Adaptivity with moving grids

Adaptivity with moving grids 65
norm is the least-squares norm given by
∫
I = |F(ξ,t) − ξ| 2 dξ. (3.39)
Ω C
Minimizing I subject to the equidistribution principle is in fact the celebrated
Monge–Kantorovich problem from differential geometry. This principle
is often called optimum transport, as it leads to a transformation, the
creation of which takes a minimum amount of work as a deviation from the
identity. Intuitively, this is likely to deliver a regular mesh, as this mesh
will be as close (in an averaged sense) to the most regular possible mesh,
i.e., a completely uniform one. Remarkably, this minimization problem has
a unique solution with a very elegant expression for the transformation.
Theorem 3.3. There exists a unique optimal mapping F(ξ,t) satisfying
the equidistribution equation. This map has the same regularity as M.
Furthermore, F(ξ,t) is the unique mapping from this class which can be
written as the gradient (with respect to ξ) ofaconvex (mesh) potential
P (ξ,t), so that
F(ξ,t)=∇ ξ P (ξ,t), ∆ ξ P (ξ,t) > 0. (3.40)
Proof. See Brenier (1991) or Caffarelli (1992, 1996) for an abstract proof
and Delzanno et al. (2008) for a proof in the context of adaptive mesh
generation.
The following is then immediate.
Lemma 3.4. The map F is irrotational so that ∇ ξ × F = 0, and the
Jacobian of F is symmetric.
Significantly, the transformation above is an example of a Legendre transformation
(Sewell 2002). Such transformations include translations and
linear maps by positive definite symmetric matrices. We now show how to
calculate such a transformation.
3.3.2. Properties of optimally transported meshes
It is immediate that if x = ∇ ξ P then
∂X
= H(P ),
∂ξ
where H(P ) is the Hessian of P . Additionally, if the measure M ∈ W 2 (Ω P )
is strictly positive on its supports (assumed to be convex), then the potential
P ∈ Wloc 2 (Ω C), and satisfies, in the classical sense, the Monge–Ampère
equation
M(∇ ξ P, t)H(P )=θ(t). (3.41)