Adaptivity with moving grids

70 C. J. Budd, W. Huang and R. D. Russell
If M is constant, then equation (3.46) without the power law scaling admits
a variables-separable solution for which L t = CL n . If n > 1 then this
equation has solutions which blow up in a finite time. The rescaling prevents
this possibility. The operator on the left of this system is a smoothing
operator, similar to the operator used in (2.14), (3.9), which reduces the
stiffness of this system when it is discretized. Observe that the term θ(t) has
not been included in (3.46). Indeed this term arises naturally as a constant
of integration. The PMA equation (3.46) has many properties in common
with the moving mesh equation (3.9). In particular, if M is independent of
time then the solution of (3.41) corresponds to a stable solution of (3.46).
Indeed, we have the following results.
Lemma 3.6.
(a) Suppose that M t = 0 and the Monge–Ampère equation (3.41) admits
a (steady) convex solution P (ξ) with associated map X(ξ) forwhich
H(P ) > 0, so that P satisfies the equation
Then:
M(∇P )H(P )=θ.
(i) the PMA equation (3.46) admits a time-dependent solution
Q(ξ,t)= θ1/n t
+ P (ξ) (3.47)
ɛ
for which ∇ ξ Q = ∇ ξ P = x(ξ);
(ii) the resulting mesh is locally stable.
(b) If M is slowly varying, then the solution of (3.46) remains ɛ-close to a
solution of (3.41) for all time.
Proof. This is given in Budd and Williams (2009) and is very similar to
the corresponding proof given for the stability of (3.9)
It is also possible to show (Budd and Williams 2009) that, throughout
the evolution of the mesh, if H(Q) and∆Q are initially positive then they
stay positive for all time. This application of the maximum principle guarantees
that the map generating the mesh is locally invertible for all time,
and hence no mesh tangling can occur. This is a very useful feature of
such r-adaptive methods, and we will see numerical examples of this in the
following calculations.
3.3.4. Discretizing the parabolic Monge–Ampère equation (3.46)
To discretize (3.46) in space we can impose a uniform grid of mesh size ∆ξ
on the computational space and assume that Q and Q t take point values