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