Adaptivity with moving grids

Adaptivity with moving grids 19
The condition (2.10) plays an important role in our subsequent analysis of
the errors of computations on both static and moving non-uniform meshes.
The ratio between lengths of adjacent elements is also used in Dorfi and
Drury (1987) and studied by Verwer, Blom, Furzeland and Zegeling (1989).
The concept of quasi-uniformity has natural extensions to higher dimensions
augmented with small angle conditions. For example, in two dimensions,
if we have a triangulation τ P then this is shape-regular, ensuring
control over small angles, if, for each element τ e ∈ τ p with area |τ e |, longest
side of length h e and interior circle of diameter ρ e ,wehaveaconstantσ 1
such that
max
τ e
h e
ρ e
≤ σ 1 . (2.11)
Such a shape-regular mesh is then quasi-uniform if there is a second constant
σ 2 for which
max τe∈τp |τ e |
min τe∈τP |τ e | ≤ σ 2. (2.12)
As in the one-dimensional case, quasi-uniform meshes have similar error
estimates to uniform ones (Johnson 1987). However, it is often much harder
to achieve this for time-dependent problems.
2.3. Mesh calculation, mesh tangling and mesh racing
The function F must be determined as part of the process of calculating τ P .
This map can be calculated either explicitly or implicitly. In the explicit
method, an equation is derived for F which is expressed in terms of the
position of the mesh points. This (usually large and nonlinear) system is
then solved to find F and hence to determine the location of the mesh.
This procedure lies at the heart of a number of equidistribution positionbased
methods for calculating the mesh, such as the moving mesh partial
differential equation, optimal transport and variational methods.Typically
such methods cluster the mesh points where high precision is required, and
the location of points of density of the mesh points moves as the solution
evolves (in a similar manner to a longitudinal wave passing down the length
of a spring, whilst the coils of the spring do not move very far from their
equilibrium positions).
In an alternative procedure, the velocity v of the mesh points in τ P is
determined. This is given by
v = F t . (2.13)
The mesh point positions are updated using this velocity. This approach
is very closely linked with particle and Lagrangian methods and includes
methods such as GCL, MFE and the deformation map method. (Using the
analogy above, such methods are like moving the whole spring.)