Adaptivity with moving grids

Adaptivity with moving grids 99
scale of (T − t), a length scale of L =(T − t) 1/2 , and a solution scale of
U =(T − t) −1/2 . For the moving mesh PDE based on a simple monitor
function of the form M(u) to be invariant under the action of the unitary
multiplicative group, we require that
M(u) =M(|u|).
The condition (5.14) then implies that
M((T − t) −1/2 u)=
Both conditions are satisfied if
1
(T − t) M(u).
M(u) =|u| 2 . (5.16)
Calculations using a regularized form of this monitor function are described
in Section 5.1.2.
Scale-invariant moving mesh methods for a general class of scale-invariant
PDEs can also be constructed in higher dimensions, say x ∈ R n . We consider
two cases, firstly the method of Ceniceros and Hou (2001), and then the
optimal transport method. The first of these describes a two-dimensional
moving mesh generated by the PDE system
x t = ∇ ξ .(M∇ ξ x),
y t = ∇ ξ .(M∇ ξ y).
This system scales in an identical manner to both MMPDE5 and MMPDE6,
and consequently is scale-invariant provided that the monitor function M
satisfies the condition (5.14). In the case of optimal transport in an n-
dimensional system, the PMA equation gives a mesh from ∇ ξ Q, where Q
satisfies the PDE
(I − γ∆)Q t =(M(u)H(Q)) 1/n . (5.17)
Now, if the underlying problem has the natural scaling symmetry x → λx,
then this is equivalent to the scaling symmetry Q → LQ. It is immediate
that
H(LQ) =L n H(Q).
It then follows immediately that (5.17) is invariant under the action of the
scaling symmetries provided that the monitor function satisfies the functional
equation
M(Uu,Lx) 1/n = 1 M(u, x). (5.18)
T
We note that, in two dimensions, the scaling structure of the function Q
also implies that the mesh skewness s, as defined in Section 3 by the relation
s = ∆(Q)2
H(Q) − 2,