Adaptivity with moving grids

84 C. J. Budd, W. Huang and R. D. Russell
4.1. Moving mesh finite element methods
The moving finite element method was originally developed by Miller and
Miller (1981) and Miller (1981), and represents a very important class of
velocity-based moving mesh methods, with much underlying theory and
many applications. See, for example, Adjerid and Flaherty (1986), Baines
et al. (2005), Beckett et al. (2001a), Cao, Huang and Russell (1999a), Carlson
and Miller (1998a, 1998b), Di et al. (2005), Lang et al. (2003), Li et al.
(2002) and Wathen and Baines (1985). A very complete survey of these and
related methods is given in Baines (1994) and we will only describe them
briefly here. The MFE method determines a mesh velocity ẋ = v through
a variational principle coupled to the solution of a PDE by using a finite
element method. Specifically, we consider the time-dependent PDE
∂u
= Lu, (4.1)
∂t
with L a spatial differential operator. The continuous version of the MFE
determines the solution and the mesh together by minimizing the residual
given by
[
min I v, Du ] ∫ ( ) Du
2
≡
v, Du Dt
Dt
Ω P
Dt − ∇u · v −Lu W dx. (4.2)
Here the function W is a weight function for which
W =1
in the usual version of MFE described in Miller and Miller (1981) and Miller
(1981); alternatively, we can take
1
W =
1+|∇u| 2 ,
for the weighted form of MFE described in Carlson and Miller (1998a,
1998b). Observe that MFE is naturally trying to advect the mesh along
with the solution flow. Of course, in practice this equation is discretized
using a Galerkin method.
This method is elegant, and when tuned correctly works well (Baines
1994). However, it does have significant disadvantages. One of these is that
the functional derivative of I with respect to v can become singular, and
regularization is needed in practice.
4.2. Geometric conservation law (GCL) methods
The geometric conservation law (GCL) methods are based upon a direct differentiation
of the equidistribution equation with respect to time, to derive
an equation for the mesh velocity. In its simplest form the method assumes
that the monitor function M is normalized so that the integral of M over