More generally,
it should be noted that whereas the adaptation factor often changes a great
deal in a mesh, the skewness generally does not. To control terms in the
error expression (2.2) arising from large solution gradients, it is generally
more important to vary the adaptation factor. If this results in a locally
larger value of the skewness then this can usually be tolerated.
2.2.3. Mesh smoothness and regularity
The smoothness or regularity of a mesh is a measure of how much the mesh
elements vary over the mesh. This can be important since the accuracy and
error in the numerical solution of partial differential equations generally depend
upon the type of discretization, the quality of the mesh, the treatment
of boundary conditions, and so on. A uniform mesh has the highest degree
of regularity, which can lead to particularly low error estimates on such
meshes. It is sometimes claimed that only uniform meshes have such low
estimates, but in fact, as we shall see, they share this with sufficiently regular
meshes. Although there is generally no simple relationship between the
smoothness of the mesh and the error (see Veldman and Rinzema (1992)),
for most problems and most discretization methods, abrupt variations in the
mesh will cause a deterioration in the convergence rate and an increase in
the error (Thompson et al. 1985), or indeed in the accuracy of the approximation
of a function over the mesh. Moreover, most discrete approximations
of spatial differential operators have much larger condition numbers on an
abruptly varying mesh than they do on a gradually varying one, and these
ill-conditioned approximations may result in stiffness in the time integration
for time-dependent problems.
The smoothness of a mesh can be expressed in terms of the regularity of
the underlying mesh function F.
Definition. Ameshτ P has degree of regularity r if F ∈ C r (Ω C ).
The regularity of F can often be achieved by determining F as a solution
of a PDE system or a minimizer of a functional as in variational mesh
generation methods. In many cases it is possible to have strong control
over the derivatives of F allowing guaranteed regularity of the mesh τ P .It
should be noted that an obvious way to determine a mesh is to prescribe
the Jacobian function J exactly (Brackbill and Saltzman 1982). However,