Adaptivity with moving grids

Adaptivity with moving grids 9
methods. The first group is referred to as velocity-based since the methods
directly target the mesh velocity and obtain mesh point locations by integrating
the velocity field. Methods in this group are more or less motivated
by the Lagrange method in fluid dynamics, where the mesh coordinates,
defined to follow fluid particles, are obtained by integrating flow velocity. A
major effort in the development of these methods has been to avoid mesh
tangling, an undesired property of the Lagrange method. This type of
method includes those developed in Anderson and Rai (1983), Cao, Huang
and Russell (2002), Liao and Anderson (1992), Miller and Miller (1981),
Miller (1981), Petzold (1987) and Yanenko et al. (1976). The method of
Yanenko et al. (1976) is of Lagrange type. In the work of Anderson and Rai
(1983), mesh movement is based on attraction and repulsion pseudo-forces
between nodes motivated by a spring model in mechanics. The moving finite
element method (MFE) of Miller and Miller (1981) and Miller (1981)
has aroused considerable interest. It computes the solution and the mesh
simultaneously by minimizing the residual of the PDEs written in a finite
element form. Penalty terms are added to avoid possible singularities in the
mesh movement equations; see Carlson and Miller (1998a, 1998b). A way
of treating the singularities but without using penalty functions has been
proposed by Wathen and Baines (1985). Liao and Anderson (1992) and
Cai, Fleitas, Jiang and Liao (2004) use a deformation map approach. Cao
et al. (2002) develop the GCL method based on the geometric conservation
law (see Section 4). Similar ideas have been used by Baines, Hubbard
and Jimack (2005) and Baines, Hubbard, Jimack and Jones (2006) for fluid
flow problems.
The second group of moving mesh methods is referred to as location-based
because the methods directly control the location of mesh points. Methods
in this group typically employ an adaptation functional and determine the
mesh or the coordinate transformation as a minimizer of the functional. For
example, the method of Dorfi and Drury (1987) can be linked to a functional
associated with equidistribution principle (Huang et al. 1994). The
moving mesh PDE (MMPDE) method developed in Cao, Huang and Russell
(1999b), Huang et al. (1994) and Huang and Russell (1997a, 1999) moves
the mesh through the gradient flow equation of an adaptation functional,
which includes the energy of a harmonic mapping (Dvinsky 1991) as a special
example. A combination of the MMPDE method with local refinement
is studied in Lang, Cao, Huang and Russell (2003). Li, Tang and Zhang
(2002) and Tang and Tang (2003) also use the energy of a harmonic mapping
as their adaptation functional, but discretize the physical PDE in the
rezoning approach.
So far a number of moving mesh methods and a variety of variants have
been developed and successfully applied to practical problems; see the review
articles of Cao et al. (2003), Eisman (1985, 1987), Hawken, Gottlieb